Racemic drugs are not  drug combinations  in the accepted sense of two or more co-formulated therapeutic agents, but combinations of isomeric substances whose pharmacological activity may reside predominantly in one specific enantiomeric form. In case of stereoselectivity in action only one of the components in the racemic mixture is truly active. The other isomer, the distomer, should be regarded as impurity or isomeric ballast, [ 31 ]  a term coined by Ariëns, not contributing to the effects aimed at. In contrast to the pharmacokinetic properties of an enantiomeric pair, differences in pharmacodynamic activity tend to be more obvious. There is a wide spectrum of possibilities of distomer actions, many of which are confirmed experimentally. [ 32 ] [ 33 ] [ 34 ]   Selected examples of the distomer actions (viz. equipotent, less active, inactive, antagonistic,  chiral inversion ) are presented in the table below.


Since there is a frequent large pharmacokinetic and pharmacodynamic differences between enantiomers of a chiral drug it is not surprising that enantiomers may result in stereoselective toxicity. They can reside in the pharmacologically active enantiomer (eutomer) or in the inactive one (distomer). [ 41 ] [ 42 ] [ 43 ]  The toxicologic differences between enantiomers of have also been demonstrated. The following are examples of some of the chiral drugs where their toxic/undesirable side-effects dwell almost in the distomer. This would seem to be a clear cut case of going for a  chiral switch .


Penicillamine  is a chiral drug with one chiral center and exists as a pair of enantiomers. (S)-penicillamine is the eutomer with the desired antiarthritic activity while the (R)-penicillamine is extremely toxic. [ 44 ]


Ketamine is a widely used anaesthetic agent. It is a chiral molecule that is administered as a racemate. Studies show that (S)-(+)-ketamine is the active anaesthetic and the undesired side-effects (hallucination and agitation) reside in the distomer, (R)-(-)-ketamine. [ 45 ] [ 46 ] [ 47 ]


The initial use of racemic dopa for the treatment of Parkinson's disease resulted in a number of adverse effects viz. nausea, vomiting, anorexia, involuntary movements and granulocytopenia. The use of L-dopa [the (S)-enantiomer] resulted in reducing the required dose, and adverse effects. The granulocytopenia was not observed with the single enantiomer. [ 48 ] [ 49 ] [ 50 ]


The antitubercular agent Ethambutol contains two constitutionally symmetrical stereogenic centers in its structure and exists in three stereoisomeric forms. An enantiomeric pair (S,S)- and (R,R)-ethambutol, along with the achiral stereoisomer called  meso -form, it holds a diastereomeric relationship with the optically active stereoisomers. The activity of the drug resides in the (S,S)-enantiomer which is 500 and 12 fold more potent than the (R,R)-ethambutol and the  meso- form. The drug had initially been introduced for clinical use as the racemate and was changed to the (S,S)-enantiomer, as a result of optic neuritis leading to blindness. Toxicity is related to both dose and duration of treatment.  All the three stereoisomers were almost equipotent with respect to side effects. Hence the use of S,S)-enantiomer greatly enhanced the risk/benefit ratio. [ 51 ] [ 52 ]


Thalidomide is a classical example highlighting the alleged role of chirality in drug toxicity. Thalidomide was a racemic therapeutic and prescribed to pregnant women to control nausea and vomiting. The drug was withdrawn from world market when it became evident that the use in pregnancy causes phocomelia (clinical conditions where babies are born with deformed hand and limbs). Later in late 1970s studies indicated that the (R)- enantiomer is an effective  sedative , the (S)-enantiomer harbors teratogenic effect and causes fetal abnormalities. [ 53 ] [ 54 ] [ 55 ]  Later studies established that under biological conditions the ( R) -thalidomide, good partner, undergoes an  in vivo  metabolic inversion to the ( S) -thalidomide, evil partner and vice versa. It is a  bidirectional  chiral   inversion . Hence the argument that the thalidomide tragedy could have been avoided by using a single enantiomer is ambiguous and pointless. [ 56 ] [ 57 ] [ 58 ]


The salient features are presented in the table below.


Unichiral indicates configurationally homogeneous substance (i.e. made up of chiral molecules of one and the same configuration). Other commonly used synonyms are  enantiopure drugs  and enantiomerically pure drugs. Monochiral drugs has also been suggested as another synonym. [ 59 ]  Professor Eliel, Wilen, and Gal expressed their deep concern over the misuse of the term "homochiral" in articles to denote enantiomerically pure drugs, which is incorrect. [ 60 ] [ 61 ] [ 62 ]  Homochiral means objects or molecules of the same handedness. Hence should be used only for comparison of two or more objects of like "chirality". For instance, left hands of different individuals, or say R-naproxen and R-ibuprofen.


Globally drug companies and regulatory agencies have an inclination towards the development of unichiral drugs as a consequence of the increased understanding of the differing biological properties of individual enantiomers of a racemic therapeutics.  Most of these unichiral drugs are the consequence of  chiral switch  approach. The table below list selected unichiral drugs used in drug therapy.


A company may go in for developing a racemic drug against an enantiomer by providing adequate reasoning. The rationale why a company might pursue developing racemic drugs [ 63 ] [ 64 ] [ 65 ]  could include expensive separation of enantiomers, eutomer racemizes in solution (e.g. oxazepam), [ 66 ]  activities of the enantiomeric pair are different but supplementary, distomer is inactive, but separation is exorbitant. Insignificant/low toxicity of the distomer, high therapeutic index, mutually beneficial, pharmacological activities of both the enantiomers, and if the development of an enantiomer takes huge amount of time for a drug of emergency need e.g., cancer, AIDS, etc.


Chiral purity is a measure of the purity of a chiral drug. Other synonyms employed include  enantiomeric excess , enantiomer purity, enantiomeric purity, and optical purity. Optical purity is an obsolete term since today most of the chiral purity measurements are done using chromatographic techniques (not based on optical principles). Enantiomeric excess tells the extent (in %) to which the chiral substance contains one enantiomer over the other. For a racemic drug the enantiomeric excess will be 0%. There are number of  chiral analysis  tools such as polarimetry, NMR spectroscopy with the use of chiral shift reagents, chiral GC (gas chromatography), chiral HPLC (high performance liquid chromatography), chiral TLC (thin-layer chromatography) [ 67 ]  and other chiral chromatographic techniques, that are employed to evaluate chiral purity. [ 68 ]  Assessing the purity of a unichiral drug or enantiopure drug is of great importance from a drug safety and efficacy perspective.


Chiral  inversion  is the process of conversion of one  enantiomer  of a chiral molecule to its mirror-image version with no other change in the molecule. [ 1 ] [ 2 ] [ 3 ] [ 4 ]


Chiral inversion happens depending on various factors (viz. biological-, solvent-, light-, temperature- induced, etc.) and the  energy barrier  energy barrier associated with the stereogenic element present in the chiral molecule. 2-Arylpropionic acid  nonsteroidal anti-inflammatory drugs  (NSAIDs) provide one of the best  pharmaceutical examples of chiral inversion.  Chirality  is attributed to a molecule due to the presence of a stereogenic element (viz. center, planar, helical, or axis). Many pharmaceutical drugs are chiral and have a labile (configurationally unstable) stereogenic element. Chiral compounds with  stereogenic center  are found to have high energy barriers for inversion and generally undergo biologically mediated chiral inversion.  While compounds with  helical  or  planar chirality  have low energy barriers and chiral inversions are often caused by solvent, light, temperature. [ 5 ]  When this happens, the configuration of the chiral molecule may rapidly change reversibly or irreversibly depending on the conditions. The chiral inversion has been intensively studied in the context of the pharmacological and toxicological consequences. [ 6 ]  Other than NSAIDs, chiral drugs with different chemical structures can also show this effect.


Chiral drugs  have different effects on the body depending on whether one enantiomer or both enantiomers act on different biological targets. As a result, chiral inversion can change how a pharmaceutical drug works in the body. From a pharmacological and toxicological point of view, it is very important to learn more about chiral inversion, the things that make it happen, and the tools used to figure out chiral inversion.


Essentially there are two types of chiral inversion, unidirectional and bidirectional. [ 7 ]  Inversion process is dependent on species and substrate.


It is well documented that (R)-enantiomers of profens in the presence of coenzyme A (CoA), adenosine triphosphate (ATP) and Mg +2  are converted to active (S)-forms. The pathways of chiral inversion is illustrated taking  ibuprofen  as the prototype, in the scheme below. [ 14 ] [ 25 ]


The pathway consists mainly of three steps:


Because the acyl-CoA thioester (profenyl-CoA) changes the structure of triglycerides and phospholipids, metabolic chiral inversion may cause toxic effects. [ 10 ]


Chiral drugs with stereo-labile configuration are likely to undergo interconversion of the enantiomers that may be enzymatic (biological) or non-enzymatic. Enzyme-mediated conversion is the process of chiral inversion that happens in a living organism. Non-enzymatic inversion of drugs is important and relevant in the pharmaceutical manufacturing process. This may have impact on the shelf-life of a drug and the economic feasibility of the resolution. Inversion can also happen without enzymes when precolumn derivatization is used in enantioselective chromatographic separation techniques. Racemization can also happen in the acidic environment of the stomach and other bodily fluids.


