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With the continued success of Backstreet Boys and *NSYNC, American and British groups like 98 Degrees, Westlife, O-Town, A1, Blue, and Busted gained quick popularity both domestically and internationally. International boy bands would also occasionally spring up, such as the Moldovan band O-Zone, and Overground. American Christian boy band Plus One also enjoyed brief remarkable success during this time.
At the height of boy band popularity in North America, MTV created their own parody boyband, 2gether. Like the Monkees in the 1960s, they were a manufactured act composed of actors. 2gether played off of the idea that every successful boy band must have five distinct personality types: the bad boy, the shy one, the young one, the older brother type, and a heart throb.
Since 2001, the dominance of traditional boy bands on pop charts began to fade in the western hemisphere, although Gil Kaufman of MTV has described "new boy bands" that are "more likely to resemble My Chemical Romance, Sum 41, and Simple Plan.
In 2001, Taiwanese boy band F4 (called JVKV since 2007) blew up big as a result of the success of their TV drama "Meteor Garden". According to "Forbes", F4 has sold 3.5 million copies of their first two albums all over Asia as of July 2003.
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With their success, many other Taiwanese boy bands emerged around this time, such as 5566 and Fahrenheit. In South Korea, Shinhwa also spread hallyu wave throughout Asia such as Japan, Thailand, Singapore, Taiwan, Hong Kong and China. Also in 2001, a new all-male pop band and dance group boyband hailing from Japan called Exile debuted under Avex Group's label Rhythm Zone with 14 members, putting them on par with Super Junior, a South Korean boy band, who at the time, had had 13 members at its peak.
Japanese boy band Arashi has sold over 30 million copies of their records since their first release in 1999. They had the yearly best-selling single in Japan in 2008 and 2009. In 2003 SMAP released the single "Sekai ni Hitotsu Dake no Hana" that has become the third best-selling single ever in Japan, with over three million copies sold.
In North America, the Jonas Brothers rose to fame from promotion on the Disney Channel in 2008. Other boy bands like JLS and Mindless Behavior also emerged and experienced remarkable success around this time. However, apart from them, boy bands have not seen the commercial boom experienced in the genre from the mid to late nineties in North America.
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The mid 2000s, especially the United Kingdom and the rest of Europe, saw the continued longevity of nineties boy bands such as Backstreet Boys and Westlife (before they disbanded in 2012), and the successful comeback of Take That in 2005, Boyzone in 2007, and New Kids on the Block in 2008. Some sections of the press have referred to these acts, particularly those who have reformed after a previous split, such as Take That, Boyzone, and 98 Degrees, as 'man bands'.
2010s and 2020s: Big Time Rush, One Direction and rise of K-pop.
In the early 2010s, there was somewhat of a resurgence of boy band popularity in countries where the trend had not maintained, with the emergence of new boy bands like Big Time Rush, the Wanted, and One Direction and the formation of supergroup NKOTBSB which comprised members of New Kids on the Block and Backstreet Boys. NKOTBSB's success inspired boy bands who were fairly popular during the 1990s and 2000s to make a comeback, such as A1, Blue, 98 Degrees, Five, 911, and O-Town. Like 2gether and the Monkees, Big Time Rush was a manufactured act created for a television show. One Direction were often credited as sparking a resurgence in the popularity and interest boy bands alongside being credited with forming part of a new "British Invasion" in the United States. Their Where We Are Tour was the highest-grossing tour by a vocal group in history and after the release of their fourth album, "Four", they became the only group in the 58-year history of the "Billboard" 200 to have their first four albums debut at number one.
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In Southeast Asia, local boy bands also emerged as a result of the continued success of Korean and Japanese boy bands. After the debut of the Philippines supergroup SB19 in 2018, the group appeared at number six on Billboard Year-End Social 50 chart, making them the first Southeast Asian act to reach the top 10 of the magazine's annual chart.
In South Korea, boy bands have been commercially successful. On the Circle Chart year-end albums chart of 2022, 7 of the top 10 and 13 of the top 20 albums are by boy bands or by subunits/members of boy bands. Seventeen's "FML" is the best-selling album of all time in South Korea, with more than 6.2 million copies sold, and BTS's "" became the first album released since 2001 to sell more than 1 million copies.
In 2013, Billboard started covering music releases in K-pop, though K-pop had been entering the charts as early as 2009. By 2017, BTS crossed into the international music market, furthering the Korean Wave in the United States and becoming the first Korean group to receive a certification by the Recording Industry Association of America (RIAA) with their single "Mic Drop". The band is the first Korean act to top the U.S. "Billboard" 200 with their studio album ' (2018) and have since hit the top of the U.S. charts with their albums ' (2018), ' (2019), ' (2020), "Be" (2020) and "Proof". "Love Yourself: Answer" also broke South Korea's Gaon Album Chart's all-time monthly record previously set by "Love Yourself: Tear" and became the first Korean album certified Gold in the United States. SuperM later became the first K-pop group to debut at No. 1 in the U.S. "Billboard" 200. In 2020, BTS "Dynamite" debuted atop the "Billboard" Hot 100, making them the first all-South Korean act in Hot 100 history to debut at number one. It garnered the band their first Grammy nomination, for Best Pop Duo/Group Performance at 63rd Annual Grammy Awards, making them the first K-pop act to be nominated for one.
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On 12 December 2024, South Korean boy group Stray Kids become the first act ever to debut at No. 1 on the U.S. Billboard 200 chart with their first six charting albums Oddinary (2022), Maxident (2022), 5-Star (2023), Rock-Star (2023), Ate (2024) and Hop (2024).
In Japan, Arashi was the best-selling music artist in Japan from 2013 through 2017 by value of sales and also having the yearly best-selling album in the country in 2010, 2011, 2013, 2015 and 2016. Other successful Japanese boy bands in this decade include Sandaime J Soul Brothers, the second best-selling music artist of 2016 in the country and Kanjani Eight, the fifth best-selling music artist of that year in Japan.
In Norway, the boy band Ballinciaga gained commercial success in 2022 with dance and party songs like "Dans På Bordet" and "Beklager (Guttaklubben)". The group is also known for keeping their identities anonymous by wearing pink-colored masks in the public. Other Norwegian boy bands that gained commercial success in the country in 2020s are Undergrunn and in 2022 and 2024 respectively.
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Key factors of the concept.
Seen as important to a "boy band" group's commercial success is the group's image, carefully controlled by managing all aspects of the group's attire, promotional materials (which are frequently supplied to teen magazines), and music videos. The key factor of a boy band is being trendy. This means that the band conforms to the most recent fashion and musical trends in the popular music scene. Typically, each member of the group will have some distinguishing feature and be portrayed as having a particular personality stereotype, such as "the baby", "the bad boy", or "the shy one". While managing the portrayal of popular musicians is as old as popular music, the particular pigeonholing of band members is a defining characteristic of boy and girl bands. In K-pop, officially designated positions within the group are common, such as "leader", "maknae" (Korean: 막내, English: "the youngest"), "visual", "center" "vocalist", "rapper" and "dancer". The latter three are based on the members' specialized skills and are further divided into "main-", "lead-" and "sub-" (vocalist/rapper/dancer) positions, with the members occupying the "main" positions often being considered the most skilled and having the most parts in songs or being highlighted during solo dance parts. The "leader" is the spokesperson who represents the group in public, and is in charge of mediating between group members, as well as between the group and the label.
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In most cases, their music is written, arranged and produced by a producer who works with the band at all times and controls the group's sound – if necessary, to the point of hiring session singers to record guide vocals for each member of the group to sing individually if the members cannot harmonize well together. However, for clarity of each voice, recording each voice individually is most commonly the norm with most modern vocal groups. In recent years, auto-tune has become a popular tool in vocal producing, some boy bands have come under fire for that reason. Some have also come under fire for lip syncing in their performances as well, for example New Kids on the Block.
A typical boy band performance features elaborately choreographed dancing, with the members taking turns singing and/or rapping. Boy bands generally do not compose or produce their own material, unless the members lobby hard enough for creative control. However, some bands were created around the talent of a songwriter within the group like Gary Barlow of Take That or Tony Mortimer of East 17. It is not uncommon to find extra songs on an album written by one or more of the band members; however, their producers rarely use these as singles.
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Since the 21st century, however, boy bands have been expected to write or at least contribute in some part lyrically to songs. Apart from the groups mentioned above who all had at least one primary songwriter from their beginning, other groups soon caught up. At the close of the nineties, groups like Backstreet Boys and *NSYNC who had previously used writers like Max Martin during their early albums began writing their own songs. Newer groups from late 2000s such as JLS have all made a point from early interviews that they write their own songs and hold their own image as this is an important part of marketing. Some bands like The Wanted have even spent time learning the craft of songwriting. There has also been a rising trend of so-called "songwriter-" or "producer idols" (Korean: "Hangul" 작곡돌, "rev. Rom." jakgok-dol) in K-pop since the early 2000s. Nowadays, it is not uncommon for groups to have at least one member who is heavily involved in the songwriting and producing of the groups' music. In many cases, these members are the rappers in the group, who have often gained songwriting and producing experience while being active as amateur or underground rappers before joining the group. There is also a higher expectation for rappers to write their own lyrics due to self-expression being a core value of the hip-hop genre. There are cases of "producer idols" writing or producing for other artists outside of their solo or respective group work as well, such as BTS' RM and Suga.
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Individuals can also go on to achieve greater success as a solo artist coming out of a boy band having used the groups popularity to build on. Usually this signals the end of the group until potential future reunions. Examples of this include Justin Timberlake from NSYNC, and Robbie Williams from Take That. Some boy band members have gone on to successful careers elsewhere in the media. In K-pop, it is expected and common practice for members to embark on solo endeavors as musical artists or in other entertainment sectors, such as acting, or as variety personalities, alongside their group career after a few years. At the latest, this happens around the time the eldest member reaches the age of 28 (in exceptional cases 30) and is drafted for mandatory military service, forcing the group into a temporary hiatus of at least 18 months. The other members then often go on to pursue solo endeavors and reconvene as a group while no member is serving, or after all members have completed their service.
Music genres.
Although most boy bands consist of R&B or pop influences, other music genres, most notably country music and folk music, are also represented. South 65 and Marshall Dyllon, for example, were both country music boy bands. Il Divo, created by Simon Cowell in 2004, are a vocal group that performs operatic pop in several (mainly Italian) languages. Since then operatic/classical boy bands have become quite popular and common, especially in the UK. Since 2001, there has been some crossover with power pop and pop punk from bands that play live instruments. Just recently some boy bands decided to go back to their original doo-wop roots, most notably, The Overtones.
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Controversy.
Since the 2000s, groups such as Backstreet Boys and LFO have disliked the term "boy band" and have preferred to be known as a "male vocal group". Being categorized among boy bands was also the main reason the Moffatts split up. Boy bands have been accused by the music press of emphasizing the appearance and marketing of the group above the quality of music, deliberately trying to appeal to a preteen audience and for conforming to trends instead of being original. Such criticisms can become extremely scathing. Boy bands are often seen as being short-lived, although some acts such as the Jackson 5, Backstreet Boys, Human Nature, New Edition, SMAP, Shinhwa, Take That and Westlife have sustained lasting careers. |
B-tree
In computer science, a B-tree is a self-balancing tree data structure that maintains sorted data and allows searches, sequential access, insertions, and deletions in logarithmic time. The B-tree generalizes the binary search tree, allowing for nodes with more than two children. Unlike other self-balancing binary search trees, the B-tree is well suited for storage systems that read and write relatively large blocks of data, such as databases and file systems.
History.
While working at Boeing Research Labs, Rudolf Bayer and Edward M. McCreight invented B-trees to efficiently manage index pages for large random-access files. The basic assumption was that indices would be so voluminous that only small chunks of the tree could fit in main memory. Bayer and McCreight's paper "Organization and maintenance of large ordered indices" was first circulated in July 1970 and later published in "Acta Informatica".
Bayer and McCreight never explained what, if anything, the "B" stands for; "Boeing", "balanced", "between", "broad", "bushy", and "Bayer" have been suggested. McCreight, when asked "I want to know what B in B-Tree stands for," answered:"Everybody does!"
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"So you just have no idea what a lunchtime conversation can turn into. So there we were, Rudy and I, at lunch. We had to give the thing a name... We were working for Boeing at the time, but we couldn't use the name without talking to the lawyers. So there's a B."
"It has to do with Balance. There's another B."
"Rudy was the senior author. Rudy (Bayer) was several years older than I am, and had ... many more publications than I did. So there's another B."
"And so at the lunch table, we never did resolve whether there was one of those that made more sense than the rest."
"What Rudy likes to say is, the more you think about what the B in B-Tree means, the better you understand B-Trees!"
Definition.
