README.md

---
license: cc-by-sa-4.0
pretty_name: Weight Systems Defining Five-Dimensional Reflexive and Non-Reflexive Polyhedra
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
---

# Dataset of Weight Systems Defining Five-Dimensional Reflexive and Non-Reflexive Polyhedra

This dataset contains all weight systems defining five-dimensional reflexive and
non-reflexive polyhedra, instrumental in the study of Calabi-Yau fourfolds in mathematics
and theoretical physics. The data was compiled by Friedrich Schöller and Harald Skarke 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/).

## Dataset Details

The dataset consists of two subsets: weight systems defining reflexive polyhedra and
weight systems defining non-reflexive polyhedra. 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 polyhedron 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 polyhedron.

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 polyhedron.
- `vertex_count:` Vertex count of the polyhedron.
- `facet_count:` Facet count of the polyhedron.
- `point_count:` Lattice point count of the polyhedron.
- `dual_point_count:` Lattice point count of the dual polyhedron (only for reflexive
  polyhedra).
- `h11:` Hodge number \\( h^{1,1} \\) (only for reflexive polyhedra).
- `h12:` Hodge number \\( h^{1,2} \\) (only for reflexive polyhedra).
- `h13:` Hodge number \\( h^{1,3} \\) (only for reflexive polyhedra).

## 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
in the Git LFS storage directory, as well as 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
```

## Citation

Please cite the following research paper when referencing this dataset:

```
@article{Scholler:2018apc,
    author = {Sch\"oller, 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"
}
```