Enzyme-mediated (biological) chiral inversion of organic compounds is caused by highly chiral endogenous molecules found in receptors, enzymes, and other structures. [ 8 ]  While enzyme inhibitors suppress enzyme activity, enzyme inducers boost enzyme concentration and activity. The primary determinants of inter-individual variability in drug metabolism in humans are thought to include genetic polymorphism and a variety of other variables, including age, gender, biological conditions, pregnancy, illnesses, stress, nutrition, and drugs. For instance, Reichel et al. [ 26 ]  reported that a 2-arylpropionyl-coenzyme-A epimerase was molecularly cloned and expressed as a crucial enzyme in the inversion metabolism of ibuprofen. Ibuprofen's chiral inversion by enzymes has been documented in humans. [ 27 ]


The liver, gastrointestinal tract (GIT), lungs, kidney, and brain are among the tissues that participate in the chiral inversion of medicines. The liver has been shown to be the most crucial organ in the development of this mechanism. [ 28 ]   Although some studies contend that rat liver homogenates lack the enzymatic mechanisms necessary to invert the R-enantiomers of flurbiprofen, naproxen, suprofen, and ibuprofen, the liver may also be involved in the inversion of R-ibuprofen in rats. [ 15 ]  On the other hand, it was noted that certain medicines underwent chiral inversion without the involvement of the liver (hepatocytes). Although liver did not play a substantial role in the inversion of benoxaprofen, studies using benoxaprofen and ketoprofen show that one of the primary sites of inversion in rats is the GI tract. [ 15 ]


Chiral inversion is a very important part of designing and making drugs. Because this process can change how chiral drugs work in the body and can cause side effects that can be serious or even fatal. Traditionally, chiral inversions have been studied with NMR spectroscopy at different temperatures and chiroptical methods like polarimetry. But strong, complementary methods based on dynamic chromatography (GC, HPLC, SFC, CEC, and MEKC) and electrophoresis have been made and used to figure out how the enantiomeric composition of stereo-labile chiral compounds changes over time. [ 29 ]   Most of the time, liquid chromatographic methods are used to do enantioselective analysis of chiral drugs. When an analyte with one stereogenic center or axis is separated well, the chromatogram will show two peaks. But if the analyte is stereo-labile, the peaks tend to merge. [ 30 ]  How much coalescence there is will depend on how fast chiral inversion and enantioresolution happen. Over time, the peaks will merge into a flat area. Dynamic chromatography shows how the elution profile changes over time. This makes it useful for figuring out how pH, temperature, and solvents affect chiral inversion, which can happen on the stationary phase, in the injector, or in the detector. [ 29 ]


Multidimensional approaches have been used to improve separation and detection. Table below shows a list of common methods and experiments used to figure out chiral inversion. Any of these methods can then be used to determine chiral inversion. Which instrument is used to analyze a chiral compound depends on its physical and chemical properties (i.e., the solubility, vapor pressure, thermal and solvent stability, and detection). [ 29 ]


For example, capillary electrophoresis or liquid chromatography could be used if the analyte can be ionized and has a high vapor pressure, but it is also soluble in polar solvents. [ 31 ]  On the other hand, gas chromatography is the best way to test a substance that is stable at high temperatures but has a low vapor pressure. When compared to gas or liquid chromatography, supercritical fluid chromatography is a better way to measure chiral inversion because it uses mass spectrometers and a green method. [ 32 ]


Enantiomers of a chiral drug often interact in an enantioselective way in a chiral environment. This may be offered by different biotic substances (viz. proteins, nucleic acids, phospholipids and oligosaccharides). They are made up of chiral building blocks that are put together in space in handed conformations. These biological targets function as receptors for the drug enantiomers. So, at the binding sites of these receptors, enantiomers will be seen as different chemical species. The three point attachment model (Easson & Stedman model) [ 33 ]  can be used to see how chiral discrimination works. Figure depicts how the enantiomers of a drug interact with receptors in a way that depends on the drug's shape.
This model was made for chiral drugs with a single stereogenic center. It says that there are three binding sites in the receptor (B', C' and D') that match the drug's pharmacophoric groups (B, C, D). A three-point fit (good fit) is possible for the eutomer at BB', CC' and DD'(Fig. A). Even though the distomer is the wrong enantiomer, it can fit either a one-point interaction (bad fit), or a two-point attachment (CC' and DD') with the same receptor site as shown in (Fig. B).


Eutomer  is the version that works the way you want it to, and  distomer  is the version that doesn't work or works in a way you don't want it to. [ 34 ] [ 35 ]  Most of the time, the mirror-image versions have different binding affinities. In the eutomer, the ligands or moiety around a stereogenic element have more binding energy than in the distomer. When the eutomer goes through chiral inversion, it loses its ability to bind to a biological receptor. Because of these enantiospecific interactions, therapeutic and toxicological properties are enantioselective [ 27 ] [ 6 ]  So, the stereo-stability of chiral drugs may have big effects on the process of making new drugs, especially when it comes to how pharmaceutical, pharmacokinetic, and pharmacodynamic information is read and understood. At every stage of designing, making, and testing a drug for safety, chiral inversion must be taken into account.


In the study of  abstract polytopes , a  chiral polytope  is a polytope that is as symmetric as possible without being mirror-symmetric, formalized in terms of the  action  of the  symmetry group  of the polytope on its  flags .


The more technical definition of a chiral polytope is a polytope that has two orbits of  flags  under its  group of symmetries , with
adjacent flags in different orbits. This implies that it must be  vertex-transitive ,  edge-transitive , and  face-transitive , as each vertex, edge, or face must be represented by flags in both orbits; however, it cannot be mirror-symmetric, as every mirror symmetry of the polytope would exchange some pair of adjacent flags. [ 1 ]


For the purposes of this definition, the symmetry group of a polytope may be defined in either of two different ways: it can refer to the symmetries of a polytope as a geometric object (in which case the polytope is called  geometrically chiral ) or it can refer to the symmetries of the polytope as a combinatorial structure (the automorphisms of an  abstract polytope ). Chirality is meaningful for either type of symmetry but the two definitions classify different polytopes as being chiral or nonchiral. [ 2 ]


Geometrically chiral polytopes are relatively exotic compared to the more ordinary regular polytopes. It is not possible for a geometrically chiral polytope to be convex, [ 3 ]  and many geometrically chiral polytopes of note are  skew .


In three dimensions, it is not possible for a geometrically chiral polytope to have finitely many finite faces. For instance, the snub cube is vertex-transitive, but its flags have more than two orbits, and it is neither edge-transitive nor face-transitive, so it is not symmetric enough to meet the formal definition of chirality. The  quasiregular polyhedra  and their duals, such as the  cuboctahedron  and the  rhombic dodecahedron , provide another interesting type of near-miss: they have two orbits of flags, but are mirror-symmetric, and not every adjacent pair of flags belongs to different orbits. However, despite the nonexistence of finite chiral three-dimensional polyhedra, there exist infinite three-dimensional chiral  skew polyhedra  of types {4,6}, {6,4}, and {6,6}. [ 2 ]


In four dimensions, there are a geometrically chiral finite polytopes. One example is Roli's cube, a  skew polytope  on the skeleton of the  4-cube . [ 4 ] [ 5 ]


In  chemistry , a molecule or ion is called  chiral  ( / ˈ k aɪ r əl / ) if it cannot be superposed on its  mirror image  by any combination of  rotations ,  translations , and some  conformational  changes. This geometric property is called  chirality  ( / k aɪ ˈ r æ l ɪ t i / ). [ 1 ] [ 2 ] [ 3 ] [ 4 ]  The terms are derived from  Ancient Greek   χείρ  ( cheir ) 'hand'; which is the  canonical  example of an object with this property.


A chiral molecule or ion exists in two  stereoisomers  that are mirror images of each other, [ 5 ]  called  enantiomers ; they are often distinguished as either "right-handed" or "left-handed" by their  absolute configuration  or some other criterion. The two enantiomers have the same chemical properties, except when reacting with other chiral compounds. They also have the same  physical  properties, except that they often have opposite  optical activities . A homogeneous mixture of the two enantiomers in equal parts is said to be  racemic , and it usually differs chemically and physically from the pure enantiomers.


Chiral molecules will usually have a stereogenic element from which chirality arises. The most common type of stereogenic element is a stereogenic center, or stereocenter. In the case of organic compounds, stereocenters most frequently take the form of a carbon atom with four distinct (different) groups attached to it in a tetrahedral geometry. Less commonly, other atoms like N, P, S, and Si can also serve as stereocenters, provided they have four distinct substituents (including lone pair electrons) attached to them.


A given stereocenter has two possible configurations (R and S), which give rise to stereoisomers ( diastereomers  and  enantiomers ) in molecules with one or more stereocenter. For a chiral molecule with one or more stereocenter, the enantiomer corresponds to the stereoisomer in which every stereocenter has the opposite configuration. An organic compound with only one stereogenic carbon is always chiral. On the other hand, an organic compound with multiple stereogenic carbons is typically, but not always, chiral. In particular, if the stereocenters are configured in such a way that the molecule can take a conformation having a plane of symmetry or an inversion point, then the molecule is achiral and is known as a  meso  compound .


Molecules with chirality arising from one or more stereocenters are classified as possessing central chirality. There are two other types of stereogenic elements that can give rise to chirality, a stereogenic axis ( axial chirality ) and a stereogenic plane ( planar chirality ). Finally, the inherent curvature of a molecule can also give rise to chirality ( inherent chirality ). These types of chirality are far less common than central chirality.  BINOL  is a typical example of an axially chiral molecule, while  trans -cyclooctene  is a commonly cited example of a planar chiral molecule. Finally,  helicene  possesses helical chirality, which is one type of inherent chirality.


Chirality is an important concept for  stereochemistry  and  biochemistry . Most substances relevant to  biology  are chiral, such as  carbohydrates  ( sugars ,  starch , and  cellulose ), all but one of the  amino acids  that are the building blocks of  proteins , and the  nucleic acids . Naturally occurring  triglycerides  are often chiral, but not always. In living organisms, one typically finds only one of the two enantiomers of a chiral compound. For that reason, organisms that consume a chiral compound usually can metabolize only one of its enantiomers. For the same reason, the two enantiomers of a chiral  pharmaceutical  usually have vastly different  potencies  or effects.


The chirality of a molecule is based on the  molecular symmetry  of its conformations. A conformation of a molecule is chiral if and only if it belongs to the  C n ,  D n ,  T ,  O ,  I   point groups  (the chiral point groups). However, whether the molecule itself is considered to be chiral depends on whether its chiral conformations are persistent isomers that could be isolated as separated enantiomers, at least in principle, or the  enantiomeric  conformers rapidly interconvert at a given temperature and timescale through low-energy conformational changes (rendering the molecule achiral). For example, despite having chiral  gauche  conformers that belong to the  C 2  point group,  butane  is considered achiral at room temperature because rotation about the central C–C bond rapidly interconverts the enantiomers (3.4 kcal/mol barrier). Similarly,  cis -1,2-dichlorocyclohexane consists of  chair conformers  that are nonidentical mirror images, but the two can interconvert via the cyclohexane chair flip (~10 kcal/mol barrier). As another example, amines with three distinct substituents (R 1 R 2 R 3 N:) are also regarded as achiral molecules because their enantiomeric pyramidal conformers rapidly undergo  pyramidal inversion .


However, if the temperature in question is low enough, the process that interconverts the enantiomeric chiral conformations becomes slow compared to a given timescale.  The molecule would then be considered to be chiral at that temperature. The relevant timescale is, to some degree, arbitrarily defined: 1000 seconds is sometimes employed, as this is regarded as the lower limit for the amount of time required for chemical or chromatographic separation of enantiomers in a practical sense.  Molecules that are chiral at room temperature due to restricted rotation about a single bond (barrier to rotation ≥ ca. 23 kcal/mol) are said to exhibit  atropisomerism .