According to Knuth's definition, a B-tree of order "m" is a tree which satisfies the following properties:
Each internal node's keys act as separation values which divide its subtrees. For example, if an internal node has 3 child nodes (or subtrees) then it must have 2 keys: "a"1 and "a"2. All values in the leftmost subtree will be less than "a"1, all values in the middle subtree will be between "a"1 and "a"2, and all values in the rightmost subtree will be greater than "a"2.
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A B-tree of depth "n"+1 can hold about "U" times as many items as a B-tree of depth "n", but the cost of search, insert, and delete operations grows with the depth of the tree. As with any balanced tree, the cost grows much more slowly than the number of elements.
Some balanced trees store values only at leaf nodes and use different kinds of nodes for leaf nodes and internal nodes. B-trees keep values in every node in the tree except leaf nodes.
Differences in terminology.
The literature on B-trees is not uniform in its terminology.
Bayer and McCreight (1972), Comer (1979), and others define the order of B-tree as the minimum number of keys in a non-root node. Folk and Zoellick points out that terminology is ambiguous because the maximum number of keys is not clear. An order 3 B-tree might hold a maximum of 6 keys or a maximum of 7 keys. Knuth (1998) avoids the problem by defining the order to be the maximum number of children (which is one more than the maximum number of keys).
The term leaf is also inconsistent. Bayer and McCreight (1972) considered the leaf level to be the lowest level of keys, but Knuth considered the leaf level to be one level below the lowest keys. There are many possible implementation choices. In some designs, the leaves may hold the entire data record; in other designs, the leaves may only hold pointers to the data record. Those choices are not fundamental to the idea of a B-tree.
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For simplicity, most authors assume there are a fixed number of keys that fit in a node. The basic assumption is the key size is fixed and the node size is fixed. In practice, variable-length keys may be employed.
Informal description.
Node structure.
As with other trees, B-trees can be represented as a collection of three types of nodes: "root", "internal" (a.k.a. interior), and "leaf".
Note the following variable definitions:
In B-trees, the following properties are maintained for these nodes:
Each internal node in a B-tree has the following format:
Each leaf node in a B-tree has the following format:
The node bounds are summarized in the table below:
Insertion and deletion.
To maintain the predefined range of child nodes, internal nodes may be joined or split.
Usually, the number of keys is chosen to vary between and formula_2, where is the minimum number of keys, and formula_3 is the minimum degree or branching factor of the tree. The factor of 2 will guarantee that nodes can be split or combined.
If an internal node has formula_2 keys, then adding a key to that node can be accomplished by splitting the hypothetical formula_5 key node into two key nodes and moving the key that would have been in the middle to the parent node. Each split node has the required minimum number of keys. Similarly, if an internal node and its neighbor each have keys, then a key may be deleted from the internal node by combining it with its neighbor. Deleting the key would make the internal node have formula_6 keys; joining the neighbor would add keys plus one more key brought down from the neighbor's parent. The result is an entirely full node of formula_2 keys.
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A B-tree is kept balanced after insertion by splitting a would-be overfilled node, of formula_5 keys, into two -key siblings and inserting the mid-value key into the parent. Depth only increases when the root is split, maintaining balance. Similarly, a B-tree is kept balanced after deletion by merging or redistributing keys among siblings to maintain the -key minimum for non-root nodes. A merger reduces the number of keys in the parent potentially forcing it to merge or redistribute keys with its siblings, and so on. The only change in depth occurs when the root has two children, of and (transitionally) formula_6 keys, in which case the two siblings and parent are merged, reducing the depth by one.
This depth will increase slowly as elements are added to the tree, but an increase in the overall depth is infrequent, and results in all leaf nodes being one more node farther away from the root.
Comparison to other trees.
Because a range of child nodes is permitted, B-trees do not need re-balancing as frequently as other self-balancing search trees, but may waste some space, since nodes are not entirely full.
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B-trees have substantial advantages over alternative implementations when the time to access the data of a node greatly exceeds the time spent processing that data, because then the cost of accessing the node may be amortized over multiple operations within the node. This usually occurs when the node data are in secondary storage such as disk drives. By maximizing the number of keys within each internal node, the height of the tree decreases and the number of expensive node accesses is reduced. In addition, rebalancing of the tree occurs less often. The maximum number of child nodes depends on the information that must be stored for each child node and the size of a full disk block or an analogous size in secondary storage. While 2–3 B-trees are easier to explain, practical B-trees using secondary storage need a large number of child nodes to improve performance.
Variants.
The term B-tree may refer to a specific design or a general class of designs. In the narrow sense, a B-tree stores keys in its internal nodes but need not store those keys in the records at the leaves. The general class includes variations such as the B+ tree, the B* tree and the B*+ tree.
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B-tree usage in databases.
Sorted file search time.
Sorting and searching algorithms can be characterized by the number of comparison operations that must be performed using order notation. A binary search of a sorted table with records, for example, can be done in roughly comparisons. If the table had 1,000,000 records, then a specific record could be located with at most 20 comparisons: .
Large databases have historically been kept on disk drives. The time to read a record on a disk drive far exceeds the time needed to compare keys once the record is available due to seek time and a rotational delay. The seek time may be 0 to 20 or more milliseconds, and the rotational delay averages about half the rotation period. For a 7200 RPM drive, the rotation period is 8.33 milliseconds. For a drive such as the Seagate ST3500320NS, the track-to-track seek time is 0.8 milliseconds and the average reading seek time is 8.5 milliseconds. For simplicity, assume reading from disk takes about 10 milliseconds.
The time to locate one record out of a million in the example above would take 20 disk reads times 10 milliseconds per disk read, which is 0.2 seconds.
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The search time is reduced because individual records are grouped together in a disk block. A disk block might be 16 kilobytes. If each record is 160 bytes, then 100 records could be stored in each block. The disk read time above was actually for an entire block. Once the disk head is in position, one or more disk blocks can be read with little delay. With 100 records per block, the last 6 or so comparisons don't need to do any disk reads—the comparisons are all within the last disk block read.
To speed up the search further, the time to do the first 13 to 14 comparisons (which each required a disk access) must be reduced.
Index performance.
A B-tree index can be used to improve performance. A B-tree index creates a multi-level tree structure that breaks a database down into fixed-size blocks or pages. Each level of this tree can be used to link those pages via an address location, allowing one page (known as a node, or internal page) to refer to another with leaf pages at the lowest level. One page is typically the starting point of the tree, or the "root". This is where the search for a particular key would begin, traversing a path that terminates in a leaf. Most pages in this structure will be leaf pages which refer to specific table rows.
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Because each node (or internal page) can have more than two children, a B-tree index will usually have a shorter height (the distance from the root to the farthest leaf) than a Binary Search Tree. In the example above, initial disk reads narrowed the search range by a factor of two. That can be improved by creating an auxiliary index that contains the first record in each disk block (sometimes called a sparse index). This auxiliary index would be 1% of the size of the original database, but it can be searched quickly. Finding an entry in the auxiliary index would tell us which block to search in the main database; after searching the auxiliary index, we would have to search only that one block of the main database—at a cost of one more disk read.
In the above example the index would hold 10,000 entries and would take at most 14 comparisons to return a result. Like the main database, the last six or so comparisons in the auxiliary index would be on the same disk block. The index could be searched in about eight disk reads, and the desired record could be accessed in 9 disk reads.
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Creating an auxiliary index can be repeated to make an auxiliary index to the auxiliary index. That would make an aux-aux index that would need only 100 entries and would fit in one disk block.
Instead of reading 14 disk blocks to find the desired record, we only need to read 3 blocks. This blocking is the core idea behind the creation of the B-tree, where the disk blocks fill out a hierarchy of levels to make up the index. Reading and searching the first (and only) block of the aux-aux index which is the root of the tree identifies the relevant block in aux-index in the level below. Reading and searching that aux-index block identifies the relevant block to read, until the final level, known as the leaf level, identifies a record in the main database. Instead of 150 milliseconds, we need only 30 milliseconds to get the record.
The auxiliary indices have turned the search problem from a binary search requiring roughly disk reads to one requiring only disk reads where is the blocking factor (the number of entries per block: entries per block in our example; reads).
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In practice, if the main database is being frequently searched, the aux-aux index and much of the aux index may reside in a disk cache, so they would not incur a disk read. The B-tree remains the standard index implementation in almost all relational databases, and many nonrelational databases use them too.
Insertions and deletions.
If the database does not change, then compiling the index is simple to do, and the index need never be changed. If there are changes, managing the database and its index require additional computation.
Deleting records from a database is relatively easy. The index can stay the same, and the record can just be marked as deleted. The database remains in sorted order. If there are a large number of lazy deletions, then searching and storage become less efficient.
Insertions can be very slow in a sorted sequential file because room for the inserted record must be made. Inserting a record before the first record requires shifting all of the records down one. Such an operation is just too expensive to be practical. One solution is to leave some spaces. Instead of densely packing all the records in a block, the block can have some free space to allow for subsequent insertions. Those spaces would be marked as if they were "deleted" records.
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Both insertions and deletions are fast as long as space is available on a block. If an insertion won't fit on the block, then some free space on some nearby block must be found and the auxiliary indices adjusted. The best case is that enough space is available nearby so that the amount of block reorganization can be minimized. Alternatively, some out-of-sequence disk blocks may be used.
Usage in databases.
The B-tree uses all of the ideas described above. In particular, a B-tree:
In addition, a B-tree minimizes waste by making sure the interior nodes are at least half full. A B-tree can handle an arbitrary number of insertions and deletions.
Best case and worst case heights.
Let be the height of the classic B-tree (see for the tree height definition). Let be the number of entries in the tree. Let "m" be the maximum number of children a node can have. Each node can have at most keys.
It can be shown (by induction for example) that a B-tree of height "h" with all its nodes completely filled has entries. Hence, the best case height (i.e. the minimum height) of a B-tree is:
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Let formula_11 be the minimum number of children an internal (non-root) node must have. For an ordinary B-tree, formula_12
Comer (1979) and Cormen et al. (2001) give the worst case height (the maximum height) of a B-tree as
Algorithms.
Search.
Searching is similar to searching a binary search tree. Starting at the root, the tree is recursively traversed from top to bottom. At each level, the search reduces its field of view to the child pointer (subtree) whose range includes the search value. A subtree's range is defined by the values, or keys, contained in its parent node. These limiting values are also known as separation values.
Binary search is typically (but not necessarily) used within nodes to find the separation values and child tree of interest.
Insertion.
All insertions start at a leaf node. To insert a new element, search the tree to find the leaf node where the new element should be added. Insert the new element into that node with the following steps:
If the splitting goes all the way up to the root, it creates a new root with a single separator value and two children, which is why the lower bound on the size of internal nodes does not apply to the root. The maximum number of elements per node is "U"−1. When a node is split, one element moves to the parent, but one element is added. So, it must be possible to divide the maximum number "U"−1 of elements into two legal nodes. If this number is odd, then "U"=2"L" and one of the new nodes contains ("U"−2)/2 = "L"−1 elements, and hence is a legal node, and the other contains one more element, and hence it is legal too. If "U"−1 is even, then "U"=2"L"−1, so there are 2"L"−2 elements in the node. Half of this number is "L"−1, which is the minimum number of elements allowed per node.
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An alternative algorithm supports a single pass down the tree from the root to the node where the insertion will take place, splitting any full nodes encountered on the way pre-emptively. This prevents the need to recall the parent nodes into memory, which may be expensive if the nodes are on secondary storage. However, to use this algorithm, we must be able to send one element to the parent and split the remaining "U"−2 elements into two legal nodes, without adding a new element. This requires "U" = 2"L" rather than "U" = 2"L"−1, which accounts for why some textbooks impose this requirement in defining B-trees.
Deletion.
There are two popular strategies for deletion from a B-tree.
The algorithm below uses the former strategy.
There are two special cases to consider when deleting an element:
The procedures for these cases are in order below.
Deletion from an internal node.
Each element in an internal node acts as a separation value for two subtrees, therefore we need to find a replacement for separation. Note that the largest element in the left subtree is still less than the separator. Likewise, the smallest element in the right subtree is still greater than the separator. Both of those elements are in leaf nodes, and either one can be the new separator for the two subtrees. Algorithmically described below:
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Rebalancing after deletion.
Rebalancing starts from a leaf and proceeds toward the root until the tree is balanced. If deleting an element from a node has brought it under the minimum size, then some elements must be redistributed to bring all nodes up to the minimum. Usually, the redistribution involves moving an element from a sibling node that has more than the minimum number of nodes. That redistribution operation is called a rotation. If no sibling can spare an element, then the deficient node must be merged with a sibling. The merge causes the parent to lose a separator element, so the parent may become deficient and need rebalancing. The merging and rebalancing may continue all the way to the root. Since the minimum element count doesn't apply to the root, making the root be the only deficient node is not a problem. The algorithm to rebalance the tree is as follows:
Sequential access.
