Duplicate from calabi-yau-data/ws-5d

Co-authored-by: Friedrich Schöller <schoeller@users.noreply.huggingface.co>

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+auto/

README.md

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+---
+license: cc-by-sa-4.0
+pretty_name: Weight Systems Defining Five-Dimensional IP Lattice Polytopes
+configs:
+  - config_name: non-reflexive
+    data_files:
+      - split: full
+        path: non-reflexive/*.parquet
+  - config_name: reflexive
+    data_files:
+      - split: full
+        path: reflexive/*.parquet
+size_categories:
+  - 100B<n<1T
+tags:
+  - physics
+  - math
+---
+
+# Weight Systems Defining Five-Dimensional IP Lattice Polytopes
+
+This dataset contains all weight systems defining five-dimensional reflexive and
+non-reflexive IP lattice polytopes, instrumental in the study of Calabi-Yau fourfolds in
+mathematics and theoretical physics. The data was compiled by Harald Skarke and Friedrich
+Schöller in [arXiv:1808.02422](https://arxiv.org/abs/1808.02422). More information is
+available at the [Calabi-Yau data website](http://hep.itp.tuwien.ac.at/~kreuzer/CY/). The
+dataset can be explored using the [search
+frontend](http://rgc.itp.tuwien.ac.at/fourfolds/). See below for a short mathematical
+exposition on the construction of polytopes.
+
+Please cite the paper when referencing this dataset:
+
+```
+@article{Scholler:2018apc,
+    author = {Schöller, Friedrich and Skarke, Harald},
+    title = "{All Weight Systems for Calabi-Yau Fourfolds from Reflexive Polyhedra}",
+    eprint = "1808.02422",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-th",
+    doi = "10.1007/s00220-019-03331-9",
+    journal = "Commun. Math. Phys.",
+    volume = "372",
+    number = "2",
+    pages = "657--678",
+    year = "2019"
+}
+```
+
+## Dataset Details
+
+The dataset consists of two subsets: weight systems defining reflexive (and therefore IP)
+polytopes and weight systems defining non-reflexive IP polytopes. Each subset is split
+into 4000 files in Parquet format. Rows within each file are sorted lexicographically by
+weights. There are 185,269,499,015 weight systems defining reflexive polytopes and
+137,114,261,915 defining non-reflexive polytopes, making a total of 322,383,760,930 IP
+weight systems.
+
+Each row in the dataset represents a polytope and contains the six weights defining it,
+along with the vertex count, facet count, and lattice point count. The reflexive dataset
+also includes the Hodge numbers \\( h^{1,1} \\), \\( h^{1,2} \\), and \\( h^{1,3} \\) of
+the corresponding Calabi-Yau manifold, and the lattice point count of the dual polytope.
+
+For any Calabi-Yau fourfold, the Euler characteristic \\( \chi \\) and the Hodge number
+\\( h^{2,2} \\) can be derived as follows:
+
+$$ \chi = 48 + 6 (h^{1,1} − h^{1,2} + h^{1,3}) $$
+
+$$ h^{2,2} = 44 + 4 h^{1,1} − 2 h^{1,2} + 4 h^{1,3} $$
+
+This dataset is licensed under the
+[CC BY-SA 4.0 license](http://creativecommons.org/licenses/by-sa/4.0/).
+
+### Data Fields
+
+- `weight0` to `weight5`: Weights of the weight system defining the polytope.
+- `vertex_count`: Vertex count of the polytope.
+- `facet_count`: Facet count of the polytope.
+- `point_count`: Lattice point count of the polytope.
+- `dual_point_count`: Lattice point count of the dual polytope (only for reflexive
+  polytopes).
+- `h11`: Hodge number \\( h^{1,1} \\) (only for reflexive polytopes).
+- `h12`: Hodge number \\( h^{1,2} \\) (only for reflexive polytopes).
+- `h13`: Hodge number \\( h^{1,3} \\) (only for reflexive polytopes).
+
+## Usage
+
+The dataset can be used without downloading it entirely, thanks to the streaming
+capability of the `datasets` library. The following Python code snippet demonstrates how
+to stream the dataset and print the first five rows:
+
+```python
+from datasets import load_dataset
+
+dataset = load_dataset("calabi-yau-data/ws-5d", name="reflexive", split="full", streaming=True)
+