A chiral compound can contain no  improper axis of rotation  ( S n ), which includes planes of symmetry and inversion center. Chiral molecules are always dissymmetric (lacking  S n ) but not always asymmetric (lacking all symmetry elements except the trivial identity). Asymmetric molecules are always chiral. [ 6 ]


The following table shows some examples of chiral and achiral molecules, with the  Schoenflies notation  of the  point group  of the molecule. In the achiral molecules, X and Y (with no subscript) represent achiral groups, whereas X R  and X S  or Y R  and Y S  represent  enantiomers . Note that there is no meaning to the orientation of an  S 2  axis, which is just an inversion. Any orientation will do, so long as it passes through the center of inversion. Also note that higher symmetries of chiral and achiral molecules also exist, and symmetries that do not include those in the table, such as the chiral  C 3  or the achiral  S 4 .


An example of a molecule that does not have a mirror plane or an inversion and yet would be considered achiral is 1,1-difluoro-2,2-dichlorocyclohexane (or 1,1-difluoro-3,3-dichlorocyclohexane). This may exist in many conformers ( conformational isomers ), but none of them has a mirror plane. In order to have a mirror plane, the  cyclohexane  ring would have to be flat, widening the bond angles and giving the conformation a very high energy. This compound would not be considered chiral because the chiral conformers interconvert easily.


An achiral molecule having chiral conformations could theoretically form a mixture of right-handed and left-handed crystals, as often happens with  racemic  mixtures of chiral molecules (see  Chiral resolution#Spontaneous resolution and related specialized techniques ), or as when achiral liquid  silicon dioxide  is cooled to the point of becoming chiral  quartz .


A  stereogenic center  (or  stereocenter ) is an atom such that swapping the positions of two ligands (connected groups) on that atom results in a molecule that is stereoisomeric to the original.  For example, a common case is a  tetrahedral  carbon bonded to four distinct groups  a ,  b ,  c , and  d  (C abcd ), where swapping any two groups (e.g., C bacd ) leads to a stereoisomer of the original, so the central C is a stereocenter.  Many chiral molecules have point chirality, namely a single chiral stereogenic center that coincides with an atom. This stereogenic center usually has four or more bonds to different groups, and may be carbon (as in many biological molecules), phosphorus (as in many  organophosphates ), silicon, or a metal (as in many chiral  coordination compounds ). However, a stereogenic center can also be a trivalent atom whose bonds are not in the same plane, such as  phosphorus  in  P-chiral phosphines  (PRR′R″) and  sulfur  in  S-chiral sulfoxides  (OSRR′), because a lone-pair of electrons is present instead of a fourth bond.


Similarly, a stereogenic axis (or plane) is defined as an axis (or plane) in the molecule such that the swapping of any two ligands attached to the axis (or plane) gives rise to a stereoisomer.  For instance, the  C 2 -symmetric species   1,1′-bi-2-naphthol  (BINOL) and 1,3-dichloro allene  have stereogenic axes and exhibit  axial chirality , while ( E )- cyclooctene  and many  ferrocene  derivatives bearing two or more substituents have stereogenic planes and exhibit  planar chirality .


Chirality can also arise from isotopic differences between atoms, such as in the  deuterated   benzyl alcohol  PhCHDOH; which is chiral and optically active ([ α ] D  = 0.715°), even though the non-deuterated compound PhCH 2 OH is not. [ 7 ]


If two enantiomers easily interconvert, the pure enantiomers may be practically impossible to separate, and only the racemic mixture is observable. This is the case, for example, of most amines with three different substituents (NRR′R″), because of the low  energy barrier  for  nitrogen inversion .


When the optical  rotation  for an enantiomer is too low for practical measurement, the species is said to exhibit  cryptochirality .


Chirality is an intrinsic part of the identity of a molecule, so the  systematic name  includes details of the  absolute configuration  ( R/S ,  D/L , or  other designations ).


Many biologically active molecules are chiral, including the naturally occurring  amino acids  (the building blocks of  proteins ) and  sugars .


The origin of this  homochirality  in  biology  is the subject of much debate. [ 13 ]  Most scientists believe that Earth life's "choice" of chirality was purely random, and that if carbon-based life forms exist elsewhere in the universe, their chemistry could theoretically have opposite chirality. However, there is some suggestion that early amino acids could have formed in comet dust. In this case, circularly polarised radiation (which makes up 17% of stellar radiation) could have caused the selective destruction of one chirality of amino acids, leading to a selection bias which ultimately resulted in all life on Earth being homochiral. [ 14 ] [ 15 ]


Enzymes , which are chiral, often distinguish between the two enantiomers of a chiral substrate. One could imagine an enzyme as having a glove-like cavity that binds a substrate. If this glove is right-handed, then one enantiomer will fit inside and be bound, whereas the other enantiomer will have a poor fit and is unlikely to bind.


L -forms of amino acids tend to be tasteless, whereas  D -forms tend to taste sweet. [ 13 ]   Spearmint  leaves contain the  L -enantiomer of the chemical  carvone  or  R -(−)-carvone and  caraway  seeds contain the  D -enantiomer or  S -(+)-carvone. [ 9 ]  The two smell different to most people because our olfactory  receptors  are chiral.


Chirality is important in context of ordered phases as well, for example the addition of a small amount of an optically active molecule to a nematic phase (a phase that has long range orientational order of molecules) transforms that phase to a chiral nematic phase (or cholesteric phase). Chirality in context of such phases in polymeric fluids has also been studied in this context. [ 16 ]


Chirality is a symmetry property, not a property of any part of the periodic table. Thus many inorganic materials, molecules, and ions are chiral.  Quartz  is an example from the mineral kingdom. Such noncentric materials are of interest for applications in  nonlinear optics .


In the areas of  coordination chemistry  and  organometallic chemistry , chirality is pervasive and of practical importance. A famous example is  tris(bipyridine)ruthenium(II)  complex in which the three bipyridine ligands adopt a chiral propeller-like arrangement. [ 17 ]  The two enantiomers of complexes such as [Ru(2,2′-bipyridine) 3 ] 2+  may be designated as Λ (capital  lambda , the Greek version of "L") for a left-handed twist of the propeller described by the ligands, and Δ (capital  delta , Greek "D") for a right-handed twist (pictured). Also cf.  dextro- and levo- (laevo-) .


Chiral ligands confer chirality to a metal complex, as illustrated by metal- amino acid  complexes. If the metal exhibits catalytic properties, its combination with a chiral ligand is the basis of  asymmetric catalysis . [ 18 ]


The term  optical activity  is derived from the interaction of chiral materials with polarized light. In a solution, the (−)-form, or  levorotatory  form, of an optical isomer  rotates  the plane of a beam of  linearly polarized light   counterclockwise . The (+)-form, or  dextrorotatory  form, of an optical isomer does the opposite. The rotation of light is measured using a  polarimeter  and is expressed as the optical rotation.


Enantiomers can be separated by  chiral resolution . This often involves forming crystals of a salt composed of one of the enantiomers and an acid or base from the so-called  chiral pool  of naturally occurring chiral compounds, such as  malic acid  or the amine  brucine . Some racemic mixtures spontaneously crystallize into right-handed and left-handed crystals that can be separated by hand.  Louis Pasteur  used this method to separate left-handed and right-handed  sodium ammonium tartrate  crystals in 1849. Sometimes it is possible to seed a racemic solution with a right-handed and a left-handed crystal so that each will grow into a large crystal.


Liquid chromatography (HPLC and TLC) may also be used as an analytical method for the direct separation of  enantiomers  and the control of enantiomeric purity, e.g.  active pharmaceutical ingredients  (APIs) which are chiral. [ 19 ] [ 20 ]


The rotation of plane polarized light by chiral substances was first observed by  Jean-Baptiste Biot  in 1812, [ 25 ]  and gained considerable importance in the  sugar industry , analytical chemistry, and pharmaceuticals.  Louis Pasteur  deduced in 1848 that this phenomenon has a molecular basis. [ 26 ] [ 27 ]  The term  chirality  itself was coined by  Lord Kelvin  in 1894. [ 28 ]  Different enantiomers or diastereomers of a compound were formerly called  optical isomers  due to their different optical properties. [ 29 ]  At one time, chirality was thought to be restricted to organic chemistry, but this misconception was overthrown by the resolution of a purely inorganic compound, a cobalt complex called  hexol , by  Alfred Werner  in 1911. [ 30 ]


In the early 1970s, various groups established that the  human olfactory organ  is capable of distinguishing chiral compounds. [ 9 ] [ 31 ] [ 32 ]


In  geometry , a figure is  chiral  (and said to have  chirality ) if it is not identical to its  mirror image , or, more precisely, if it cannot be mapped to its mirror image by  rotations  and  translations  alone. An object that is not chiral is said to be  achiral .


A chiral object and its mirror image are said to be  enantiomorphs . The word  chirality  is derived from the Greek  χείρ  (cheir), the hand, the most familiar chiral object; the word  enantiomorph  stems from the Greek  ἐναντίος  (enantios) 'opposite' +  μορφή  (morphe) 'form'.


Some chiral three-dimensional objects, such as the  helix , can be assigned a right or left  handedness , according to the  right-hand rule .


Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral if you can wear them  inside-out . [ 1 ]


The J-, L-, S- and Z-shaped  tetrominoes  of the popular video game  Tetris  also exhibit chirality, but only in a two-dimensional space. Individually they contain no mirror symmetry in the plane.


A figure is achiral if and only if its  symmetry group  contains at least one  orientation-reversing  isometry. (In Euclidean geometry any  isometry  can be written as

 v
 ↦
 A
 v
 +
 b

 {\displaystyle v\mapsto Av+b}

  with an  orthogonal matrix

 A

 {\displaystyle A}

  and a vector

 b

 {\displaystyle b}

 . The  determinant  of

 A

 {\displaystyle A}

  is either 1 or −1 then. If it is −1 the isometry is orientation-reversing, otherwise it is orientation-preserving.


A general definition of chirality based on group theory exists. [ 2 ]  It does not refer to any orientation concept: an  isometry  is direct if and only if it is a product of squares of isometries, and if not, it is an indirect isometry. The resulting chirality definition works in spacetime. [ 3 ] [ 4 ]


In two dimensions, every figure which possesses an  axis of symmetry  is achiral, and it can be shown that every  bounded  achiral figure must have an axis of symmetry. (An  axis of symmetry  of a figure

 F

 {\displaystyle F}

  is a line

 L

 {\displaystyle L}

 , such that

 F

 {\displaystyle F}

  is invariant under the mapping

 (
 x
 ,
 y
 )
 ↦
 (
 x
 ,
 −
 y
 )

 {\displaystyle (x,y)\mapsto (x,-y)}

 , when

 L

 {\displaystyle L}

  is chosen to be the

 x

 {\displaystyle x}

 -axis of the coordinate system.) For that reason, a  triangle  is achiral if it is  equilateral  or  isosceles , and is chiral if it is  scalene .