While freshly loaded databases tend to have good sequential behaviour, this behaviour becomes increasingly difficult to maintain as a database grows, resulting in more random I/O and performance challenges.
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Initial construction.
A common special case is adding a large amount of "pre-sorted" data into an initially empty B-tree. While it is quite possible to simply perform a series of successive inserts, inserting sorted data results in a tree composed almost entirely of half-full nodes. Instead, a special "bulk loading" algorithm can be used to produce a more efficient tree with a higher branching factor.
When the input is sorted, all insertions are at the rightmost edge of the tree, and in particular any time a node is split, we are guaranteed that no more insertions will take place in the left half. When bulk loading, we take advantage of this, and instead of splitting overfull nodes evenly, split them as "unevenly" as possible: leave the left node completely full and create a right node with zero keys and one child (in violation of the usual B-tree rules).
At the end of bulk loading, the tree is composed almost entirely of completely full nodes; only the rightmost node on each level may be less than full. Because those nodes may also be less than "half" full, to re-establish the normal B-tree rules, combine such nodes with their (guaranteed full) left siblings and divide the keys to produce two nodes at least half full. The only node which lacks a full left sibling is the root, which is permitted to be less than half full.
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In filesystems.
In addition to its use in databases, the B-tree (or ) is also used in filesystems to allow quick random access to an arbitrary block in a particular file. The basic problem is turning the file block formula_14 address into a disk block address.
Some operating systems require the user to allocate the maximum size of the file when the file is created. The file can then be allocated as contiguous disk blocks. In that case, to convert the file block address formula_14 into a disk block address, the operating system simply adds the file block address formula_14 to the address of the first disk block constituting the file. The scheme is simple, but the file cannot exceed its created size.
Other operating systems allow a file to grow. The resulting disk blocks may not be contiguous, so mapping logical blocks to physical blocks is more involved.
MS-DOS, for example, used a simple File Allocation Table (FAT). The FAT has an entry for each disk block, and that entry identifies whether its block is used by a file and if so, which block (if any) is the next disk block of the same file. So, the allocation of each file is represented as a linked list in the table. In order to find the disk address of file block formula_14, the operating system (or disk utility) must sequentially follow the file's linked list in the FAT. Worse, to find a free disk block, it must sequentially scan the FAT. For MS-DOS, that was not a huge penalty because the disks and files were small and the FAT had few entries and relatively short file chains. In the FAT12 filesystem (used on floppy disks and early hard disks), there were no more than 4,080 entries, and the FAT would usually be resident in memory. As disks got bigger, the FAT architecture began to confront penalties. On a large disk using FAT, it may be necessary to perform disk reads to learn the disk location of a file block to be read or written.
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TOPS-20 (and possibly TENEX) used a 0 to 2 level tree that has similarities to a B-tree. A disk block was 512 36-bit words. If the file fit in a 512 (29) word block, then the file directory would point to that physical disk block. If the file fit in 218 words, then the directory would point to an aux index; the 512 words of that index would either be NULL (the block isn't allocated) or point to the physical address of the block. If the file fit in 227 words, then the directory would point to a block holding an aux-aux index; each entry would either be NULL or point to an aux index. Consequently, the physical disk block for a 227 word file could be located in two disk reads and read on the third.
Apple's filesystem HFS+ and APFS, Microsoft's NTFS, AIX (jfs2) and some Linux filesystems, such as Bcachefs, Btrfs and ext4, use B-trees.
B*-trees are used in the HFS and Reiser4 file systems.
DragonFly BSD's HAMMER file system uses a modified B+-tree.
Performance.
A B-tree grows slower with growing data amount, than the linearity of a linked list. Compared to a skip list, both structures have the same performance, but the B-tree scales better for growing "n". A T-tree, for main memory database systems, is similar but more compact.
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Variations.
Access concurrency.
Lehman and Yao showed that all the read locks could be avoided (and thus concurrent access greatly improved) by linking the tree blocks at each level together with a "next" pointer. This results in a tree structure where both insertion and search operations descend from the root to the leaf. Write locks are only required as a tree block is modified. This maximizes access concurrency by multiple users, an important consideration for databases and/or other B-tree-based ISAM storage methods. The cost associated with this improvement is that empty pages cannot be removed from the btree during normal operations. (However, see for various strategies to implement node merging, and source code at.)
United States Patent 5283894, granted in 1994, appears to show a way to use a 'Meta Access Method' to allow concurrent B+ tree access and modification without locks. The technique accesses the tree 'upwards' for both searches and updates by means of additional in-memory indexes that point at the blocks in each level in the block cache. No reorganization for deletes is needed and there are no 'next' pointers in each block as in Lehman and Yao.
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Parallel algorithms.
Since B-trees are similar in structure to red-black trees, parallel algorithms for red-black trees can be applied to B-trees as well.
Maple tree.
A Maple tree is a B-tree developed for use in the Linux kernel to reduce lock contention in virtual memory management.
(a,b)-tree.
(a,b)-trees are generalizations of B-trees. B-trees require that each internal node have a minimum of formula_18 children and a maximum of formula_19 children, for some preset value of formula_19. In contrast, an (a,b)-tree allows the minimum number of children for an internal node to be set arbitrarily low. In an (a,b)-tree, each internal node has between and children, for some preset values of and .
External links.
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British Museum
The British Museum is a public museum dedicated to human history, art and culture located in the Bloomsbury area of London. Its permanent collection of eight million works is the largest in the world. It documents the story of human culture from its beginnings to the present. Established in 1753, the British Museum was the first public national museum.
In 2023, the museum received 5,820,860 visitors, an increase of 42% from 2022. It was the most popular attraction in the United Kingdom according to the Association of Leading Visitor Attractions (ALVA).
At its beginning, the museum was largely based on the collections of the Anglo-Irish physician and scientist Sir Hans Sloane. It opened to the public in 1759, in Montagu House, on the site of the current building. The museum's expansion over the following 250 years was largely a result of British colonisation and resulted in the creation of several branch institutions, or independent spin-offs, the first being the Natural History Museum in 1881. Some of its best-known acquisitions, such as the Greek Elgin Marbles and the Egyptian Rosetta Stone, are subject to long-term disputes and repatriation claims.
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In 1973, the British Library Act 1972 detached the library department from the British Museum, but it continued to host the now separated British Library in the same Reading Room and building as the museum until 1997. The museum is a non-departmental public body sponsored by the Department for Culture, Media and Sport. Like all UK national museums, it charges no admission fee except for loan exhibitions.
History.
Sir Hans Sloane.
Although today principally a museum of cultural art objects and antiquities, the British Museum was founded as a "universal museum". Its foundations lie in the will of the Anglo-Irish physician and naturalist Sir Hans Sloane (1660–1753), a London-based doctor and scientist from Ulster. During the course of his lifetime, and particularly after he married the widow of a wealthy Jamaican planter, Sloane gathered a large collection of curiosities, and not wishing to see his collection broken up after death, he bequeathed it to King George II, for the nation, for a sum of £20,000 () to be paid to his heirs by Parliament—intentionally far less than the estimated value of the artefacts, contemporarily estimated at £50,000 () or more according to some sources, and up to £80,000 () or more by others.
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At that time, Sloane's collection consisted of around 71,000 objects of all kinds including some 40,000 printed books, 7,000 manuscripts, extensive natural history specimens including 337 volumes of dried plants, prints and drawings including those by Albrecht Dürer and antiquities from Sudan, Egypt, Greece, Rome, the Ancient Near and Far East and the Americas.
Foundation (1753).
On 7 June 1753, King George II gave his royal assent to the Act of Parliament which established the British Museum. The British Museum Act 1753 also added two other libraries to the Sloane collection, namely the Cottonian Library, assembled by Sir Robert Cotton, dating back to Elizabethan times, and the Harleian Library, the collection of the Earls of Oxford. They were joined in 1757 by the "Old Royal Library", now the Royal manuscripts, assembled by various British monarchs. Together these four "foundation collections" included many of the most treasured books now in the British Library including the Lindisfarne Gospels and the sole surviving manuscript of "Beowulf".
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The British Museum was the first of a new kind of museum – national, belonging to neither church nor king, freely open to the public and aiming to collect everything. Sloane's collection, while including a vast miscellany of objects, tended to reflect his scientific interests. The addition of the Cotton and Harley manuscripts introduced a literary and antiquarian element, and meant that the British Museum now became both National Museum and library.
Cabinet of curiosities (1753–1778).
The body of trustees decided on a converted 17th-century mansion, Montagu House, as a location for the museum, which it bought from the Montagu family for £20,000. The trustees rejected Buckingham House, which was later converted into the present day Buckingham Palace, on the grounds of cost and the unsuitability of its location.
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Indolence and energy (1778–1800).
From 1778, a display of objects from the South Seas brought back from the round-the-world voyages of Captain James Cook and the travels of other explorers fascinated visitors with a glimpse of previously unknown lands. The bequest of a collection of books, engraved gems, coins, prints and drawings by Clayton Mordaunt Cracherode in 1800 did much to raise the museum's reputation; but Montagu House became increasingly crowded and decrepit and it was apparent that it would be unable to cope with further expansion.
The museum's first notable addition towards its collection of antiquities, since its foundation, was by Sir William Hamilton (1730–1803), British Ambassador to Naples, who sold his collection of Greek and Roman artefacts to the museum in 1784 together with a number of other antiquities and natural history specimens. A list of donations to the museum, dated 31 January 1784, refers to the Hamilton bequest of a "Colossal Foot of an Apollo in Marble". It was one of two antiquities of Hamilton's collection drawn for him by Francesco Progenie, a pupil of Pietro Fabris, who also contributed a number of drawings of Mount Vesuvius sent by Hamilton to the Royal Society in London.
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Growth and change (1800–1825).
In the early 19th century the foundations for the extensive collection of sculpture began to be laid and Greek, Roman and Egyptian artefacts dominated the antiquities displays. After the defeat of the French campaign in the Battle of the Nile, in 1801, the British Museum acquired more Egyptian sculptures and in 1802 King George III presented the Rosetta Stone – key to the deciphering of hieroglyphs. Gifts and purchases from Henry Salt, British consul general in Egypt, beginning with the Colossal bust of Ramesses II in 1818, laid the foundations of the collection of Egyptian Monumental Sculpture. Many Greek sculptures followed, notably the first purpose-built exhibition space, the Charles Towneley collection, much of it Roman sculpture, in 1805. In 1806, Thomas Bruce, 7th Earl of Elgin, ambassador to the Ottoman Empire from 1799 to 1803 removed the large collection of marble sculptures from the Parthenon, on the Acropolis in Athens and transferred them to the UK. In 1816 these masterpieces of western art were acquired by the British Museum by Act of Parliament and deposited in the museum thereafter. The collections were supplemented by the Bassae frieze from Phigaleia, Greece in 1815. The Ancient Near Eastern collection also had its beginnings in 1825 with the purchase of Assyrian and Babylonian antiquities from Mary Mackintosh Rich, the widow of Assyriologist Claudius James Rich.
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In 1802 a buildings committee was set up to plan for expansion of the museum, and further highlighted by the donation in 1822 of the King's Library, personal library of King George III's, comprising 65,000 volumes, 19,000 pamphlets, maps, charts and topographical drawings. The neoclassical architect, Sir Robert Smirke, was asked to draw up plans for an eastern extension to the museum "... for the reception of the Royal Library, and a Picture Gallery over it ..." and put forward plans for today's quadrangular building, much of which can be seen today. The dilapidated Old Montagu House was demolished and work on the King's Library Gallery began in 1823. The extension, the East Wing, was completed by 1831. However, following the founding of the National Gallery, London in 1824, the proposed Picture Gallery was no longer needed, and the space on the upper floor was given over to the Natural history collections.
The first Synopsis of the British Museum was published in 1808. This described the contents of the museum, and the display of objects room by room, and updated editions were published every few years.
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The largest building site in Europe (1825–1850).
As Sir Robert Smirke's grand neo-classical building gradually arose, the museum became a construction site. The King's Library, on the ground floor of the East Wing, was handed over in 1827, and was described as one of the finest rooms in London. Although it was not fully open to the general public until 1857, special openings were arranged during The Great Exhibition of 1851.
In 1840, the museum became involved in its first overseas excavations, Charles Fellows's expedition to Xanthos, in Asia Minor, whence came remains of the tombs of the rulers of ancient Lycia, among them the Nereid and Payava monuments. In 1857, Charles Newton was to discover the 4th-century BC Mausoleum of Halikarnassos, one of the Seven Wonders of the Ancient World. In the 1840s and 1850s the museum supported excavations in Assyria by A.H. Layard and others at sites such as Nimrud and Nineveh. Of particular interest to curators was the eventual discovery of Ashurbanipal's great library of cuneiform tablets, which helped to make the museum a focus for Assyrian studies.
