---
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
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.

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("cy-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/cy-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 a polytope is the polytope itself,
i.e., \\( \nabla^{**} = \nabla \\).

### Weight Systems

Weight systems provide a means to describe simple polytopes known as *simplexes*. More
broadly, *combined weight systems*, which are collections of individual weight systems,
can describe any polytope. A combined weight system is a matrix consisting of real
numbers. The construction process is outlined as follows:

Consider a polytope in \\(\mathbb{R}^n\\) with vertex count \\(k\\), where \\(k\\) is
bigger than \\(n\\). It is possible to position \\(n\\) of these vertices at arbitrary
(linearly independent) locations through a linear transformation. The placement of the
remaining \\(k - n\\) vertices is then determined. Their positions are the defining
properties of a polytope. To specify these positions independently of the applied linear
transformation, one can use the following system of equations. If \\(\mathbf{v}_0,
\mathbf{v}_1, \dots \mathbf{v}_{k-1}\\) are the vertices of the polytope, these relations
fix \\(k - n\\) vertices in terms of the other \\(n\\):

$$
  \sum_{i=0}^{k-1} q_i^{(j)} \mathbf{v}_i
  = 0 \quad \text{for } 0 \le j \le k - n - 1 \;,
$$

where \\(q_i^{(j)}\\) is the matrix of real numbers, the combined weight system. In cases
where \\(k = n + 1\\), \\(j\\) is limited to the value zero, reducing the matrix to a
single weight system \\(q_i\\). In this scenario, the polytope is a simplex, and the
equation simplifies to:

$$ \sum_{i=0}^n q_i \mathbf{v}_i = 0 \;. $$

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 polytope.

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.

Typically, the dual of a lattice polytope defined by a weight system is not a lattice
polytope. However, our interest lies in a different construction than simply considering
polytopes defined by (combined) weight systems, as described above. In this construction,
they are just the starting point. We start with the polytope \\(\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
\\(\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 |

We will now construct these polytopes from their corresponding weight system. Fixing the
first two vertices of the polytopes

$$
  \mathbf{v}_0 = (1, 0) \quad \text{and} \quad
  \mathbf{v}_1 = (0, 1) \;,
$$

one can obtain the position of the third vertex by solving the weight system equation from
before:

$$
  \mathbf{v}_2 = - \frac{q_0 \mathbf{v}_0 + q_1 \mathbf{v}_1}{q_2} \;.
$$

The resulting 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;">

One may notice that a simpler description could be obtained by fixing \\(\mathbf{v}_2 =
(1, 0)\\) instead of \\(\mathbf{v}_0\\), which would avoid fractional vertex coordinates.
However, this approach would not illustrate the general case in higher dimensions, where
this is not possible since there is not always a weight equal to 1.

### 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 |