Consider the following pattern:


This figure is chiral, as it is not identical to its mirror image:


But if one prolongs the pattern in both directions to infinity, one receives an (unbounded) achiral figure which has no axis of symmetry. Its symmetry group is a  frieze group  generated by a single  glide reflection .


In three dimensions, every figure that possesses a  mirror plane of symmetry   S 1 , an inversion center of symmetry  S 2 , or a higher  improper rotation  (rotoreflection)  S n  axis of symmetry [ 5 ]  is achiral. (A  plane of symmetry  of a figure

 F

 {\displaystyle F}

  is a plane

 P

 {\displaystyle P}

 , such that

 F

 {\displaystyle F}

  is invariant under the mapping

 (
 x
 ,
 y
 ,
 z
 )
 ↦
 (
 x
 ,
 y
 ,
 −
 z
 )

 {\displaystyle (x,y,z)\mapsto (x,y,-z)}

 , when

 P

 {\displaystyle P}

  is chosen to be the

 x

 {\displaystyle x}

 -

 y

 {\displaystyle y}

 -plane of the coordinate system. A  center of symmetry  of a figure

 F

 {\displaystyle F}

  is a point

 C

 {\displaystyle C}

 , such that

 F

 {\displaystyle F}

  is invariant under the mapping

 (
 x
 ,
 y
 ,
 z
 )
 ↦
 (
 −
 x
 ,
 −
 y
 ,
 −
 z
 )

 {\displaystyle (x,y,z)\mapsto (-x,-y,-z)}

 , when

 C

 {\displaystyle C}

  is chosen to be the origin of the coordinate system.) Note, however, that there are achiral figures lacking both plane and center of symmetry. An example is the figure


which is invariant under the orientation reversing isometry

 (
 x
 ,
 y
 ,
 z
 )
 ↦
 (
 −
 y
 ,
 x
 ,
 −
 z
 )

 {\displaystyle (x,y,z)\mapsto (-y,x,-z)}

  and thus achiral, but it has neither plane nor center of symmetry. The figure


also is achiral as the origin is a center of symmetry, but it lacks a plane of symmetry.


Achiral figures can have a  center axis .


A  knot  is called  achiral  if it can be continuously deformed into its mirror image, otherwise it is called a  chiral knot . For example, the  unknot  and the  figure-eight knot  are achiral, whereas the  trefoil knot  is chiral.


A  chiral  phenomenon is one that is not identical to its  mirror image  (see the article on  mathematical chirality ). The  spin  of a  particle  may be used to define a  handedness , or helicity, for that particle, which, in the case of a massless particle, is the same as chirality. A   symmetry transformation  between the two is called  parity  transformation. Invariance under parity transformation by a  Dirac fermion  is called  chiral symmetry .


The helicity of a particle is positive ("right-handed") if the direction of its  spin  is the same as the direction of its motion. It is negative ("left-handed") if the directions of spin and motion are opposite. So a standard  clock , with its spin vector defined by the rotation of its hands, has left-handed helicity if tossed with its face directed forwards.


Mathematically,  helicity  is the sign of the projection of the  spin   vector  onto the  momentum   vector : "left" is negative, "right" is positive.


The  chirality  of a particle is more abstract: It is determined by whether the particle transforms in a right- or left-handed  representation  of the  Poincaré group . [ a ]


For massless particles –  photons ,  gluons , and (hypothetical)  gravitons  – chirality is the same as  helicity ; a given massless particle appears to spin in the same direction along its axis of motion regardless of point of view of the observer.


For massive particles – such as  electrons ,  quarks , and  neutrinos  – chirality and helicity must be distinguished: In the case of these particles, it is possible for an observer to change to a  reference frame  moving faster than the spinning particle, in which case the particle will then appear to move backwards, and its helicity (which may be thought of as "apparent chirality") will be reversed. That is, helicity is a  constant of motion , but it is not  Lorentz invariant . Chirality is Lorentz invariant, but is not a constant of motion: a massive left-handed spinor, when propagating, will evolve into a right handed spinor over time, and vice versa.


A  massless  particle moves with the  speed of light , so no real observer (who must always travel at less than the  speed of light ) can be in any reference frame where the particle appears to reverse its relative direction of spin, meaning that all real observers see the same helicity. Because of this, the direction of spin of massless particles is not affected by a change of inertial reference frame (a  Lorentz boost ) in the direction of motion of the particle, and the sign of the projection (helicity) is fixed for all reference frames: The helicity of massless particles is a  relativistic invariant  (a quantity whose value is the same in all inertial reference frames) which always matches the massless particle's chirality.


The discovery of  neutrino oscillation  implies that  neutrinos have mass , so the  photon  is the only confirmed massless particle;  gluons  are expected to also be massless, although this has not been conclusively tested. [ b ]  Hence, these are the only two particles now known for which helicity could be identical to chirality, and only the  photon  has been confirmed by measurement. All other observed particles have mass and thus may have different helicities in different reference frames. [ c ]


Particle physicists have only observed or inferred left-chiral  fermions  and right-chiral antifermions engaging in the  charged weak interaction . [ 1 ]  In the case of the weak interaction, which can in principle engage with both left- and right-chiral fermions, only two left-handed  fermions  interact. Interactions involving right-handed or opposite-handed fermions have not been shown to occur, implying that the universe has a preference for left-handed chirality. This preferential treatment of one chiral realization over another violates parity, as first noted by  Chien Shiung Wu  in her famous experiment known as the  Wu experiment . This is a striking observation, since parity is a symmetry that holds for all other  fundamental interactions .


Chirality for a  Dirac fermion   ψ  is defined through the  operator  γ 5 , which has  eigenvalues  ±1; the eigenvalue's sign is equal to the particle's chirality: +1 for right-handed, −1 for left-handed. Any Dirac field can thus be projected into its left- or right-handed component by acting with the  projection operators    1 / 2  (1 −  γ 5 )  or   1 / 2  (1 +  γ 5 )  on  ψ .


The coupling of the charged weak interaction to fermions is proportional to the first projection operator, which is responsible for this interaction's  parity symmetry  violation.


A common source of confusion is due to conflating the  γ 5 , chirality operator with the  helicity  operator. Since the helicity of massive particles is frame-dependent, it might seem that the same particle would interact with the weak force according to one frame of reference, but not another. The resolution to this paradox is that  the chirality operator is equivalent to helicity for massless fields only , for which helicity is not frame-dependent. By contrast, for massive particles, chirality is not the same as helicity, or, alternatively,  helicity is not Lorentz invariant, so there is no frame dependence of the weak interaction: a particle that couples to the weak force in one frame does so in every frame.


A theory that is asymmetric with respect to chiralities is called a  chiral theory , while a non-chiral (i.e., parity-symmetric) theory is sometimes called a  vector theory . Many pieces of the  Standard Model  of physics are non-chiral, which is traceable to   anomaly cancellation  in chiral theories.  Quantum chromodynamics  is an example of a vector theory, since both chiralities of all quarks appear in the theory, and couple to gluons in the same way.


The  electroweak theory , developed in the mid 20th century, is an example of a chiral theory. Originally, it assumed that  neutrinos were massless , and assumed the existence of only left-handed  neutrinos  and right-handed antineutrinos. After the observation of  neutrino oscillations , which imply that  neutrinos are massive  (like all other  fermions ) the revised  theories of the electroweak interaction  now include both right- and left-handed  neutrinos . However, it is still a chiral theory, as it does not respect parity symmetry.


The exact nature of the  neutrino  is still unsettled and so the  electroweak theories  that have been proposed are somewhat different, but most accommodate the chirality of  neutrinos  in the same way as was already done for all other  fermions .


Vector  gauge theories  with massless Dirac fermion fields   ψ   exhibit chiral symmetry, i.e., rotating the left-handed and the right-handed components independently makes no difference to the theory. We can write this as the action of rotation on the fields:


or


With  N   flavors , we have unitary rotations instead:  U( N ) L  × U( N ) R .


More generally, we write the right-handed and left-handed states as a projection operator acting on a spinor. The right-handed and left-handed projection operators are


and


Massive fermions do not exhibit chiral symmetry, as the mass term in the  Lagrangian ,    m ψ ψ ,  breaks chiral symmetry explicitly.


Spontaneous chiral symmetry breaking  may also occur in some theories, as it most notably does in  quantum chromodynamics .


The chiral symmetry transformation can be divided into a component that treats the left-handed and the right-handed parts equally, known as  vector symmetry , and a component that actually treats them differently, known as  axial symmetry . [ 2 ]  (cf.  Current algebra .) A scalar field model encoding chiral symmetry and its  breaking  is the  chiral model .


The most common application is expressed as equal treatment of clockwise and counter-clockwise rotations from a fixed frame of reference.


The general principle is often referred to by the name  chiral symmetry . The rule is absolutely valid in the  classical mechanics  of  Newton  and  Einstein , but results from  quantum mechanical  experiments show a difference in the behavior of left-chiral versus right-chiral  subatomic particles .


Consider  quantum chromodynamics  (QCD) with two  massless   quarks   u  and  d  (massive fermions do not exhibit chiral symmetry). The Lagrangian reads


In terms of left-handed and right-handed spinors, it reads


(Here,  i  is the imaginary unit and

 ⧸

 D

 {\displaystyle \displaystyle {\not }D}

  the  Dirac operator .)


Defining


it can be written as


The Lagrangian is unchanged under a rotation of  q L  by any 2×2 unitary matrix  L , and  q R  by any 2×2 unitary matrix  R .


This symmetry of the Lagrangian is called  flavor chiral symmetry , and denoted as  U(2) L  × U(2) R .   It decomposes into


The singlet vector symmetry,  U(1) V ,  acts as


and thus invariant under  U(1)  gauge symmetry. This corresponds to  baryon number  conservation.


The singlet axial group   U(1) A   transforms as the following global transformation


However, it does not correspond to a conserved quantity, because the associated axial current is not conserved. It is explicitly violated by a  quantum anomaly .