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Sir Thomas Grenville (1755–1846), a trustee of the British Museum from 1830, assembled a library of 20,240 volumes, which he left to the museum in his will. The books arrived in January 1847 in twenty-one horse-drawn vans. The only vacant space for this large library was a room originally intended for manuscripts, between the Front Entrance Hall and the Manuscript Saloon. The books remained here until the British Library moved to St Pancras in 1998.
Collecting from the wider world (1850–1875).
The opening of the forecourt in 1852 marked the completion of Robert Smirke's 1823 plan, but already adjustments were having to be made to cope with the unforeseen growth of the collections. Infill galleries were constructed for Assyrian sculptures and Sydney Smirke's Round Reading Room, with space for a million books, opened in 1857. Because of continued pressure on space the decision was taken to move natural history to a new building in South Kensington, which would later become the British Museum of Natural History.
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Roughly contemporary with the construction of the new building was the career of a man sometimes called the "second founder" of the British Museum, the Italian librarian Anthony Panizzi. Under his supervision, the British Museum Library (now part of the British Library) quintupled in size and became a well-organised institution worthy of being called a national library, the largest library in the world after the National Library of Paris. The quadrangle at the centre of Smirke's design proved to be a waste of valuable space and was filled at Panizzi's request by a circular Reading Room of cast iron, designed by Smirke's brother, Sydney Smirke.
Until the mid-19th century, the museum's collections were relatively circumscribed but, in 1851, with the appointment to the staff of Augustus Wollaston Franks to curate the collections, the museum began for the first time to collect British and European medieval antiquities, prehistory, branching out into Asia and diversifying its holdings of ethnography. A real coup for the museum was the purchase in 1867, over French objections, of the Duke of Blacas's wide-ranging and valuable collection of antiquities. Overseas excavations continued and John Turtle Wood discovered the remains of the 4th century BC Temple of Artemis at Ephesos, another Wonder of the Ancient World.
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Scholarship and legacies (1875–1900).
The natural history collections were an integral part of the British Museum until their removal to the new British Museum of Natural History in 1887, nowadays the Natural History Museum in South Kensington. With the departure and the completion of the new White Wing (fronting Montague Street) in 1884, more space was available for antiquities and ethnography and the library could further expand. This was a time of innovation as electric lighting was introduced in the Reading Room and exhibition galleries.
The William Burges collection of armoury was bequeathed to the museum in 1881. In 1882, the museum was involved in the establishment of the independent Egypt Exploration Fund (now Society) the first British body to carry out research in Egypt. A bequest from Miss Emma Turner in 1892 financed excavations in Cyprus. In 1897 the death of the great collector and curator, A. W. Franks, was followed by an immense bequest of 3,300 finger rings, 153 drinking vessels, 512 pieces of continental porcelain, 1,500 netsuke, 850 inro, over 30,000 bookplates and miscellaneous items of jewellery and plate, among them the Oxus Treasure.
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In 1898 Baron Ferdinand de Rothschild bequeathed the Waddesdon Bequest, the glittering contents from his New Smoking Room at Waddesdon Manor. This consisted of almost 300 pieces of "objets d'art et de vertu" which included exquisite examples of jewellery, plate, enamel, carvings, glass and maiolica, among them the Holy Thorn Reliquary, probably created in the 1390s in Paris for John, Duke of Berry. The collection was in the tradition of a "Schatzkammer" such as those formed by the Renaissance princes of Europe. Baron Ferdinand's will was most specific, and failure to observe the terms would make it void, the collection should be
These terms are still observed, and the collection occupies room 2a.
New century, new building (1900–1925).
By the last years of the 19th century, The British Museum's collections had increased to the extent that its building was no longer large enough. In 1895 the trustees purchased the 69 houses surrounding the museum with the intention of demolishing them and building around the west, north and east sides of the museum. The first stage was the construction of the northern wing beginning 1906.
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All the while, the collections kept growing. Emil Torday collected in Central Africa, Aurel Stein in Central Asia, D. G. Hogarth, Leonard Woolley and T. E. Lawrence excavated at Carchemish. Around this time, the American collector and philanthropist J. Pierpont Morgan donated a substantial number of objects to the museum, including William Greenwell's collection of prehistoric artefacts from across Europe which he had purchased for £10,000 in 1908. Morgan had also acquired a major part of Sir John Evans's coin collection, which was later sold to the museum by his son J. P. Morgan Jr. in 1915. In 1918, because of the threat of wartime bombing, some objects were evacuated via the London Post Office Railway to Holborn, the National Library of Wales (Aberystwyth) and a country house near Malvern. On the return of antiquities from wartime storage in 1919 some objects were found to have deteriorated. A conservation laboratory was set up in May 1920 and became a permanent department in 1931. It is today the oldest in continuous existence. In 1923, the British Museum welcomed over one million visitors.
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Disruption and reconstruction (1925–1950).
New mezzanine floors were constructed and book stacks rebuilt in an attempt to cope with the flood of books. In 1931, the art dealer Sir Joseph Duveen offered funds to build a gallery for the Parthenon sculptures. Designed by the American architect John Russell Pope, it was completed in 1938. The appearance of the exhibition galleries began to change as dark Victorian reds gave way to modern pastel shades.
Following the retirement of George Francis Hill as Director and Principal Librarian in 1936, he was succeeded by John Forsdyke.
As tensions with Nazi Germany developed and it appeared that war may be imminent Forsdyke came to the view that with the likelihood of far worse air-raids than that experienced in World War I that the museum had to make preparations to remove its most valuable items to secure locations. Following the Munich crisis Forsdyke ordered 3,300 No-Nail Boxes and stored them in the basement of Duveen Gallery. At the same time he began identifying and securing suitable locations. As a result, the museum was able to quickly commence relocating selected items on 24 August 1939, (a mere day after the Home Secretary advised them to do so), to secure basements, country houses, Aldwych Underground station and the National Library of Wales. Many items were relocated in early 1942 from their initial dispersal locations to a newly developed facility at Westwood Quarry in Wiltshire.
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The evacuation was timely, for in 1940 the Duveen Gallery was severely damaged by bombing. Meanwhile, prior to the war, the Nazis had sent a researcher to the British Museum for several years with the aim of "compiling an anti-Semitic history of Anglo-Jewry".
After the war, the museum continued to collect from all countries and all centuries: among the most spectacular additions were the 2600 BC Mesopotamian treasure from Ur, discovered during Leonard Woolley's 1922–34 excavations. Gold, silver and garnet grave goods from the Anglo-Saxon ship burial at Sutton Hoo (1939) and late Roman silver tableware from Mildenhall, Suffolk (1946). The immediate post-war years were taken up with the return of the collections from protection and the restoration of the museum after the Blitz. Work also began on restoring the damaged Duveen Gallery.
A new public face (1950–1975).
In 1953, the museum celebrated its bicentenary. Many changes followed: the first full-time in-house designer and publications officer were appointed in 1964, the Friends organisation was set up in 1968, an Education Service established in 1970 and publishing house in 1973. In 1963, a new Act of Parliament introduced administrative reforms. It became easier to lend objects, the constitution of the board of trustees changed and the Natural History Museum became fully independent. By 1959 the Coins and Medals office suite, completely destroyed during the war, was rebuilt and re-opened, attention turned towards the gallery work with new tastes in design leading to the remodelling of Robert Smirke's Classical and Near Eastern galleries. In 1962 the Duveen Gallery was finally restored and the Parthenon Sculptures were moved back into it, once again at the heart of the museum.
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By the 1970s, the museum was again expanding. More services for the public were introduced; visitor numbers soared, with the temporary exhibition "Treasures of Tutankhamun" in 1972, attracting 1,694,117 visitors, the most successful in British history. In the same year the Act of Parliament establishing the British Library was passed, separating the collection of manuscripts and printed books from the British Museum. This left the museum with antiquities; coins, medals and paper money; prints and drawings; and ethnography. A pressing problem was finding space for additions to the library which now required an extra of shelving each year. The Government suggested a site at St Pancras for the new British Library but the books did not leave the museum until 1997.
The Great Court emerges (1975–2000).
The departure of the British Library to a new site at St Pancras, finally achieved in 1998, provided the space needed for the books. It also created the opportunity to redevelop the vacant space in Robert Smirke's 19th-century central quadrangle into the Queen Elizabeth II Great Court – the largest covered square in Europe – which opened in 2000. The ethnography collections, which had been housed in the short-lived Museum of Mankind at 6 Burlington Gardens from 1970, were returned to new purpose-built galleries in the museum in 2000.
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The museum again readjusted its collecting policies as interest in "modern" objects: prints, drawings, medals and the decorative arts reawakened. Ethnographical fieldwork was carried out in places as diverse as New Guinea, Madagascar, Romania, Guatemala and Indonesia and there were excavations in the Near East, Egypt, Sudan and the UK. The Weston Gallery of Roman Britain, opened in 1997, displayed a number of recently discovered hoards which demonstrated the richness of what had been considered an unimportant part of the Roman Empire. The museum turned increasingly towards private funds for buildings, acquisitions and other purposes. In 2000, the British Museum was awarded National Heritage Museum of the Year.
The British Museum today.
Today the museum no longer houses collections of natural history, and the books and manuscripts it once held now form part of the independent British Library. The museum nevertheless preserves its universality in its collections of artefacts representing the cultures of the world, ancient and modern. The original 1753 collection has grown to over 13 million objects at the British Museum, 70 million at the Natural History Museum and 150 million at the British Library.
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The Round Reading Room, which was designed by the architect Sydney Smirke, opened in 1857. For almost 150 years researchers came here to consult the museum's vast library. The Reading Room closed in 1997 when the national library (the British Library) moved to a new building at St Pancras. Today it has been transformed into the Walter and Leonore Annenberg Centre.
With the bookstacks in the central courtyard of the museum empty, the demolition for Lord Foster's glass-roofed Great Court could begin. The Great Court, opened in 2000, while undoubtedly improving circulation around the museum, was criticised for having a lack of exhibition space at a time when the museum was in serious financial difficulties and many galleries were closed to the public. At the same time the African collections that had been temporarily housed in 6 Burlington Gardens were given a new gallery in the North Wing funded by the Sainsbury family – with the donation valued at £25 million.
The museum's online database had nearly 4,500,000 individual object entries in 2,000,000 records at the start of 2023. In 2022–23 there were 27 million visits to the website. This compares with 19.5 millions website visits in 2013.
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There were 5,820,860 visits to the museum in 2023, a 42% increase on 2022. The museum was the most visited tourist attraction in Britain in 2023. The number of visits, however, has not recovered to the level reached before the Covid pandemic.
A number of films have been shot at the British Museum.
Governance.
Director.
The British Museum is a non-departmental public body sponsored by the Department for Culture, Media and Sport through a three-year funding agreement. Its head is the Director of the British Museum. The British Museum was run from its inception by a 'principal librarian' (when the book collections were still part of the museum), a role that was renamed 'director and principal librarian' in 1898, and 'director' in 1973 (on the separation of the British Library).
Trustees.
A board of 25 trustees (with the director as their accounting officer for the purposes of reporting to Government) is responsible for the general management and control of the museum, in accordance with the British Museum Act 1963 and the Museums and Galleries Act 1992. Prior to the 1963 Act, it was chaired by the Archbishop of Canterbury, the Lord Chancellor and the Speaker of the House of Commons. The board was formed on the museum's inception to hold its collections in trust for the nation without actually owning them themselves, and now fulfil a mainly advisory role. Trustee appointments are governed by the regulatory framework set out in the code of practice on public appointments issued by the Office of the Commissioner for Public Appointments.
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Building.
The Greek Revival façade facing Great Russell Street is a characteristic building of Sir Robert Smirke, with 44 columns in the Ionic order high, closely based on those of the temple of Athena Polias at Priene in Asia Minor. The pediment over the main entrance is decorated by sculptures by Sir Richard Westmacott depicting "The Progress of Civilisation", consisting of fifteen allegorical figures, installed in 1852.
The construction commenced around the courtyard with the East Wing (The King's Library) in 1823–1828, followed by the North Wing in 1833–1838, which originally housed among other galleries a reading room, now the Wellcome Gallery. Work was also progressing on the northern half of the West Wing (The Egyptian Sculpture Gallery) 1826–1831, with Montagu House demolished in 1842 to make room for the final part of the West Wing, completed in 1846, and the South Wing with its great colonnade, initiated in 1843 and completed in 1847, when the Front Hall and Great Staircase were opened to the public. The museum is faced with Portland stone, but the perimeter walls and other parts of the building were built using Haytor granite from Dartmoor in South Devon, transported via the unique Haytor Granite Tramway.