+for row in dataset.take(5):
+    print(row)
+```
+
+When cloning the Git repository with Git Large File Storage (LFS), data files are stored
+both in the Git LFS storage directory and in the working tree. To avoid occupying double
+the disk space, use a filesystem that supports copy-on-write, and run the following
+commands to clone the repository:
+
+```bash
+# Initialize Git LFS
+git lfs install
+
+# Clone the repository without downloading LFS files immediately
+GIT_LFS_SKIP_SMUDGE=1 git clone https://huggingface.co/datasets/calabi-yau-data/ws-5d
+
+# Change to the repository directory
+cd ws-5d
+
+# Test deduplication (optional)
+git lfs dedup --test
+
+# Download the LFS files
+git lfs fetch
+
+# Create working tree files as clones of the files in the Git LFS storage directory using
+# copy-on-write functionality
+git lfs dedup
+```
+
+## Construction of Polytopes
+
+This is an introduction to the mathematics involved in the construction of polytopes
+relevant to this dataset. For more details and precise definitions, consult the paper
+[arXiv:1808.02422](https://arxiv.org/abs/1808.02422) and references therein.
+
+### Polytopes
+
+A polytope is the convex hull of a finite set of points in \\(n\\)-dimensional Euclidean
+space, \\(\mathbb{R}^n\\). This means it is the smallest convex shape that contains all
+these points. The minimal collection of points that define a particular polytope are its
+vertices. Familiar examples of polytopes include triangles and rectangles in two
+dimensions, and cubes and octahedra in three dimensions.
+
+A polytope is considered an *IP polytope* (interior point polytope) if the origin of
+\\(\mathbb{R}^n\\) is in the interior of the polytope, not on its boundary or outside it.
+
+For any IP polytope \\(\nabla\\), its dual polytope \\(\nabla^*\\) is defined as the set
+of points \\(\mathbf{y}\\) satisfying
+
+$$
+  \mathbf{x} \cdot \mathbf{y}
+  \ge -1 \quad \text{for all } \mathbf{x} \in \nabla \;.
+$$
+
+This relationship is symmetric: the dual of the dual of an IP polytope is the polytope
+itself, i.e., \\( \nabla^{**} = \nabla \\).
+
+### Weight Systems
+
+Weight systems provide a means to describe simple polytopes known as *simplices*. A weight
+system is a tuple of real numbers. The construction process is outlined as follows:
+
+Consider an \\(n\\)-dimensional simplex in \\(\mathbb{R}^n\\), i.e., a polytope in
+\\(\mathbb{R}^n\\) with vertex count \\(n + 1\\) and \\(n\\) of its edges extending in
+linearly independent directions. It is possible to position \\(n\\) of its vertices at
+arbitrary (linearly independent) locations through a linear transformation. The placement
+of the remaining vertex is then determined. Its position is the defining property of the
+simplex. To specify the position independently of the applied linear transformation, one
+can use the following equation. If \\(\mathbf{v}_0, \mathbf{v}_1, \dots, \mathbf{v}_n\\)
+are the vertices of the simplex, this relation fixes one vertex in terms of the other
+\\(n\\):
+
+$$ \sum_{i=0}^n q_i \mathbf{v}_i = 0 \;, $$
+
+where \\(q_i\\) is the tuple of real numbers, the weight system.
+
+It is important to note that scaling all weights in a weight system by a common factor
+results in an equivalent weight system that defines the same simplex.
+
+The condition that a simplex is an IP simplex is equivalent to the condition that all
+weights in its weight system are bigger than zero.
+
+For this dataset, the focus is on a specific construction of lattice polytopes described
+in subsequent sections.
+
+### Lattice Polytopes
+
+A lattice polytope is a polytope with vertices at the points of a regular grid, or
+lattice. Using linear transformations, any lattice polytope can be transformed so that its
+vertices have integer coordinates, hence they are also referred to as integral
+polytopes.
+
+The dual of a lattice with points \\(L\\) is the lattice consisting of all points
+\\(\mathbf{y}\\) that satisfy
+
+$$