The remaining chiral symmetry  SU(2) L  × SU(2) R  turns out to be  spontaneously broken  by a  quark condensate

 ⟨

 q
 ¯

 R

 a

 q

 L

 b

 ⟩
 =
 v

 δ

 a
 b

 {\displaystyle \textstyle \langle {\bar {q}}_{\text{R}}^{a}q_{\text{L}}^{b}\rangle =v\delta ^{ab}}

  formed through nonperturbative action of QCD gluons, into the diagonal vector subgroup  SU(2) V  known as  isospin . The  Goldstone bosons  corresponding to the three broken generators are the three  pions .
As a consequence, the effective theory of QCD bound states like the baryons, must now include mass terms for them, ostensibly disallowed by unbroken chiral symmetry. Thus, this  chiral symmetry breaking  induces the bulk of hadron masses, such as those for the  nucleons  — in effect, the bulk of the mass of all visible matter.


In the real world, because of the nonvanishing and differing masses of the quarks,  SU(2) L  × SU(2) R  is only an approximate symmetry [ 3 ]  to begin with, and therefore the pions are not massless, but have small masses: they are  pseudo-Goldstone bosons . [ 4 ]


For more  "light" quark species,  N   flavors   in general, the corresponding chiral symmetries are  U( N ) L  × U( N ) R′ ,  decomposing into


and exhibiting a very analogous  chiral symmetry breaking  pattern.


Most usually,  N  = 3  is taken, the u, d, and s quarks taken to be light (the  eightfold way ), so then approximately massless for the symmetry to be meaningful to a lowest order, while the other three quarks are sufficiently heavy to barely have a residual chiral symmetry be visible for practical purposes.


In  theoretical physics , the  electroweak  model breaks  parity  maximally. All its  fermions  are chiral  Weyl fermions , which means that the charged  weak gauge bosons W +  and W −  only couple to left-handed quarks and leptons. [ d ]


Some theorists found this objectionable, and so conjectured a  GUT  extension of the  weak force  which has new, high energy  W′ and Z′ bosons , which  do  couple with right handed quarks and leptons:


to


Here,  SU(2) L  (pronounced " SU(2)  left") is  SU(2) W  from above, while  B−L  is the  baryon number  minus the  lepton number . The electric charge formula in this model is given by


where

 T

 3
 L

 {\displaystyle \ T_{\rm {3L}}\ }

  and

 T

 3
 R

 {\displaystyle \ T_{\rm {3R}}\ }

  are the left and right  weak isospin  values of the fields in the theory.


There is also the  chromodynamic   SU(3) C . The idea was to restore parity by introducing a  left-right symmetry . This is a  group extension  of

 Z

 2

 {\displaystyle \mathbb {Z} _{2}}

  (the left-right symmetry) by


to the  semidirect product


This has two  connected components  where

 Z

 2

 {\displaystyle \mathbb {Z} _{2}}

  acts as an  automorphism , which is the composition of an  involutive   outer automorphism  of  SU(3) C  with the interchange of the left and right copies of  SU(2)  with the reversal of  U(1) B−L . It was shown by  Mohapatra  &  Senjanovic  (1975) [ 5 ]  that  left-right symmetry  can be  spontaneously broken  to give a chiral low energy theory, which is the Standard Model of Glashow, Weinberg, and Salam, and also connects the small observed neutrino masses to the breaking of left-right symmetry via the  seesaw mechanism .


In this setting, the chiral  quarks


are unified into an  irreducible representation  ("irrep")


The  leptons  are also unified into an  irreducible representation


The  Higgs bosons  needed to implement the breaking of left-right symmetry down to the Standard Model are


This then provides three  sterile neutrinos  which are perfectly consistent with current [update]   neutrino oscillation  data. Within the seesaw mechanism, the sterile neutrinos become superheavy without affecting physics at low energies.


Because the left–right symmetry is spontaneously broken, left–right models predict  domain walls . This left-right symmetry idea first appeared in the  Pati–Salam model  (1974) [ 6 ]  and Mohapatra–Pati models (1975). [ 7 ]


Chirality timeline  presents a timeline of landmark events that unfold the developments happened in the field of  chirality .


Many molecules come in two forms that are mirror images of each other, just like our hands. This type of molecule is called chiral. In nature, one of these forms is usually more common than the other. In our cells, one of these mirror images of a molecule fits "like a glove," while the other may be harmful. [ 1 ] [ 2 ]


In nature, molecules with chirality include  hormones ,  DNA ,  antibodies , and  enzymes . For example,  (R)-limonene  smells like oranges, while  (S)-limonene  smells like lemons. Both molecules have the same chemical formula, but their spatial orientations are different, which makes a big difference in their biological properties. Chiral molecules in the receptors in our noses can tell the difference between these things. Chirality affects biochemical reactions, and the way a drug works depends on what kind of enantiomer it is. Many drugs are chiral and it is important that the shape of the drug matches the shape of the cell receptor it is meant to affect. Mismatching will make the drug less effective, which could be a matter of life and death, as happened with  thalidomide  in the 1960s. [ 3 ] [ 4 ]


It has long been known that structural factors, particularly chirality and stereochemistry, have a big impact on  pharmacological efficacy  and  pharmacokinetic behavior . Since more than a century ago, pertinent information pertaining to chirality has been accumulating in numerous fields, in particular, physics, chemistry and biology, at an accelerated rate, giving rise to more comprehensive and in-depth reasoning, conceptions, and ideas. [ 5 ] [ 6 ] [ 7 ] [ 8 ]  This page offers a chronology of significant contributions that have been made in the journey of chirality [1809 to 2021].


Harold Scott MacDonald  " Donald "  Coxeter   CC   FRS   FRSC  (9 February 1907 – 31 March 2003) [ 2 ]  was a British-Canadian  geometer  and mathematician. He is regarded as one of the greatest geometers of the 20th century. [ 3 ]


Coxeter was born in England and educated at the  University of Cambridge , with student visits to  Princeton University . He worked for 60 years at the  University of Toronto  in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the  Royal Society of Canada , the  Royal Society , and the  Order of Canada .


He was an author of 12 books, including  The Fifty-Nine Icosahedra  (1938) and  Regular Polytopes  (1947). Many concepts in geometry and  group theory  are named after him, including the  Coxeter graph ,  Coxeter groups ,  Coxeter's loxodromic sequence of tangent circles ,  Coxeter–Dynkin diagrams , and the  Todd–Coxeter algorithm .


Coxeter was born in  Kensington , England, to Harold Samuel Coxeter and Lucy ( née  Gee ). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with  nitrous oxide ), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the  Royal Academy of Arts . A maternal cousin was the architect Sir  Giles Gilbert Scott . [ 4 ] [ 2 ]


In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. [ 5 ]  He felt that  mathematics and music  were intimately related, outlining his ideas in a 1962 article on "Music and Mathematics" in the  Canadian Music Journal . [ 5 ]


He was educated at  King Alfred School, London , and  St George's School, Harpenden , where his best friend was John Flinders Petrie, later a mathematician for whom  Petrie polygons  were named. He was accepted at  King's College, Cambridge , in 1925, but decided to spend a year studying in hopes of gaining admittance to  Trinity College , where the standard of mathematics was higher. [ 2 ]  Coxeter won an entrance scholarship and went to Trinity in 1926 to read mathematics. There he earned his BA (as  Senior Wrangler ) in 1928, and his doctorate in 1931. [ 5 ] [ 6 ]  In 1932 he went to  Princeton University  for a year as a  Rockefeller Fellow , where he worked with  Hermann Weyl ,  Oswald Veblen , and  Solomon Lefschetz . [ 6 ]  Returning to Trinity for a year, he attended  Ludwig Wittgenstein 's seminars on the  philosophy of mathematics . [ 5 ]  In 1934 he spent a further year at Princeton as a Procter Fellow. [ 6 ]


In 1936 Coxeter moved to the University of Toronto. In 1938 he and  P. Du Val , H. T. Flather, and  John Flinders Petrie  published  The Fifty-Nine Icosahedra  with  University of Toronto Press . In 1940 Coxeter edited the eleventh edition of  Mathematical Recreations and Essays , [ 7 ]  originally published by  W. W. Rouse Ball  in 1892. He was elevated to  professor  in 1948. He was elected a Fellow of the  Royal Society of Canada  in 1948 and a  Fellow of the Royal Society  in 1950. He met  M. C. Escher  in 1954 and the two became lifelong friends; his work on geometric figures helped inspire some of Escher's works, particularly the  Circle Limit  series based on  hyperbolic   tessellations . He also inspired some of the innovations of  Buckminster Fuller . [ 6 ]  Coxeter,  M. S. Longuet-Higgins  and  J. C. P. Miller  were the first to publish the full  list of uniform polyhedra  (1954). [ 8 ]


He worked for 60 years at the  University of Toronto  and published twelve books.


Coxeter was a  vegetarian . He attributed his longevity to his vegetarian diet, daily exercise such as fifty press-ups and standing on his head for fifteen minutes each morning, and consuming a nightly cocktail made from  Kahlúa  (a coffee liqueur),  peach schnapps , and  soy milk . [ 4 ]


Since 1978, the  Canadian Mathematical Society  have awarded the  Coxeter–James Prize  in his honor.


He was made a  Fellow of the Royal Society  in 1950 and in 1997 he was awarded their  Sylvester Medal . [ 6 ]  In 1990, he became a Foreign Member of the  American Academy of Arts and Sciences [ 9 ]  and in 1997 was made a  Companion  of the  Order of Canada . [ 10 ]


In 1973 he received the  Jeffery–Williams Prize . [ 6 ]


A  festschrift  in his honour,  The Geometric Vein , was published in 1982. It contained 41 essays on geometry, based on a symposium for Coxeter held at Toronto in 1979. [ 11 ]   A second such volume,  The Coxeter Legacy , was published in 2006 based on a Toronto Coxeter symposium held in 2004. [ 12 ]


The  Edinburgh Handedness Inventory  is a measurement scale used to assess the dominance of a person's right or left hand in everyday activities, sometimes referred to as  laterality . The inventory can be used by an observer assessing the person, or by a person self-reporting hand use. The latter method tends to be less reliable due to a person over-attributing tasks to the dominant hand.