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In 1846 Robert Smirke was replaced as the museum's architect by his brother Sydney Smirke, whose major addition was the Round Reading Room 1854–1857; at in diameter it was then the second widest dome in the world, the Pantheon in Rome being slightly wider.
The next major addition was the White Wing 1882–1884 added behind the eastern end of the South Front, the architect being Sir John Taylor.
In 1895, Parliament gave the museum trustees a loan of £200,000 to purchase from the Duke of Bedford all 69 houses which backed onto the museum building in the five surrounding streets – Great Russell Street, Montague Street, Montague Place, Bedford Square and Bloomsbury Street. The trustees planned to demolish these houses and to build around the west, north and east sides of the museum new galleries that would completely fill the block on which the museum stands. The architect Sir John James Burnet was petitioned to put forward ambitious long-term plans to extend the building on all three sides. Most of the houses in Montague Place were knocked down a few years after the sale. Of this grand plan only the Edward VII galleries in the centre of the North Front were ever constructed, these were built 1906–14 to the design by J.J. Burnet, and opened by King George V and Queen Mary in 1914. They now house the museum's collections of Prints and Drawings and Oriental Antiquities. There was not enough money to put up more new buildings, and so the houses in the other streets are nearly all still standing.
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The Duveen Gallery, sited to the west of the Egyptian, Greek & Assyrian sculpture galleries, was designed to house the Elgin Marbles by the American Beaux-Arts architect John Russell Pope. Although completed in 1938, it was hit by a bomb in 1940 and remained semi-derelict for 22 years, before reopening in 1962. Other areas damaged during World War II bombing included: in September 1940 two unexploded bombs hit the Edward VII galleries, the King's Library received a direct hit from a high explosive bomb, incendiaries fell on the dome of the Round Reading Room but did little damage; on the night of 10 to 11 May 1941 several incendiaries fell on the south-west corner of the museum, destroying the book stack and 150,000 books in the courtyard and the galleries around the top of the Great Staircase – this damage was not fully repaired until the early 1960s.
The Queen Elizabeth II Great Court is a covered square at the centre of the British Museum designed by the engineers Buro Happold and the architects Foster and Partners. The Great Court opened in December 2000 and is the largest covered square in Europe. The roof is a glass and steel construction, built by an Austrian steelwork company, with 1,656 uniquely shaped panes of glass. At the centre of the Great Court is the Reading Room vacated by the British Library, its functions now moved to St Pancras.
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Today, the British Museum has grown to become one of the largest museums in the world, covering an area of over 92,000 m2 (990,000 sq. ft). In addition to 21,600 m2 (232,000 sq. ft) of on-site storage space, and 9,400 m2 (101,000 sq. ft) of external storage space. Altogether, the British Museum showcases on public display less than 1% of its entire collection, approximately 50,000 items.
There are nearly one hundred galleries open to the public, representing of exhibition space, although the less popular ones have restricted opening times. However, the lack of a large temporary exhibition space led to the £135 million World Conservation and Exhibitions Centre to provide one and to concentrate all the museum's conservation facilities into one centre. This project was announced in July 2007, with the architects Rogers Stirk Harbour and Partners. It was granted planning permission in December 2009 and was completed in time for the Viking exhibition in March 2014. In 2017, the World Conservation and Exhibitions Centre was shortlisted for the Stirling Prize for excellence in architecture.
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Blythe House in West Kensington was used by the museum for off-site storage of small and medium-sized artefacts until the British Museum Archeological Collection, a purpose built storage facility near Reading, was opened in 2024. Another site Franks House in East London is used for storage and work on the "Early Prehistory" – Palaeolithic and Mesolithic – and some other collections.
Departments.
Department of Egypt and Sudan.
The British Museum houses the world's largest and most comprehensive collection of Egyptian antiquities (with over 100,000 pieces) outside the Egyptian Museum in Cairo. A collection of immense importance for its range and quality, it includes objects of all periods from virtually every site of importance in Egypt and the Sudan. Together, they illustrate every aspect of the cultures of the Nile Valley (including Nubia), from the Predynastic Neolithic period () through Coptic (Christian) times (12th century AD), and up to the present day, a time-span over 11,000 years.
Egyptian antiquities have formed part of the British Museum collection ever since its foundation in 1753 after receiving 160 Egyptian objects from Sir Hans Sloane. After the defeat of the French forces under Napoleon at the Battle of the Nile in 1801, the Egyptian antiquities collected were confiscated by the British army and presented to the British Museum in 1803. These works, which included the famed Rosetta Stone, were the first important group of large sculptures to be acquired by the museum. Thereafter, the UK appointed Henry Salt as consul in Egypt who amassed a huge collection of antiquities, some of which were assembled and transported with great ingenuity by the famous Italian explorer Giovanni Belzoni. Most of the antiquities Salt collected were purchased by the British Museum and the Musée du Louvre.
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By 1866 the collection consisted of some 10,000 objects. Antiquities from excavations started to come to the museum in the latter part of the 19th century as a result of the work of the Egypt Exploration Fund under the efforts of E.A. Wallis Budge. Over the years more than 11,000 objects came from this source, including pieces from Amarna, Bubastis and Deir el-Bahari. Other organisations and individuals also excavated and donated objects to the British Museum, including Flinders Petrie's Egypt Research Account and the British School of Archaeology in Egypt, as well as the University of Oxford Expedition to Kawa and Faras in Sudan.
Active support by the museum for excavations in Egypt continued to result in important acquisitions throughout the 20th century until changes in antiquities laws in Egypt led to the suspension of policies allowing finds to be exported, although divisions still continue in Sudan. The British Museum conducted its own excavations in Egypt where it received divisions of finds, including Asyut (1907), Mostagedda and Matmar (1920s), Ashmunein (1980s) and sites in Sudan such as Soba, Kawa and the Northern Dongola Reach (1990s). The size of the Egyptian collections now stand at over 110,000 objects.
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In autumn 2001 the eight million objects forming the museum's permanent collection were further expanded by the addition of six million objects from the Wendorf Collection of Egyptian and Sudanese Prehistory. These were donated by Professor Fred Wendorf of Southern Methodist University in Texas, and comprise the entire collection of artefacts and environmental remains from his excavations at Prehistoric sites in the Sahara Desert between 1963 and 1997. Other fieldwork collections have recently come from Dietrich and Rosemarie Klemm (University of Munich) and William Adams (University of Kentucky).
The seven permanent Egyptian galleries at the British Museum, which include its largest exhibition space (Room 4, for monumental sculpture), can display only 4% of its Egyptian holdings. The second-floor galleries have a selection of the museum's collection of 140 mummies and coffins, the largest outside Cairo. A high proportion of the collection comes from tombs or contexts associated with the cult of the dead, and it is these pieces, in particular the mummies, that remain among the most eagerly sought-after exhibits by visitors to the museum.
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Highlights of the collections include:
Predynastic and Early Dynastic period ()
Old Kingdom (2690–2181 BC)
Middle Kingdom (2134–1690 BC)
Second Intermediate Period (1650–1550 BC)
New Kingdom (1549–1069 BC)
Third Intermediate Period (1069–664 BC)
Late Period (664–332 BC)
Ptolemaic dynasty (305–30 BC)
Roman Period (30 BC – 641 AD)
Department of Greece and Rome.
The British Museum has one of the world's largest and most comprehensive collections of antiquities from the Classical world, with over 100,000 objects. These mostly range in date from the beginning of the Greek Bronze Age (about 3200 BC) to the establishment of Christianity as the official religion of the Roman Empire, with the Edict of Milan under the reign of the Roman emperor Constantine I in 313 AD. Archaeology was in its infancy during the nineteenth century and many pioneering individuals began excavating sites across the Classical world, chief among them for the museum were Charles Newton, John Turtle Wood, Robert Murdoch Smith and Charles Fellows.
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The Greek objects originate from across the Ancient Greek world, from the mainland of Greece and the Aegean Islands, to neighbouring lands in Asia Minor and Egypt in the eastern Mediterranean and as far as the western lands of Magna Graecia that include Sicily and southern Italy. The Cycladic, Minoan and Mycenaean cultures are represented, and the Greek collection includes important sculpture from the Parthenon in Athens, as well as elements of two of the Seven Wonders of the Ancient World, the Mausoleum at Halicarnassus and the Temple of Artemis at Ephesos.
Beginning from the early Bronze Age, the department also houses one of the widest-ranging collections of Italic and Etruscan antiquities outside Italy, as well as extensive groups of material from Cyprus and non-Greek colonies in Lycia and Caria on Asia Minor. There is some material from the Roman Republic, but the collection's strength is in its comprehensive array of objects from across the Roman Empire, with the exception of Britain (which is the mainstay of the Department of Prehistory and Europe).
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The collections of ancient jewellery and bronzes, Greek vases (many from graves in southern Italy that were once part of Sir William Hamilton's and Chevalier Durand's collections), Roman glass including the famous Cameo glass Portland Vase, Roman gold glass (the second largest collection after the Vatican Museums), Roman mosaics from Carthage and Utica in North Africa that were excavated by Nathan Davis, and silver hoards from Roman Gaul (some of which were bequeathed by the philanthropist and museum trustee Richard Payne Knight), are particularly important. Cypriot antiquities are strong too and have benefited from the purchase of Sir Robert Hamilton Lang's collection as well as the bequest of Emma Turner in 1892, which funded many excavations on the island. Roman sculptures (many of which are copies of Greek originals) are particularly well represented by the Townley collection as well as residual sculptures from the famous Farnese collection.
Objects from the Department of Greece and Rome are located throughout the museum, although many of the architectural monuments are to be found on the ground floor, with connecting galleries from Gallery 5 to Gallery 23. On the upper floor, there are galleries devoted to smaller material from ancient Italy, Greece, Cyprus and the Roman Empire.
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The current collection includes:
Temple of Hephaestus
Parthenon
Propylaea
Erechtheion
Temple of Athena Nike
Choragic Monument of Thrasyllos
Tower of the Winds
Temple of Poseidon, Sounion
Temple of Nemesis, Rhamnus
Temple of Bassae
Sanctuary of Apollo at Daphni
Temple of Athena Polias, Priene
Mausoleum at Halicarnassus
Temple of Artemis in Ephesus
Knidos in Asia Minor
Xanthos in Asia Minor
Temple of Zeus, Salamis in Cyprus
Wider collection
Prehistoric Greece and Italy (3300 BC – 8th century BC)
Etruscan (8th century BC – 1st century BC)
Ancient Greece (8th century BC – 4th century AD)
Ancient Rome (1st century BC – 4th century AD)
The collection encompasses architectural, sculptural and epigraphic items from many other sites across the classical world including Amathus, Atripalda, Aphrodisias, Delos, Iasos, Idalion, Lindus, Kalymnos, Kerch, Rhamnous, Salamis, Sestos, Sounion, Tomis and Thessaloniki.
Department of the Middle East.
With a collection numbering some 330,000 works, the British Museum possesses the world's largest and most important collection of Mesopotamian antiquities outside Iraq. A collection of immense importance, the holdings of Assyrian sculpture, Babylonian and Sumerian antiquities are among the most comprehensive in the world with entire suites of rooms panelled in alabaster Assyrian palace reliefs from Nimrud, Nineveh and Khorsabad.
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The collections represent the civilisations of the ancient Near East and its adjacent areas. These cover Mesopotamia, Persia, the Arabian Peninsula, Anatolia, the Caucasus, parts of Central Asia, Syria, the Holy Land and Phoenician settlements in the western Mediterranean from the prehistoric period and include objects from the 7th century.
The first significant addition of Mesopotamian objects was from the collection of Claudius James Rich in 1825. The collection was later dramatically enlarged by the excavations of A. H. Layard at the Assyrian sites of Nimrud and Nineveh between 1845 and 1851. At Nimrud, Layard discovered the North-West Palace of Ashurnasirpal II, as well as three other palaces and various temples. He later uncovered the Palace of Sennacherib at Nineveh with 'no less than seventy-one halls'. As a result, a large numbers of Lamassus, palace reliefs, stelae, including the Black Obelisk of Shalmaneser III, were brought to the British Museum.
Layard's work was continued by his assistant, Hormuzd Rassam and in 1852–1854 he went on to discover the North Palace of Ashurbanipal at Nineveh with many magnificent reliefs, including the famous Lion Hunt of Ashurbanipal and Lachish reliefs. He also discovered the Royal Library of Ashurbanipal, a large collection of cuneiform tablets of enormous importance that today number around 130,000 pieces. W. K. Loftus excavated in Nimrud between 1850 and 1855 and found a remarkable hoard of ivories in the Burnt Palace. Between 1878 and 1882 Rassam greatly improved the museum's holdings with exquisite objects including the Cyrus Cylinder from Babylon, the bronze gates from Balawat, important objects from Sippar, and a fine collection of Urartian bronzes from Toprakkale including a copper figurine of a winged, human-headed bull.