+  \mathbf{x} \cdot \mathbf{y} \in \mathbb{Z} \quad \text{for all } \mathbf{x} \in L \;.
+$$
+
+*Reflexive polytopes* are a specific type of lattice polytope characterized by having a
+dual that is also a lattice polytope, with vertices situated on the dual lattice. These
+polytopes play a central role in the context of this dataset.
+
+The weights of a lattice polytope are always rational. This characteristic enables the
+rescaling of a weight system so that its weights become integers without any common
+divisor. This rescaling has been performed in this dataset.
+
+The construction of the lattice polytopes from this dataset works as follows: We start
+with the simplex \\(\nabla\\), arising from a weight system as previously described. Then,
+we define the polytope \\(\Delta\\) as the convex hull of the intersection of
+\\(\nabla^*\\) with the points of the dual lattice. In the context of this dataset, the
+polytope \\(\Delta\\) is referred to as ‘the polytope’. Correspondingly,
+\\(\Delta^{\!*}\\) is referred to as ‘the dual polytope’. The lattice of \\(\nabla\\) and
+\\(\Delta^{\!*}\\) is taken to be the coarsest lattice possible, such that \\(\nabla\\) is
+a lattice polytope, i.e., the lattice generated by the vertices of \\(\nabla\\). This
+construction is exemplified in the following sections.
+
+A weight system is considered an IP weight system if the corresponding \\(\Delta\\) is an
+IP polytope; that is, the origin is within its interior. Since only IP polytopes have
+corresponding dual polytopes, this condition is essential for the polytope \\(\Delta\\) to
+be classified as reflexive.
+
+### Two Dimensions
+
+In two dimensions, all IP weight systems define reflexive polytopes and every vertex of
+\\(\nabla^*\\) lies on the dual lattice, making \\(\Delta\\) and \\(\nabla^*\\) identical.
+There are exactly three IP weight systems that define two-dimensional polytopes
+(polygons). Each polytope is reflexive and has three vertices and three facets (edges):
+
+| weight system | number of points of \\(\nabla\\) | number of points of \\(\nabla^*\\) |
+|--------------:|---------------------------------:|-----------------------------------:|
+|     (1, 1, 1) |                                4 |                                 10 |
+|     (1, 1, 2) |                                5 |                                  9 |
+|     (1, 2, 3) |                                7 |                                  7 |
+
+The polytopes and their duals are depicted below. Lattice points are indicated by dots.
+<img src="pictures/ws-2d.png" style="display: block; margin-left: auto; margin-right: auto; width:520px;">
+
+### General Dimension
+
+In higher dimensions, the situation becomes more complex. Not all IP polytopes are
+reflexive, and generally, \\(\Delta \neq \nabla^*\\).
+
+This example shows the construction of the three-dimensional polytope \\(\Delta\\) with
+weight system (2, 3, 4, 5) and its dual \\(\Delta^{\!*}\\). Lattice points lying on the
+polytopes are indicated by dots. \\(\Delta\\) has 7 vertices and 13 lattice points,
+\\(\Delta^{\!*}\\) also has 7 vertices, but 16 lattice points.
+<img src="pictures/ws-3d-2-3-4-5.png" style="display: block; margin-left: auto; margin-right: auto; width:450px;">
+
+The counts of reflexive single-weight-system polytopes by dimension \\(n\\) are:
+
+| \\(n\\) | reflexive single-weight-system polytopes |
+|--------:|-----------------------------------------:|
+|       2 |                                        3 |
+|       3 |                                       95 |
+|       4 |                                  184,026 |
+|       5 |           (this dataset) 185,269,499,015 |
+
+One should note that distinct weight systems may well lead to the same polytope (we have
+not checked how often this occurs). In particular it seems that polytopes with a small
+number of lattice points are generated many times.

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