The Edinburgh Handedness Inventory was published in 1971 by Richard Carolus Oldfield [ 1 ]  and has been used in various scientific studies [ 2 ] [ 3 ]  as well as popular literature. [ 4 ]  According to Google Scholar it has been cited tens of thousands of times. Within the very substantial literature on handedness it is used far more than any rival, such as FLANDERS, or the Annett Hand Preference Questionnaire, which is not good at eliciting either-hand responses. [ 5 ]


Nevertheless, profound dissatisfaction with the Inventory has been expressed [ 6 ] [ 7 ]  and statistical analysis of the Inventory has shown that the two-handed items broom and box are poorly correlated with the other eight items, while drawing is too highly correlated with writing to add information. [ 8 ] [ 9 ]  A major revision has been published. [ 10 ]


An  enantiopure drug  is a pharmaceutical that is available in one specific  enantiomeric  form. Most biological molecules (proteins, sugars, etc.) are present in only one of many  chiral  forms, so different enantiomers of a  chiral drug  molecule bind differently (or not at all) to target  receptors . Chirality can be observed when the geometric properties of an object is not superimposable with its mirror image. Two forms of a molecule are formed (both mirror images) from a chiral carbon, these two forms are called enantiomers. One enantiomer of a drug may have a desired beneficial effect while the other may cause serious and undesired side effects, or sometimes even beneficial but entirely different effects. [ 1 ]  The desired enantiomer is known as an eutomer while the undesired enantiomer is known as the distomer. [ 2 ]  When equal amounts of both enantiomers are found in a mixture, the mixture is known as a racemic mixture. If a mixture for a drug does not have a 1:1 ratio of its enantiomers it is a candidate for an enantiopure drug. Advances in industrial chemical processes have made it economical for pharmaceutical manufacturers to take drugs that were originally marketed as a racemic mixture and market the individual enantiomers, either by specifically manufacturing the desired enantiomer or by  resolving  a  racemic mixture . On a case-by-case basis, the  U.S. Food and Drug Administration  (FDA) has allowed single enantiomers of certain drugs to be marketed under a different name than the racemic mixture. [ 3 ]  Also case-by-case, the United States Patent Office has granted patents for single enantiomers of certain drugs. The regulatory review for marketing approval (safety and efficacy) and for patenting (proprietary rights) is independent, and differs country by country.


In 1848, Louis Pasteur became the first scientist to discover chirality and enantiomers while he was working with tartaric acid. During the experiments, he noticed that there were two crystal structures produced but these structures looked to be non-superimposable mirror images of each other; this observation of isomers that were non-superimposable mirror images became known as enantiomers. A couple years later, in 1857, Pasteur then discovered enantioselectivity when he noticed that the two enantiomer structures he had previously discovered metabolized at much different speeds. This suggested that one configuration was preferred over the other in vivo. As organic chemistry knowledge became more advanced, the discovery of enantioselectivity was used in the creation of enantiopure drugs. [ 4 ]


The formation of an enantiopure drug results from the separation of the enantiomers of a chiral drug. This separation was prompted when it was found that each enantiomer of a molecule can have different effects when used in drugs. This is because the body is very chiral selective reacting to each enantiomer differently and therefore producing different pharmaceutical effects. The use of a drug with a single enantiomer makes the drug more effective. Before a drug of a pure enantiomer can be formed, the two enantiomers must first be separated and tested. Three main techniques are used for this separation: capillary gas chromatography, high performance liquid chromatography, and capillary electrophoresis. Other technique such as chiral crystallization, enzyme-based kinetic separation, and enantioselective synthesis are also used. [ 5 ]


The body of living organisms are composed of many enantiopure chiral substances. For example, amino acids that make up the proteins in the body have the same configuration, L-absolute configuration. Because of this specificity, vital processes such as constructing proteins, rely on stereoselectivity to ensure that out of all the potential enantiomers available, the body is utilizing the correct enantiopure compound. [ 6 ]


Selectivity  is a very important part of organic synthesis. In scientific papers regarding synthesis, selectivity is often listed in data tables alongside percent yield and other reaction conditions. While selectivity is deemed important in scientific literature, it has been challenging to effectively implement selectivity in  drug development  and production. A major issue with selectivity in  pharmaceuticals  is that a large percentage of drug syntheses by nature are not selective reactions, racemic mixtures are formed as the products. Separating racemic mixtures into their respective enantiomers takes extra time, money, and energy. One way to separate enantiomers is to chemically convert them into species that can be separated: diastereomers.  Diastereomers , unlike enantiomers, have entirely different physical properties—boiling points, melting points, NMR shifts, solubilities—and they can be separated by conventional means such as  chromatography  or  recrystallization . This is a whole extra step in the synthesis process and not desirable from a manufacturing standpoint. [ 7 ]  As a result, a number of pharmaceuticals are synthesized and marketed as a racemic mixture of enantiomers in cases where the less-effective enantiomer is benign. However, by identifying and specifically purifying the enantiomer which effectively binds to its respective  binding site  in the body, less of the drug would be needed to achieve the desired effect. [ 8 ]  With the improvement of chiral technology, a rich repertoire of enantioselective chromatographic methods have become available for the separation of drug enantiomers on the analytical, [ 9 ]  preparative, [ 10 ]  and industrial scales. [ 11 ] [ 12 ]


According to the FDA, the stereoisomeric composition of a chiral drug should be known, and its effects should be well-characterized from pharmacologic, toxicologic, and clinical standpoints. In order to profile the different stereoisomers of enantiopure drugs, manufacturers are urged to develop quantitative assays for individual enantiomers in  in vivo  samples early in the development stage.


Ideally, the main pharmacologic activities of the isomers should be compared in  in vitro  systems in animals. During instances when toxic findings are present beyond the natural extensions of the pharmacologic effects of the drug, toxicologic evaluation of the individual isomers in question must be completed. [ 13 ]


When drugs are covered under patent protection, only the pharmaceutical company that holds the patent is allowed to manufacture, market, and eventually profit from them. The lifetime of the patent varies between countries and also between drugs; in the United States, most drug patents last about twenty years. [ 14 ]  Once the patent has expired, the drug can be manufactured and sold by other companies - at which point, it is referred to as a  generic drug . Its availability on the market as a generic drug removes the monopoly of the patent holder, thereby encouraging competition and causing a significant drop in drug prices, which ensures that life-saving and important drugs reach the general population at fair prices. However, the company holding the initial patent may get a new patent by forming a new version of the drug that is significantly changed compared to the original compound. [ 15 ]  Patentability of different isomers has been controversial over the past ten years and there have been a number of related legal issues. In making their determinations, courts have looked at factors including: (i) Whether the racemate was known in the prior art. (ii) The difficulty in resolving the enantiomers. (iii) The  stereoselectivity  of the relevant receptor. (iv) Other secondary considerations of non-obviousness such as commercial success, unexpected results, and satisfaction of long-felt needs in the art. The decisions made regarding these issues have varied and there is no clear answer to the legality of patenting stereoisomers. These issues have been resolved on a case-by-case basis. [ 16 ]  With the number of current pharmaceuticals currently being marketed as  racemic mixtures , it is likely that patentability will continue to be debated in the near future.


There are examples of common drugs, like ibuprofen, where the use of chiral switching has caused controversy. Ibuprofen is a racemic mixture where the S-enantiomer is known to play a major role in reducing inflammation as it inhibits COX-2 (cooxygenase 2) compared to the R-enantiomer; the fact that the S-enantiomer is stronger is what led to the chiral switching. But, when the racemic ibuprofen enters the body, a little over half of the R-enantiomers experience chiral inversion and transform into the favored S-enantiomer. This observation has led to a conclusion that the racemic and the S-enantiomer are potentially biologically equivalent. Because of this and the more recent evidence suggesting that the R-enantiomer may actually contribute to COX-2 inhibition, as well, but at a slower rate, there is still debate on whether or not the chiral switching seen in ibuprofen is really advantageous or if it is just to give patent protections to the manufacturers.


The following table lists pharmaceuticals that have been available in both  racemic  and single- enantiomer  form. These single-enantiomer drug switched from the respective racemic drug are referred to as  chiral switch .


The following are cases where the individual enantiomers have markedly different effects:


In  human biology ,  footedness  is the natural preference of one's left or right  foot  for various purposes. It is the foot equivalent of  handedness . While purposes vary, such as applying the greatest force in a certain foot to complete the action of kick as opposed to stomping, footedness is most commonly associated with the preference of a particular foot in the leading position while engaging in foot- or kicking-related sports, such as  association football  and  kickboxing . A person may thus be  left-footed ,  right-footed  or  ambipedal  (able to use both feet equally well).


In  association football , the ball is predominantly struck by the foot. Footedness may refer to the foot a player uses to kick with the greatest force and skill. Most people are right-footed, kicking with the right leg. [ 1 ]  Capable left-footed footballers are rare and therefore quite sought after. [ 2 ]  As rare are "two-footed" players, who are equally capable with both feet. Such players make up only one sixth of players in the top professional leagues in Europe. [ 3 ]  Two-footedness can be learnt, a notable case being England international  Tom Finney , but can only be properly developed in the early years. [ 3 ]  In  Australian Rules Football , several players are equally adept at using both feet to kick the ball, such as  Sam Mitchell  and Charles Bushnell (footballer, retired). [ 4 ]


In  basketball , a sport composed almost solely of right-handed players, it is common for most athletes to have a dominant left leg which they would use when jumping to complete a right-hand layup. [ citation needed ]  Hence, left-handed basketball players tend to use their right leg more as they finish a left handed layup (although both right- and left-handed players are usually able to use both hands when finishing near the basket).


In the  National Football League  (NFL)  placekickers  and  punters  who kick with their left leg are a relative rarity. As of the  2023 NFL season , only four of the league's 32 punters were left-footed.  [ 5 ]  The apparent advantage to punting with the left foot is that, because it is not as common,  return specialists  are not as experienced handling the ball spinning in the opposite direction. [ 6 ]  Left-footed placekickers are similarly uncommon.


In  boardsports  (e.g., surfing, skateboarding and snowboarding), one stands erect on a single, lightweight board that slides along the ground or on water. The need for balance causes one to position the body perpendicular to the direction of motion, with one foot leading the other. As with handedness, when this task is repetitively performed, one tends to naturally choose a particular foot for the leading position.


Boardsport riders are "footed" in one of two stances, generally called "regular" and "goofy". Riders will generally quickly choose a preferred stance that becomes permanently preferred. A "regular" stance indicates the left foot leading on the board with the right foot pushing, while a "goofy" stance leads with the right foot on the board, pushing with the left. [ 7 ]  Professionals seem to be evenly distributed between the stances. Practice can yield a high level of  ambidexterity  between the two stances, such that even seasoned participants of a boardsport have difficulty discerning the natural footedness of an unfamiliar rider in action.


To increase the difficulty, variety, and aesthetic value of tricks, riders can ride "switch stance" (abbreviated to "switch"). For example, a goofy-footed skateboarder normally performs an  ollie  with the right foot forward, but a "switch ollie" would have the rider standing with the left foot at the front of the board. In sports where switch riding is common and expected, like  street skateboarding , riders have the goal of appearing natural at, and performing the same tricks in, both regular and goofy stances. Some sports like  kitesurfing  and  windsurfing  generally require the rider to be able to switch stance depending on the wind or travel direction rather than rider preference. Each time direction is changed, the stance changes. Snowboarders who ride switch may adopt a "duck stance", where the feet are mounted turned out, or pointed away from the mid-line of the body, typically at a roughly 15-degree angle. In this position, the rider will have the leading foot facing forward in either regular or switch stance.