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In the early 20th century excavations were carried out at Carchemish, Turkey by D. G. Hogarth and Leonard Woolley, the latter assisted by T. E. Lawrence. The Mesopotamian collections were greatly augmented by excavations in southern Iraq after World War I. From Tell al-Ubaid came the bronze furnishings of a Sumerian temple, including life-sized lions and a panel featuring the lion-headed eagle Indugud found by H. R. Hall in 1919–24. Woolley went on to excavate Ur between 1922 and 1934, discovering the Royal Cemeteries of the 3rd millennium BC. Some of the masterpieces include the Standard of Ur, the Ram in a Thicket, the Royal Game of Ur, and two bull-headed lyres. The department also has three diorite statues of the ruler Gudea from the ancient state of Lagash and a series of limestone kudurru or boundary stones from different locations across ancient Mesopotamia.
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From the modern state of Syria come almost forty funerary busts from Palmyra and a group of stone reliefs from the excavations of Max von Oppenheim at Tell Halaf that was purchased in 1920. More material followed from the excavations of Max Mallowan at Chagar Bazar and Tell Brak in 1935–1938 and from Woolley at Alalakh in the years just before and after World War II. Mallowan returned with his wife Agatha Christie to carry out further digs at Nimrud in the postwar period which secured many important artefacts for the museum. The collection of Palestinian material was strengthened by the work of Kathleen Kenyon at Tell es-Sultan (Jericho) in the 1950s and the acquisition in 1980 of around 17,000 objects found at Lachish by the Wellcome-Marston expedition of 1932–1938. Archaeological digs are still taking place where permitted in the Middle East, and, depending on the country, the museum continues to receive a share of the finds from sites such as in Jordan.
The museum's collection of Islamic art, including archaeological material, numbers about 40,000 objects, one of the largest of its kind in the world. As such, it contains a broad range of pottery, paintings, tiles, metalwork, glass, seals, and inscriptions from across the Islamic world, from Spain in the west to India in the east. It is particularly famous for its collection of Iznik ceramics (the largest in the world), its large number of mosque lamps including one from the Dome of the Rock, mediaeval metalwork such as the Vaso Vescovali with its depictions of the Zodiac, a fine selection of astrolabes, and Mughal paintings and precious artwork including a large jade terrapin made for the emperor Jahangir. Thousands of objects were excavated after the war by professional archaeologists at Iranian sites such as Siraf by David Whitehouse and Alamut Castle by Peter Willey. The collection was augmented in 1983 by the Godman bequest of Iznik, Hispano-Moresque and early Iranian pottery. Artefacts from the Islamic world are on display in Gallery 34 of the museum.
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A representative selection from the Department of Middle East, including the most important pieces, are on display in 13 galleries throughout the museum and total some 4,500 objects. A whole suite of rooms on the ground floor display the sculptured reliefs from the Assyrian palaces at Nineveh, Nimrud and Khorsabad, while 8 galleries on the upper floor hold smaller material from ancient sites across the Middle East. The remainder form the study collection which ranges in size from beads to large sculptures. They include approximately 130,000 cuneiform tablets from Mesopotamia.
Highlights of the collections include:
Nimrud:
Assyrian palace reliefs from:
Sculptures and inscriptions:
Assyrian palace reliefs and sculptures from:
Royal Library of Ashurbanipal:
Khorsabad and Balawat:
Department of Prints and Drawings.
The Department of Prints and Drawings holds the national collection of Western prints and drawings. It ranks as one of the largest and best print room collections in existence alongside the Albertina in Vienna, the Paris collections and the Hermitage. The holdings are easily accessible to the general public in the Study Room, unlike many such collections. The department also has its own exhibition gallery in Room 90, where the displays and exhibitions change several times a year.
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Since its foundation in 1808, the prints and drawings collection has grown to international renown as one of the richest and most representative collections in the world. There are approximately 50,000 drawings and over two million prints. The collection of drawings covers the period from the 14th century to the present, and includes many works of the highest quality by the leading artists of the European schools. The collection of prints covers the tradition of fine printmaking from its beginnings in the 15th century up to the present, with near complete holdings of most of the great names before the 19th century. Key benefactors to the department have been Clayton Mordaunt Cracherode, Richard Payne Knight, John Malcolm, Campbell Dodgson, César Mange de Hauke and Tomás Harris. Writer and author Louis Alexander Fagan, who worked in the department 1869–1894 made significant contributions to the department in form of his "Handbook to the Department", as well as various other books about the museum in general.
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Department of Britain, Europe and Prehistory.
The Department of Britain, Europe and Prehistory is responsible for collections that cover a vast expanse of time and geography. It includes some of the earliest objects made by humans in east Africa over 2 million years ago, as well as Prehistoric and neolithic objects from other parts of the world; and the art and archaeology of Europe from the earliest times to the present day. Archeological excavation of prehistoric material took off and expanded considerably in the twentieth century and the department now has literally millions of objects from the Paleolithic and Mesolithic periods throughout the world, as well as from the Neolithic, Bronze Age and Iron Age in Europe. Stone Age material from Africa has been donated by famous archaeologists such as Louis and Mary Leakey, and Gertrude Caton–Thompson. Paleolithic objects from the Sturge, Christy and Lartet collections include some of the earliest works of art from Europe. Many Bronze Age objects from across Europe were added during the nineteenth century, often from large collections built up by excavators and scholars such as Greenwell in Britain, Tobin and Cooke in Ireland, Lukis and de la Grancière in Brittany, Worsaae in Denmark, Siret at El Argar in Spain, and Klemm and Edelmann in Germany. A representative selection of Iron Age artefacts from Hallstatt were acquired as a result of the Evans/Lubbock excavations and from Giubiasco in Ticino through the Swiss National Museum.
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In addition, the British Museum's collections covering the period AD 300 to 1100 are among the largest and most comprehensive in the world, extending from Spain to the Black Sea and from North Africa to Scandinavia; a representative selection of these has recently been redisplayed in a newly refurbished gallery. Important collections include Latvian, Norwegian, Gotlandic and Merovingian material from Johann Karl Bähr, Alfred Heneage Cocks, Sir James Curle and Philippe Delamain respectively. However, the undoubted highlight from the early mediaeval period is the magnificent items from the Sutton Hoo royal grave, generously donated to the nation by the landowner Edith Pretty. The late mediaeval collection includes a large number of seal-dies from across Europe, the most famous of which include those from the Town of Boppard in Germany, Isabella of Hainault from her tomb in Notre Dame Cathedral, Paris, Inchaffray Abbey in Scotland and Robert Fitzwalter, one of the Barons who led the revolt against King John in England. |
The late mediaeval collection includes a large number of seal-dies from across Europe, the most famous of which include those from the Town of Boppard in Germany, Isabella of Hainault from her tomb in Notre Dame Cathedral, Paris, Inchaffray Abbey in Scotland and Robert Fitzwalter, one of the Barons who led the revolt against King John in England. There is also a large collection of medieval signet rings, prominent among them is the gold signet ring belonging to Jean III de Grailly who fought in the Hundred Years' War, as well as those of Mary, Queen of Scots and Richard I of England. Other groups of artefacts represented in the department include the national collection of (c.100) icon paintings, most of which originate from the Byzantine Empire and Russia, and over 40 mediaeval astrolabes from across Europe and the Middle East. The department also includes the national collection of horology with one of the most wide-ranging assemblage of clocks, watches and other timepieces in Europe, with masterpieces from every period in the development of time-keeping. |
The department also includes the national collection of horology with one of the most wide-ranging assemblage of clocks, watches and other timepieces in Europe, with masterpieces from every period in the development of time-keeping. Choice horological pieces came from the Morgan and Ilbert collections. The department is also responsible for the curation of Romano-British objects – the museum has by far the most extensive such collection in Britain and one of the most representative regional collections in Europe outside Italy. It is particularly famous for the large number of late Roman silver treasures, many of which were found in East Anglia, the most important of which is the Mildenhall Treasure. The museum purchased many Roman-British objects from the antiquarian Charles Roach Smith in 1856. These quickly formed the nucleus of the collection. The department also includes ethnographic material from across Europe including a collection of Bulgarian costumes and shadow puppets from Greece and Turkey. A particular highlight are the three Sámi drums from northern Sweden of which only about 70 are extant.
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Objects from the Department of Britain, Europe and Prehistory are mostly found on the upper floor of the museum, with a suite of galleries numbered from 38 to 51. Most of the collection is stored in its archive facilities, where it is available for research and study.
Highlights of the collections include:
Stone Age (c. 3.4 million years BC – c. 2000 BC)
Bronze Age ()
Iron Age ()
Romano-British (43 AD – 410 AD)
Early Mediaeval ()
Mediaeval ()
Renaissance to Modern ( – present)
The many hoards of treasure include those of Esquiline, Carthage, First Cyprus, Hockwold, Hoxne, Lampsacus, Mildenhall, Vale of York and Water Newton, (4th–10th centuries AD)
Department of Asia.
The scope of the Department of Asia is extremely broad; its collections of over 75,000 objects cover the material culture of the whole Asian continent and from the Neolithic up to the present day. Until recently, this department concentrated on collecting Oriental antiquities from urban or semi-urban societies across the Asian continent. Many of those objects were collected by colonial officers and explorers in former parts of the British Empire, especially the Indian subcontinent. Examples include the collections made by individuals such as James Wilkinson Breeks, Sir Alexander Cunningham, Sir Harold Deane, Sir Walter Elliot, James Prinsep, Charles Masson, Sir John Marshall and Charles Stuart. A large number of Chinese antiquities were purchased from the Anglo-Greek banker George Eumorfopoulos in the 1930s. The large collection of some 1800 Japanese prints and paintings owned by Arthur Morrison was acquired in the early twentieth century. In the second half of the twentieth century, the museum greatly benefited from the bequest of the philanthropist PT Brooke Sewell, which allowed the department to purchase many objects and fill in gaps in the collection.
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In 2004, the ethnographic collections from Asia were transferred to the department. These reflect the diverse environment of the largest continent in the world and range from India to China, the Middle East to Japan. Much of the ethnographic material comes from objects originally owned by tribal cultures and hunter-gatherers, many of whose way of life has disappeared in the last century. Particularly valuable collections are from the Andaman and Nicobar Islands (much assembled by the British naval officer Maurice Portman), Sri Lanka (especially through the colonial administrator Hugh Nevill), Northern Thailand, south-west China, the Ainu of Hokkaido in Japan (chief among them the collection of the Scottish zoologist John Anderson), Siberia (with artefacts collected by the explorer Kate Marsden and Bassett Digby and is notable for its Sakha pieces, especially the ivory model of a summer festival at Yakutsk) and the islands of South-East Asia, especially Borneo. The latter benefited from the purchase in 1905 of the Sarawak collection put together by Dr Charles Hose, as well as from other colonial officers such as Edward A Jeffreys. In addition, a unique and valuable group of objects from Java, including shadow puppets and a gamelan musical set, was assembled by Sir Stamford Raffles.
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The principal gallery devoted to Asian art in the museum is Gallery 33 with its comprehensive display of Chinese, Indian subcontinent and South-east Asian objects. An adjacent gallery showcases the Amaravati sculptures and monuments. Other galleries on the upper floors are devoted to its Japanese, Korean, painting and calligraphy, and Chinese ceramics collections.
Highlights of the collections include:
East Asia
South Asia
Southeast Asia
Department of Africa, Oceania and the Americas.
The British Museum houses one of the world's most comprehensive collections of ethnographic material from Africa, Oceania and the Americas, representing the cultures of indigenous peoples throughout the world. Over 350,000 objects spanning thousands of years tells the history of mankind from three major continents and many rich and diverse cultures; the collecting of modern artefacts is ongoing. Many individuals have added to the department's collection over the years but those assembled by Henry Christy, Harry Beasley and William Oldman are outstanding.
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Objects from this department are mostly on display in several galleries on the ground and lower floors. Gallery 24 displays ethnographic from every continent while adjacent galleries focus on North America and Mexico. A long suite of rooms (Gallery 25) on the lower floor display African art. There are plans in place to develop permanent galleries for showcasing art from Oceania and South America.
Africa
The Sainsbury African Galleries display 600 objects from the greatest permanent collection of African arts and culture in the world. The three permanent galleries provide a substantial exhibition space for the museum's African collection comprising over 200,000 objects. A curatorial scope that encompasses both archaeological and contemporary material, including both unique masterpieces of artistry and objects of everyday life. A great addition was material amassed by Sir Henry Wellcome, which was donated by the Wellcome Historical Medical Museum in 1954.
Highlights of the African collection include objects found at megalithic circles in The Gambia, a dozen exquisite Afro-Portuguese ivories, a series of soapstone figures from the Kissi people in Sierra Leone and Liberia, hoard of bronze Kru currency rings from the Sinoe River in Liberia, Asante goldwork and regalia from Ghana including the Bowdich collection, the rare Akan Drum from the same region in west Africa, pair of door panels and lintel from the palace at Ikere-Ekiti in Yorubaland, the Benin and Igbo-Ukwu bronze sculptures, the beautiful Bronze Head of Queen Idia, a magnificent brass head of a Yoruba ruler and quartz throne from Ife, a similar terracotta head from Iwinrin Grove near Ife, the Apapa Hoard from Lagos and other mediaeval bronze hoards from Allabia and the Forçados River in southern Nigeria.
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Included is an Ikom monolith from Cross River State, several ancestral screens from the Kalabari tribe in the Niger Delta, the Torday collection of central African sculpture, textiles and weaponry from the Kuba Kingdom including three royal figures, the unique Luzira Head from Uganda, processional crosses and other ecclesiastical and royal material from Gondar and Magdala, Ethiopia following the British Expedition to Abyssinia, excavated objects from Great Zimbabwe (that includes a unique soapstone, anthropomorphic figure) and satellite towns such as Mutare including a large hoard of Iron Age soapstone figures, a rare divining bowl from the Venda peoples and cave paintings and petroglyphs from South Africa.
Oceania
The British Museum's Oceanic collections originate from the vast area of the Pacific Ocean, stretching from Papua New Guinea to Easter Island, from New Zealand to Hawaii. The three main anthropological groups represented in the collection are Polynesia, Melanesia and Micronesia – Aboriginal art from Australia is considered separately in its own right. Metal working was not indigenous to Oceania before Europeans arrived, so many of the artefacts from the collection are made from stone, shell, bone and bamboo. Prehistoric objects from the region include a bird-shaped pestle and a group of stone mortars from Papua New Guinea.
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The British Museum is fortunate in having some of the earliest Oceanic and Pacific collections, many of which were put together by members of Cook's and Vancouver's expeditions or by colonial administrators and explorers such as Sir George Grey, Sir Frederick Broome, Joseph Bradshaw, Robert Christison, Gregory Mathews, Frederick Meinertzhagen, Thomas Mitchell and Arthur Gordon, before Western culture significantly impacted on indigenous cultures. The department has also benefited greatly from the legacy of pioneering anthropologists such as AC Haddon, Bronisław Malinowski and Katherine Routledge. An artefact is a wooden Aboriginal shield, probably dating from the late eighteenth century. There is some debate as to whether this shield was found at Botany Bay or, given the nature of the wood being red mangrove which grows abundantly only 500 km north of Botany Bay, possibly obtained through trade networks or at an entirely different location.
The Wilson cabinet of curiosities from Palau is an example of pre-contact ware. Another outstanding exemplar is the mourner's dress from Tahiti given to Cook on his second voyage, one of only ten in existence. In the collection is a large war canoe from the island of Vella Lavella in the Solomon Islands, one of the last ever to be built in the archipelago.
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The Māori collection is the finest outside New Zealand with many intricately carved wooden and jade objects and the Aboriginal art collection is distinguished by its wide range of bark paintings, including two very early bark etchings collected by John Hunter Kerr. A particularly important group of objects was purchased from the London Missionary Society in 1911, that includes the unique statue of A'a from Rurutu Island, the rare idol from the isle of Mangareva and the Cook Islands deity figure. Other highlights include the huge Hawaiian statue of Kū-ka-ili-moku or god of war (one of three extant in the world) and the famous Easter Island statues Hoa Hakananai'a and Moai Hava.
Americas
The Americas collection mainly consists of 19th and 20th century items although the Paracas, Moche, Inca, Maya, Aztec, Taino and other early cultures are well represented. The Kayung totem pole, which was made in the late nineteenth century on Haida Gwaii, dominates the Great Court and provides a fitting introduction to this very wide-ranging collection that stretches from the very north of the North American continent where the Inuit population has lived for centuries, to the tip of South America where indigenous tribes have long thrived in Patagonia.
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Highlights of the collection include Aboriginal Canadian and Native American objects from North America collected by the 5th Earl of Lonsdale, the Marquis of Lorne, the explorer David Haig-Thomas and Bryan Mullanphy, Mayor of St. Louis, the Squier and Davis collection of prehistoric mound relics from North America, two carved stone bowls in the form of a seated human figure made by ancient North West Coast peoples from British Columbia, the headdress of Chief Yellow Calf from the Arapaho tribe in Wyoming, a lidded rivercane basket from South Carolina and the earliest historic example of Cherokee basketry, a selection of pottery vessels found in prehistoric dwellings at Mesa Verde and Casas Grandes, one of the enigmatic crystal skulls of unknown origin, a collection of nine turquoise Aztec mosaics from Mexico (the largest in Europe), important artefacts from Teotihuacan and Isla de Sacrificios.
There are several rare pre-Columbian manuscripts including the Codex Zouche-Nuttall and Codex Waecker-Gotter and post-colonial ones such as the Codex Aubin and Codex Kingsborough, a spectacular series of Mayan lintels from Yaxchilan excavated by the British Mayanist Alfred Maudslay, a very high quality Mayan collection that includes sculptures from Copan, Tikal, Tulum, Pusilha, Naranjo and Nebaj (including the celebrated Fenton Vase), an ornate calcite vase with jaguar handles from the Ulua Valley in Honduras, the Lord Moyne collection from the Bay Islands, Honduras and Boyle collection from Nicaragua, over 20 stone metates with zoomorphic and anthropomorphic ornamentation from Costa Rica, a group of Zemi Figures from Vere, Jamaica, and wooden duhos from the Dominican Republic and The Bahamas.
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There are a collection of Pre-Columbian human mummies from sites across South America including Ancon, Acari, Arica and Leyva, a number of prestigious pre-Columbian gold and votive objects from Colombia, three axe-shaped gold diadems found near Camaná from the Siguas culture in Peru, unique collection of Moche wooden figures and staffs from the off Peru, ethnographic objects from across the Amazon region including the Schomburgk and Maybury Lewis collections and part of the von Martius and von Spix collection, two rare Tiwanaku pottery vessels from Lake Titicaca and important items from Tierra del Fuego donated by Commander Phillip Parker King.
Department of Money and Medals.
The British Museum is home to one of the world's finest numismatic collections, comprising about a million objects, including coins, medals, tokens and paper money. The collection spans the entire history of coinage from its origins in the 7th century BC to the present day and is representative of both the East and West. The Department of Coins and Medals was created in 1861 and celebrated its 150th anniversary in 2011.
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Department of Conservation and Scientific Research.
This department was founded in 1920. Conservation has six specialist areas: ceramics & glass; metals; organic material (including textiles); stone, wall paintings and mosaics; Eastern pictorial art and Western pictorial art. The science department has and continues to develop techniques to date artefacts, analyse and identify the materials used in their manufacture, to identify the place an artefact originated and the techniques used in their creation. The department also publishes its findings and discoveries.
Libraries and archives.
This department covers all levels of education, from casual visitors, schools, degree level and beyond. The museum's various libraries hold in excess of 350,000 books, journals and pamphlets covering all areas of the museum's collection. Also the general museum archives which date from its foundation in 1753 are overseen by this department; the individual departments have their own separate archives and libraries covering their various areas of responsibility, which can be consulted by the public on application. The Anthropology Library is especially large, with 120,000 volumes. However, the Paul Hamlyn Library, which had become the central reference library of the British Museum and the only library there freely open to the general public, closed permanently in August 2011. The website and online database of the collection also provide increasing amounts of information.
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British Museum Press.
The British Museum Press (BMP) is the publishing business and a division of the British Museum Company Ltd., a company and a charity (established in 1973) wholly owned by the trustees of the British Museum.
The BMP publishes both popular and scholarly illustrated books to accompany the exhibition programme and explore aspects of the general collection. Profits from their sales goes to support the British Museum.
Scholarly titles are published in the Research Publications series, all of which are peer-reviewed. This series was started in 1978 and was originally called Occasional Papers. The series is designed to disseminate research on items in the collection. To date, over 200 books have been published in this series. Between six and eight titles are published each year in this series. They can be found on the British Museum Research Repository.
Controversies and criticism.
Contested artefacts.
It is a point of controversy whether museums should possess artefacts illegally taken from other countries, and the British Museum is a notable target for criticism. The Elgin Marbles, the Benin Bronzes, Ethiopian Tabots and the Rosetta Stone are among the most disputed objects in its collections, and organisations have been formed demanding the return of these artefacts to their native countries.
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The Elgin Marbles or Parthenon Marbles claimed by Greece have been cited by UNESCO, among others, for restitution. From 1801 to 1812, Lord Elgin's agents removed about half of the surviving sculptures from the Parthenon, as well as sculptures from the Propylaea and Erechtheum. The former director of the museum has stated, "We are indebted to Elgin for having rescued the Parthenon sculptures and others from the Acropolis from the destruction they were suffering, as well as from the damage that the Acropolis monuments, including the sculptures that he did not remove, have suffered since." The British Museum itself damaged some of the artefacts during restoration in the 1930s. In late 2022, the British Museum had entered into preliminary negotiations with the Greek government about the future of the sculptures.
There is also controversy over artefacts taken during the destruction of the Old Summer Palace in Beijing by an Anglo-French expeditionary force during the Second Opium War in 1860, an event which drew protest from Victor Hugo. The British Museum and the Victoria and Albert Museum, among others, have been asked since 2009 to open their archives for investigation by a team of Chinese investigators as a part of an international mission to document Chinese national treasures in foreign collections. In 2010 Neil MacGregor, the former Director of the British Museum, said he hoped that both British and Chinese investigators would work together on the controversial collection. In 2020 the museum appointed a curator to research the history of its collections, including disputed items.
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The British Museum has stated that the "restitutionist premise, that whatever was made in a country must return to an original geographical site, would empty both the British Museum and the other great museums of the world". The museum has also argued that the British Museum Act of 1963 prevents any object from leaving its collection once it has entered it. "The Museum owns its collections, but its Trustees are not empowered to dispose of them". Nevertheless, it has returned items such as Tasmanian Aboriginal burial remains when this was consistent with legislation regarding the disposal of items in the collections.
Nazi-looted art.
In 2002 the heirs of Arthur Feldmann, an art collector murdered in the Holocaust, requested that four old master drawings stolen by the Gestapo in 1939 be returned to the family. A UK High Court judge ruled in 2005 that it would be illegal for the British Museum to return artworks looted by the Nazis to a Jewish family, despite its willingness and moral obligation to do so. The law was changed in 2009, and again in 2022 giving museums additional powers to return looted art or provide compensation. Feldmann's heirs accepted a compensation payment for a looted drawing and stated that they were happy the drawing would remain in the British Museum collection.
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According to the British Museum Spoliation report published by the Collections Trust in 2017, "Around 30% of some 21,350 continental and British drawings acquired since 1933 have an uncertain or incomplete provenance for the 1933–1945 period". The museum lists these works on its website and investigates claims for restitution.
BP sponsorship.
Since 2016, there have been a number of protests by activist groups, trade unions and the public against the British Museum's relationship with the oil company BP which the protesters believe implicates the museum in global warming. In July 2019, Ahdaf Soueif resigned from the British Museum's board of trustees in protest against the sponsorship. In February 2020, 1,500 demonstrators, including British Museum staff, took part in a day of protest over the issue. In December 2023, it was announced that the British Museum had agreed to a new £50 million sponsorship deal with BP.
Chairman's Advisory Group.
The Chairman's Advisory Group is an informal group of business leaders who provide advice to the chairman on various issues including the museum's relationship with the British government and policy on the museum's collections. Its existence was made public after a freedom of information request by a group campaigning against the museum's links with the fossil fuel industry. The museum has declined to name the members of the advisory group as they are acting in their personal capacity.
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Thefts.
Thefts from the museum include: several historic coins and medals in the 1970s; a 17th-century Japanese Kakiemon figure in 1990; two Meiji figurines and a fragment of a gold ring in 1991; fifteen Roman coins and jewellery worth £250,000 in 1993; and a Japanese chest and two Persian books in 1996.
In July 2002 a marble head, valued at £50,000, was stolen from the Archaic Greek gallery. In 2004, 15 Chinese artefacts including jewels, ornate hairpins and fingernail guards were stolen. In 2017, it was revealed that a Cartier diamond had been missing since 2011.