When a rider rolls backwards, this is called "riding  fakie ". A "fakie" trick is performed while riding backwards but taking off on the front foot. Although it is the same foot that jumps in one's traditional stance, it is normally the back foot. A rider can also land in the fakie position.


While there are some parallels between switch stance and fakie, riding switch implies opening the shoulders more to face the direction headed, though not as much as in traditional stance, while fakie stance implies a slightly more backwards facing, closed shoulder posture. "


Nollie  (nose ollie) is when the front foot takes off when one is riding in their normal stance, the same foot that jumps when doing tricks switch. In nollie position, the body and shoulders are facing forward as much as when riding in normal stance. Generally fakie and normal are done off the tail, whereas nollie and switch are done off the nose.


In skateboarding, most tricks that are performed riding backwards — with respect to the rider's preferred stance — are exclusively categorized as "switch" (in a switch stance) or as fakie, with the general rule that tricks off the  tail  are almost always described as fakie, and those off the nose are nollie. For example, a jump using the tail rolling backwards is a "fakie  ollie " (not a "switch nollie"), and a jump off the nose is a "nollie" (not a "fakie nollie").


Mongo foot refers to the use of the rider's front foot for pushing. Normally, a skateboarder will feel more comfortable using their back foot to push, while their front foot remains on the board. In the minority case of mongo-footed skateboarders, the opposite is true. Some skateboarders who do not push mongo in their regular stance may still push mongo when riding in switch stance, rather than push with their weaker back foot. Some well-known skaters who change between mongo and normal when pushing switch include Jacob Vance,  Stevie Williams , and  Eric Koston .


Although its origins remain uncertain, it is widely believed that the term derives from the pejorative use of " mongoloid ". [ citation needed ]


In  BMX , there is a  de facto  relationship between footedness and preferences of grinding position and of mid-air turning direction. The terms "regular" and "goofy" do not indicate a foot preference as in boardsports, but rather whether the rider's footedness has the usual relationship with their grinding and mid-air turning preferences. For example, consider the following classes of riders:


Both classes are of equal size and would be considered "regular". "Goofy" would describe riders whose trick preferences do not match their footedness: a rider who prefers to grind on the opposite side as do most is considered a "goofy grinder"; one who prefers to turn the opposite direction in mid-air as do most is considered a "goofy spinner". Few riders have either goofy trait, but some riders may have both. [ 8 ]


In  human biology ,  handedness  is an individual's preferential use of one  hand , known as the  dominant hand , due to it being stronger, faster or more  dextrous . The other hand, comparatively often the weaker, less dextrous or simply less subjectively preferred, is called the  non-dominant hand . [ 2 ] [ 3 ] [ 4 ]  In a study from 1975 on 7,688 children in US grades 1–6, left handers comprised 9.6% of the sample, with 10.5% of male children and 8.7% of female children being left-handed. [ 5 ] [ 6 ] [ 7 ]  Overall, around 90% of people are right-handed. [ 8 ]  Handedness is often defined by one's writing hand, as it is fairly common for people to prefer to do a particular task with a particular hand. There are people with true  ambidexterity  (equal preference of either hand), but it is rare—most people prefer using one hand for most purposes. However, in some cultures, the use of the left hand can be considered disrespectful.


Most of the current research suggests that left-handedness has an  epigenetic  marker—a combination of genetics, biology and the environment.


Because the vast majority of the population is right-handed, many devices are designed for use by right-handed people, making their use by left-handed people more difficult. [ 9 ]  In many countries, left-handed people are or were required to write with their right hands. However, left-handed people have an advantage in  sports  that involve aiming at a target in an area of an opponent's control, as their opponents are more accustomed to the right-handed majority. As a result, they are over-represented in  baseball ,  tennis ,  fencing , [ 10 ]   cricket ,  boxing , [ 11 ] [ 12 ]  and  mixed martial arts . [ 13 ]


Handedness may be measured behaviourally (performance measures) or through questionnaires (preference measures). The  Edinburgh Handedness Inventory  has been used since 1971 but contains some dated questions and is hard to score. Revisions have been published by Veale [ 19 ]  and by Williams. [ 20 ]  The longer Waterloo Handedness Questionnaire is not widely accessible. More recently, the Flinders Handedness Survey (FLANDERS) has been developed. [ 21 ]


Some non-human primates have a preferred hand for tasks, but they do not display a strong right-biased preference like modern humans, with individuals equally split between right-handed and left-handed preferences. When exactly a right handed preference developed in the human lineage is unknown, though it is known through various means that  Neanderthals  had a right-handedness bias like modern humans. Attempts to determine handedness of early humans by analysing the morphology of lithic artefacts have been found to be unreliable. [ 22 ]


There are several theories of how handedness develops.


Handedness displays a complex inheritance pattern. For example, if both parents of a child are left-handed, there is a 26% chance of that child being left-handed. [ 23 ]  A large study of twins from 25,732 families by Medland et al. (2006) indicates that the  heritability  of handedness is roughly 24%. [ 24 ]


Two theoretical single-gene models have been proposed to explain the patterns of inheritance of handedness, by Marian Annett [ 25 ]  of the  University of Leicester , and by Chris McManus [ 23 ]  of  UCL .


However, growing evidence from  linkage  and  genome-wide association  studies suggests that genetic variance in handedness cannot be explained by a single genetic  locus . [ 26 ] [ 27 ] [ 28 ] [ 29 ] [ 30 ] [ 31 ] [ 32 ] [ 33 ] [ excessive citations ]  From these studies, McManus et al. now conclude that handedness is  polygenic  and estimate that at least 40  loci  contribute to the trait. [ 34 ]


Brandler et al. performed a  genome-wide association study  for a measure of relative hand skill and found that genes involved in the determination of  left-right asymmetry  in the body play a key role in handedness. [ 35 ]  Brandler and  Paracchini  suggest the same mechanisms that determine left-right asymmetry in the body (e.g.  nodal signaling  and  ciliogenesis ) also play a role in the development of  brain asymmetry
(handedness being a reflection of brain asymmetry for motor function). [ 36 ]


In 2019, Wiberg et al. performed a genome-wide association study and found that handedness was significantly associated with four loci, three of them in genes encoding proteins involved in brain development. [ 37 ]


Four studies have indicated that individuals who have had in-utero exposure to  diethylstilbestrol  (a synthetic  estrogen -based medication used between 1940 and 1971) were more likely to be left-handed over the clinical control group. Diethylstilbestrol animal studies "suggest that estrogen affects the developing brain, including the part that governs sexual behavior and right and left dominance". [ 38 ] [ 39 ] [ 40 ] [ 41 ]


Another theory is that  ultrasound  may sometimes affect the brains of unborn children, causing higher rates of left-handedness in children whose mothers receive ultrasound during pregnancy. Research suggests there may be a weak association between  ultrasound screening  (sonography used to check the healthy development of the fetus and mother) and left-handedness. [ 42 ]


Twin studies  indicate that genetic factors explain 25% of the variance in handedness, and environmental factors the remaining 75%. [ 43 ]  While the molecular basis of handedness  epigenetics  is largely unclear, Ocklenburg et al. (2017) found that asymmetric  methylation  of  CpG sites  plays a key role for  gene expression  asymmetries related to handedness. [ 44 ] [ 45 ]


One common handedness theory is the brain hemisphere division of labor. In most people, the left side of the brain controls speaking. The theory suggests it is more efficient for the brain to divide major tasks between the hemispheres—thus most people may use the non-speaking (right) hemisphere for perception and gross motor skills. As speech is a very complex motor control task, the specialised fine motor areas controlling speech are most efficiently used to also control fine motor movement in the dominant hand. As the right hand is controlled by the left hemisphere (and the left hand is controlled by the right hemisphere) most people are, therefore right-handed. The theory depends on left-handed people having a reversed organisation. [ 46 ]  However, the majority of left-handers have been found to have left-hemisphere language dominance—just like right-handers. [ 47 ] [ 48 ]  Only around 30% of left-handers are not left-hemisphere dominant for language. Some of those have reversed brain organisation, where the verbal processing takes place in the right-hemisphere and visuospatial processing is dominant to the left hemisphere. [ 49 ]  Others have more ambiguous bilateral organisation, where both hemispheres do parts of typically lateralised functions. When tasks designed to investigate lateralisation (preference for handedness) are averaged across a group of left-handers, the overall effect is that left-handers show the same pattern of data as right-handers, but with a reduced asymmetry. [ 50 ]  This finding is likely due to the small proportion of left-handers who have atypical brain organisation. [ citation needed ]  The majority of the evidence comes from literature assessing oral language production and comprehension. When it comes to writing, findings from recent studies were inconclusive for a difference in lateralization for writing between left-handers and right-handers.  [ 51 ]


Researchers studied fetuses in utero and determined that handedness in the womb was a very accurate predictor of handedness after birth. [ 52 ]  In a 2013 study, 39% of infants (6 to 14 months) and 97% of toddlers (18 to 24 months) demonstrated a hand preference. [ 53 ]


Infants have been observed to fluctuate heavily when choosing a hand to lead in grasping and object manipulation tasks, especially in one- versus two-handed grasping. Between 36 and 48 months, there is a significant decline in variability between handedness in one-handed grasping; it can be seen earlier in two-handed manipulation. Children of 18–36 months showed more hand preference when performing bi-manipulation tasks than with simple grasping. [ 54 ]


The decrease in handedness variability in children of 36–48 months may be attributable to preschool or kindergarten attendance due to increased single-hand activities such as writing and coloring. [ 54 ]  Scharoun and Bryden noted that right-handed preference increases with age up to the teenage years. [ 6 ]


The modern turn in handedness research has been towards emphasizing degree rather than direction of handedness as a critical variable. [ 55 ]


In his book  Right-Hand, Left-Hand , Chris McManus of  University College London  argues that the proportion of left-handers is increasing, and that an above-average quota of high achievers have been left-handed. He says that left-handers' brains are structured in a way that increases their range of abilities, and that the genes that determine left-handedness also govern development of the brain's language centers. [ 56 ]


Writing in  Scientific American , he states:


Studies in the U.K., U.S. and Australia have revealed that left-handed people differ from right-handers by only one IQ point, which is not noteworthy ... Left-handers' brains are structured differently from right-handers' in ways that can allow them to process language, spatial relations and emotions in more diverse and potentially creative ways. Also, a slightly larger number of left-handers than right-handers are especially gifted in music and math. A study of musicians in professional orchestras found a significantly greater proportion of talented left-handers, even among those who played instruments that seem designed for right-handers, such as violins. Similarly, studies of adolescents who took tests to assess mathematical giftedness found many more left-handers in the population. [ 57 ]


Left-handers are overrepresented among those with lower cognitive skills and mental impairments, with those with  intellectual disability  being roughly twice as likely to be left-handed, as well as generally lower cognitive and non-cognitive abilities amongst left-handed children. [ 58 ]  Left-handers are nevertheless also overrepresented in high IQ societies, such as  Mensa . A 2005 study found that "approximately 20% of the members of Mensa are lefthanded, double the proportion in most general populations". [ 59 ]


Ghayas & Adil (2007) found that left-handers were significantly more likely to perform better on  intelligence tests  than right-handers and that right-handers also took more time to complete the tests. [ 60 ]  In a systematic review and meta-analysis, Ntolka & Papadatou-Pastou (2018) found that right-handers had higher IQ scores, but that difference was negligible (about 1.5 points). [ 61 ]


The prevalence of  difficulties  in left-right discrimination was investigated in a cohort of 2,720 adult members of  Mensa  and  Intertel  by Storfer. [ 62 ]  According to the study, 7.2% of the men and 18.8% of the women evaluated their left-right directional sense as poor or below average; moreover participants who were relatively ambidextrous experienced problems more frequently than did those who were more strongly left- or right-handed. [ 62 ]  The study also revealed an effect of age, with younger participants reporting more problems. [ 62 ]


Nelson, Campbell, and Michel studied infants and whether developing handedness during infancy correlated with language abilities in toddlers. In the article they assessed 38 infants and followed them through to 12 months and then again once they became toddlers from 18 to 24 months. They discovered that when a child developed a consistent use of their right or left hand during infancy (such as using the right hand to put the pacifier back in, or grasping random objects with the left hand), they were more likely to have superior language skills as a toddler. Children who became lateral later than infancy (i.e., when they were toddlers) showed normal development of language and had typical language scores. The researchers used  Bayley scales  of infant and toddler development to assess the subjects. [ 63 ]


In two studies,  Diana Deutsch  found that left-handers, particularly those with mixed-hand preference, performed significantly better than right-handers in musical memory tasks. [ 64 ] [ 65 ]  There are also handedness differences in perception of musical patterns. Left-handers as a group differ from right-handers, and are more heterogeneous than right-handers, in perception of certain stereo illusions, such as the  octave illusion , the  scale illusion , and the  glissando illusion . [ 66 ]


Studies have found a positive correlation between left-handedness and several specific physical and mental disorders and health problems, including:


As handedness is a highly heritable trait associated with various medical conditions, and because many of these conditions could have presented a Darwinian fitness challenge in ancestral populations, this indicates left-handedness may have previously been rarer than it currently is, due to natural selection. However, on average, left-handers have been found to have an advantage in fighting and competitive, interactive sports, which could have increased their reproductive success in ancestral populations. [ 79 ]


In 2006, researchers from  Lafayette College  and  Johns Hopkins University  concluded that there was no statistically significant correlation between handedness and earnings for the general population, but among college-educated people, left-handers earned 10 to 15% more than their right-handed counterparts. [ 80 ]


In a 2014 study published by the  National Bureau of Economic Research ,  Harvard  economist Joshua Goodman finds that left-handed people earn 10 to 12 percent less over the course of their lives than right-handed people. Goodman attributes this disparity to higher rates of emotional and behavioral problems in left-handed people. [ 58 ]


Interactive sports such as table tennis, badminton and cricket have an overrepresentation of left-handedness, while non-interactive sports such as swimming show no overrepresentation. Smaller physical distance between participants increases the overrepresentation. In  fencing , about half the participants are left-handed. [ 81 ]  In tennis, 40% of the  seeded players  are left-handed. [ 82 ]  The term  southpaw  is sometimes used to refer to a left-handed individual, especially in  baseball  and  boxing . [ 83 ]  Some studies suggest that right handed male athletes tend to be statistically taller and heavier than left handed ones. [ 84 ]


Other, sports-specific factors may  increase or decrease  the advantage left-handers usually hold in one-on-one situations:


One advantage is a left-handed catcher's ability to frame a right-handed pitcher's breaking balls. A right-handed catcher catches a right-hander's  breaking ball  across his body, with his glove moving out of the strike zone. A left-handed catcher would be able to catch the pitch moving into the strike zone and create a better target for the umpire.


According to a meta-analysis of 144 studies, totaling 1,787,629 participants, the best estimate for the male to female  odds ratio  was 1.23, indicating that men are 23% more likely to be left-handed. For example, if the incidence of female left-handedness was 10%, then the incidence of male left-handedness would be approximately 12% (10% incidence of left-handedness among women multiplied by an odds ratio of 1:1.23 for women:men results in a 12.3% incidence of left-handedness among men). [ 94 ] [ clarification needed ]


Some studies examining the relationship between handedness and sexual orientation have reported that a disproportionate minority of homosexual people exhibit left-handedness, [ 95 ]  though findings are mixed. [ 96 ] [ 97 ] [ 98 ]


A 2001 study also found that people assigned male at birth whose  gender identity  did not align with their assigned sex, were more than twice as likely to be left-handed than a clinical  control group  (19.5% vs. 8.3%, respectively). [ 99 ]


Paraphilias  (atypical sexual interests) have also been linked to higher rates of left-handedness. A 2008 study analyzing the sexual fantasies of 200 males found "elevated paraphilic interests were correlated with elevated non-right handedness". [ 100 ]  Greater rates of left-handedness have also been documented among  pedophiles . [ 101 ] [ 102 ] [ 103 ] [ 104 ]


A 2014 study attempting to analyze the biological markers of  asexuality  asserts that non-sexual men and women were 2.4 and 2.5 times, respectively, more likely to be left-handed than their heterosexual counterparts. [ 105 ]


A study at  Durham University —which examined mortality data for  cricketers  whose handedness was a matter of public record—found that left-handed men were almost twice as likely to die in war as their right-handed contemporaries. [ 106 ]  The study theorised that this was because weapons and other equipment was designed for the right-handed. "I can sympathise with all those left-handed cricketers who have gone to an early grave trying desperately to shoot straight with a right-handed Lee Enfield .303", wrote a journalist reviewing the study in the cricket press. [ 107 ]  The findings echo those of previous American studies, which found that left-handed US sailors were 34% more likely to have a serious accident than their right-handed counterparts. [ 108 ]


A high level of handedness (whether strongly favoring right or left) is associated with poorer  episodic memory , [ 109 ] [ 110 ]  and with poorer communication between brain hemispheres, [ 111 ]  which may give poorer emotional processing, although  bilateral stimulation  may reduce such effects. [ 112 ] [ 113 ]


A high level of handedness is associated with a smaller  corpus callosum  whereas low handedness with a larger one. [ 114 ]


Left-handedness is associated with better  divergent thinking . [ 115 ]


Many tools and procedures are designed to facilitate use by right-handed people, often without realizing the difficulties incurred by the left-handed. John W. Santrock has written, "For centuries, left-handers have suffered unfair discrimination in a world designed for right-handers." [ 9 ]


Many products for left-handed use are made by specialist producers, although not available from normal suppliers. Items as simple as a knife ground for use with the right hand are less convenient for left-handers. There is a multitude of examples:  kitchen tools  such as knives, corkscrews and scissors,  garden tools , and so on. While not requiring a purpose-designed product, there are more appropriate ways for left-handers to tie shoelaces. [ 116 ]  There are companies that supply products designed specifically for left-handed use. One such is Anything Left-Handed, which in 1967 opened a shop in Soho, London; the shop closed in 2006, but the company continues to supply left-handed products worldwide by mail order. [ 117 ]


Writing from left to right as in many languages, in particular, with the left hand covers and tends to smear (depending upon ink drying) what was just written. Left-handed writers have developed various ways of holding a pen for best results. [ 118 ]  For using a  fountain pen , preferred by many left-handers,  nibs  ground to optimise left-handed use (pushing rather than pulling across the paper) without scratching are available.


McManus noted that, as the  Industrial Revolution  spread across Western Europe and the United States in the 19th century, workers needed to operate complex machines that were designed with right-handers in mind. This would have made left-handers more visible and at the same time appear less capable and more clumsy. Writing left-handed with a dip pen, in particular, was prone to blots and smearing.


Moreover, apart from inconvenience, left-handed people have historically been considered unlucky or even malicious for their difference by the right-handed majority. In many languages, including English, the word for the direction "right" also means "correct" or "proper". Throughout history, being left-handed was considered negative, or evil. [ 119 ]


The Latin adjective  sinister  means  ' left '  as well as  ' unlucky ' , and this double meaning survives in European derivatives of Latin, including the English words  sinister  (meaning both 'evil' and 'on the bearer's left on a coat of arms') and  ambisinister  meaning 'awkward or clumsy with both or either hand'. [ citation needed ]


There are many negative connotations associated with the phrase  left-handed : clumsy, awkward, unlucky, insincere, sinister, malicious, and so on. A "left-handed compliment" is one that has two meanings, one of which is unflattering to the recipient. In French,  gauche  means both  ' left '  and  ' awkward '  or  ' clumsy ' , while  droit(e)  ( cognate  to English  direct  and related to  adroit ) means both  ' right '  and  ' straight ' , as well as  ' law '  and the legal sense of  ' right ' . The name Dexter derives from the Latin for  ' right ' , as does the word  dexterity  meaning manual skill. As these are all very old words, they would tend to support theories indicating that the predominance of right-handedness is an extremely old phenomenon. [ citation needed ]


Black magic  is sometimes referred to as the " left-hand path ". [ 120 ]


Before the development of fountain pens and other writing instruments,  children were taught to write with a  dip pen . While a right-hander could smoothly drag the pen across paper from left to right, a dip pen could not easily be pushed across by the left hand without digging into the paper and making blots and stains. [ 121 ]  Even with more modern pens, writing from left to right, as in many languages, with the left hand covers and can smear what was just written when moving across the line.


Into the 20th and even the 21st century, left-handed children in  Uganda  were beaten by schoolteachers or parents for writing with their left hand, [ 122 ]  or had their left hands tied behind their backs to force them to write with their right hand. [ 123 ]  As a child, the future British king  George VI  (1895–1952) was naturally left-handed. He was forced to write with his right hand, as was common practice at the time. He was not expected to become king, so that was not a factor. [ 124 ]