In August 2023, a staff member was fired after it emerged that items including gold, jewellery and gems had been stolen over a "significant" period of time. The incident led to an investigation by the Metropolitan Police and an independent review by the museum. Some of the missing artefacts were later found to have been sold on eBay for considerably less than their estimated value. The museum had been warned of the thefts as early as 2021. The museum's director, Hartwig Fischer, resigned because of the museum's inadequate response to the warnings of theft. The number of artefacts stolen was estimated to be about 2,000. As a consequence of the thefts, the museum announced a five-year plan to digitise the complete collection and make it available to view online. By May 2024, 626 of the missing items had been recovered.
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Copyright settlement.
In August 2023, the British Museum reached a settlement with the translator Yilin Wang over her translations of poetry by Qiu Jin. The museum had used her work without credit or permission in their exhibition "China's Hidden Century" which ran between May 2023 and October 2023.
Tibet naming conventions.
In January 2025, the British Museum was criticized by Tibetan human rights groups for referring to Tibet as "Xizang," the current preferred term of the government of the People's Republic of China.
Galleries.
"Department of Ancient Egypt and Sudan"
"Department of the Middle East"
"Department of Greece and Rome"
Digital and online.
The museum has a collaboration with the Google Cultural Institute to bring the collection online.
Exhibitions.
"Forgotten Empire Exhibition" (October 2005 – January 2006)
From January to April 2012 the museum presented "", the first major exhibition on the topic of the Hajj, the pilgrimage that is one of the five pillars of Islam. |
Binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying and the coefficient of each term is a specific positive integer depending on and . For example, for ,
formula_1
The coefficient in each term is known as the binomial coefficient or (the two have the same value). These coefficients for varying and can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where gives the number of different combinations (i.e. subsets) of elements that can be chosen from an -element set. Therefore is usually pronounced as " choose ".
Statement.
According to the theorem, the expansion of any nonnegative integer power of the binomial is a sum of the form
formula_2
where each formula_3 is a positive integer known as a binomial coefficient, defined as
formula_4
This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, it can be written more concisely as
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formula_5
The final expression follows from the previous one by the symmetry of and in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical, formula_6
A simple variant of the binomial formula is obtained by substituting for , so that it involves only a single variable. In this form, the formula reads
formula_7
Examples.
The first few cases of the binomial theorem are:
formula_8
In general, for the expansion of on the right side in the th row (numbered so that the top row is the 0th row):
An example illustrating the last two points: formula_9 with formula_10.
A simple example with a specific positive value of :
formula_11
A simple example with a specific negative value of :
formula_12
Geometric explanation.
For positive values of and , the binomial theorem with is the geometrically evident fact that a square of side can be cut into a square of side , a square of side , and two rectangles with sides and . With , the theorem states that a cube of side can be cut into a cube of side , a cube of side , three rectangular boxes, and three rectangular boxes.
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In calculus, this picture also gives a geometric proof of the derivative formula_13 if one sets formula_14 and formula_15 interpreting as an infinitesimal change in , then this picture shows the infinitesimal change in the volume of an -dimensional hypercube, formula_16 where the coefficient of the linear term (in formula_17) is formula_18 the area of the faces, each of dimension :
formula_19
Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms, formula_20 and higher, become negligible, and yields the formula formula_21 interpreted as
If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral formula_22 – see proof of Cavalieri's quadrature formula for details.
Binomial coefficients.
The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written formula_23 and pronounced " choose ".
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Formulas.
The coefficient of is given by the formula
formula_24
which is defined in terms of the factorial function . Equivalently, this formula can be written
formula_25
with factors in both the numerator and denominator of the fraction. Although this formula involves a fraction, the binomial coefficient formula_26 is actually an integer.
Combinatorial interpretation.
The binomial coefficient formula_3 can be interpreted as the number of ways to choose elements from an -element set (a combination). This is related to binomials for the following reason: if we write as a product
formula_28
then, according to the distributive law, there will be one term in the expansion for each choice of either or from each of the binomials of the product. For example, there will only be one term , corresponding to choosing from each binomial. However, there will be several terms of the form , one for each way of choosing exactly two binomials to contribute a . Therefore, after combining like terms, the coefficient of will be equal to the number of ways to choose exactly elements from an -element set.
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Proofs.
Combinatorial proof.
Expanding yields the sum of the products of the form where each is or . Rearranging factors shows that each product equals for some between and . For a given , the following are proved equal in succession:
This proves the binomial theorem.
Example.
The coefficient of in
formula_32
equals formula_33 because there are three strings of length 3 with exactly two 's, namely,
formula_34
corresponding to the three 2-element subsets of , namely,
formula_35
where each subset specifies the positions of the in a corresponding string.
Inductive proof.
Induction yields another proof of the binomial theorem. When , both sides equal , since and formula_36 Now suppose that the equality holds for a given ; we will prove it for . For , let denote the coefficient of in the polynomial . By the inductive hypothesis, is a polynomial in and such that is formula_26 if , and otherwise. The identity
formula_38
shows that is also a polynomial in and , and
formula_39
since if , then and . Now, the right hand side is
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formula_40
by Pascal's identity. On the other hand, if , then and , so we get . Thus
formula_41
which is the inductive hypothesis with substituted for and so completes the inductive step.
Generalizations.
Newton's generalized binomial theorem.
Around 1665, Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to complex exponents.) In this generalization, the finite sum is replaced by an infinite series. In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using the usual formula with factorials. However, for an arbitrary number , one can define
formula_42
where formula_43 is the Pochhammer symbol, here standing for a falling factorial. This agrees with the usual definitions when is a nonnegative integer. Then, if and are real numbers with , and is any complex number, one has
formula_44
When is a nonnegative integer, the binomial coefficients for are zero, so this equation reduces to the usual binomial theorem, and there are at most nonzero terms. For other values of , the series typically has infinitely many nonzero terms.
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For example, gives the following series for the square root:
formula_45
Taking , the generalized binomial series gives the geometric series formula, valid for :
formula_46
More generally, with , we have for :
formula_47
So, for instance, when ,
formula_48
Replacing with yields:
formula_49
So, for instance, when , we have for :
formula_50
Further generalizations.
The generalized binomial theorem can be extended to the case where and are complex numbers. For this version, one should again assume and define the powers of and using a holomorphic branch of log defined on an open disk of radius centered at . The generalized binomial theorem is valid also for elements and of a Banach algebra as long as , and is invertible, and .
A version of the binomial theorem is valid for the following Pochhammer symbol-like family of polynomials: for a given real constant , define formula_51 and
formula_52
for formula_53 Then
formula_54
The case recovers the usual binomial theorem.
More generally, a sequence formula_55 of polynomials is said to be of binomial type if
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An operator formula_63 on the space of polynomials is said to be the "basis operator" of the sequence formula_55 if formula_65 and formula_66 for all formula_67. A sequence formula_55 is binomial if and only if its basis operator is a Delta operator. Writing formula_69 for the shift by formula_70 operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference formula_71 for formula_72, the ordinary derivative for formula_73, and the forward difference formula_74 for formula_75.
Multinomial theorem.
The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is
formula_76
where the summation is taken over all sequences of nonnegative integer indices through such that the sum of all is . (For each term in the expansion, the exponents must add up to ). The coefficients formula_77 are known as multinomial coefficients, and can be computed by the formula
formula_78
Combinatorially, the multinomial coefficient formula_79 counts the number of different ways to partition an -element set into disjoint subsets of sizes .
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Multi-binomial theorem.
When working in more dimensions, it is often useful to deal with products of binomial expressions. By the binomial theorem this is equal to
formula_80
This may be written more concisely, by multi-index notation, as
formula_81
General Leibniz rule.
The general Leibniz rule gives the th derivative of a product of two functions in a form similar to that of the binomial theorem:
formula_82
Here, the superscript indicates the th derivative of a function, formula_83. If one sets and , cancelling the common factor of from each term gives the ordinary binomial theorem.
History.
Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent formula_84. Greek mathematician Diophantus cubed various binomials, including formula_85. Indian mathematician Aryabhata's method for finding cube roots, from around 510 AD, suggests that he knew the binomial formula for exponent formula_86.
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The Persian mathematician al-Karajī (953–1029) wrote a now-lost book containing the binomial theorem and a table of binomial coefficients, often credited as their first appearance.
An explicit statement of the binomial theorem appears in al-Samawʾal's "al-Bāhir" (12th century), there credited to al-Karajī. Al-Samawʾal algebraically expanded the square, cube, and fourth power of a binomial, each in terms of the previous power, and noted that similar proofs could be provided for higher powers, an early form of mathematical induction. He then provided al-Karajī's table of binomial coefficients (Pascal's triangle turned on its side) up to and a rule for generating them equivalent to the recurrence relation . The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost. The binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui and also Chu Shih-Chieh. Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, although those writings are now also lost.
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In Europe, descriptions of the construction of Pascal's triangle can be found as early as Jordanus de Nemore's "De arithmetica" (13th century). In 1544, Michael Stifel introduced the term "binomial coefficient" and showed how to use them to express formula_87 in terms of formula_88, via "Pascal's triangle". Other 16th century mathematicians including Niccolò Fontana Tartaglia and Simon Stevin also knew of it. 17th-century mathematician Blaise Pascal studied the eponymous triangle comprehensively in his "Traité du triangle arithmétique".
By the early 17th century, some specific cases of the generalized binomial theorem, such as for formula_89, can be found in the work of Henry Briggs' "Arithmetica Logarithmica" (1624). Isaac Newton is generally credited with discovering the generalized binomial theorem, valid for any real exponent, in 1665, inspired by the work of John Wallis's "Arithmetic Infinitorum" and his method of interpolation. A logarithmic version of the theorem for fractional exponents was discovered independently by James Gregory who wrote down his formula in 1670.
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Applications.
Multiple-angle identities.
For the complex numbers the binomial theorem can be combined with de Moivre's formula to yield multiple-angle formulas for the sine and cosine. According to De Moivre's formula,
formula_90
Using the binomial theorem, the expression on the right can be expanded, and then the real and imaginary parts can be taken to yield formulas for and . For example, since
formula_91
But De Moivre's formula identifies the left side with formula_92, so
formula_93
which are the usual double-angle identities. Similarly, since
formula_94
De Moivre's formula yields
formula_95
In general,
formula_96
and
formula_97There are also similar formulas using Chebyshev polynomials.
Series for "e".
The number is often defined by the formula
formula_98
Applying the binomial theorem to this expression yields the usual infinite series for . In particular:
formula_99
The th term of this sum is
formula_100
As , the rational expression on the right approaches , and therefore
formula_101
This indicates that can be written as a series:
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formula_102
Indeed, since each term of the binomial expansion is an increasing function of , it follows from the monotone convergence theorem for series that the sum of this infinite series is equal to .
Probability.
The binomial theorem is closely related to the probability mass function of the negative binomial distribution. The probability of a (countable) collection of independent Bernoulli trials formula_103 with probability of success formula_104 all not happening is
In abstract algebra.
The binomial theorem is valid more generally for two elements and in a ring, or even a semiring, provided that . For example, it holds for two matrices, provided that those matrices commute; this is useful in computing powers of a matrix.
The binomial theorem can be stated by saying that the polynomial sequence is of binomial type. |
Boxing Day
Boxing Day is a holiday celebrated after Christmas Day, occurring on the second day of Christmastide (26 December). Boxing Day was once a day to donate gifts to those in need, but it has evolved to become a part of Christmas festivities, with many people choosing to shop for deals on Boxing Day. It originated in the United Kingdom and is celebrated in several Commonwealth nations. The attached bank holiday or public holiday may take place on 27 or 28 December if necessary to ensure it falls on a weekday. Boxing Day is also concurrent with the Christian festival Saint Stephen's Day.
In parts of Europe, such as east Spain, (Catalonia,Valencia and the Balearic Islands), the Czech Republic, Germany, Austria, Hungary, Netherlands, Italy, Poland, Slovakia, Slovenia, Croatia, Denmark, Finland, Romania, Sweden, Belgium, Norway, Latvia and the Republic of Ireland, 26 December is Saint Stephen's Day, which is considered the second day of Christmas.
Etymology.
There are competing theories for the origins of the term, none of which are definitive.
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The European tradition of giving money and other gifts to those in need, or in service positions, has been dated to the Middle Ages, but the exact origin is unknown; it may refer to the alms box placed in the narthex of Christian churches to collect donations for the poor. The tradition may come from a custom in the late Roman and early Christian era, wherein alms boxes placed in churches were used to collect special offerings tied to the Feast of Saint Stephen, which, in the Western Christian Churches, falls on the same day as Boxing Day, the second day of Christmastide. On this day, it is customary in some localities for the alms boxes to be opened and distributed to the poor.
The "Oxford English Dictionary" gives the earliest attestation from Britain in 1743, defining it as "the day after Christmas day", and saying "traditionally on this day tradespeople, employees, etc., would receive presents or gratuities (a "Christmas box") from their customers or employers."
The term "Christmas box" dates back to the 17th century, and among other things meant:
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