size_categories:1K<n<10K license:other Calabi-Yau Toric Geometry String Theory Polyhedra Geometry Physics region:us
---
name: Reflexive Polyhedra of Calabi-Yau Threefolds
date: '2023-09-28'
license: other
domain: Physics
tags:
- Calabi-Yau
- Toric Geometry
- String Theory
- Polyhedra
- Geometry
- Physics
pretty_name: Calabi-Yau 3-Folds
size_categories:
- 1K<n<10K
---

# Dataset Card for Reflexive Polyhedra of Calabi-Yau Threefolds

## Table of Contents

- [Dataset Description](#dataset-description)
  - [General Information](#general-information)
  - [Dataset Origin](#dataset-origin)
- [Dataset Characteristics](#dataset-characteristics)
  - [Schema](#schema)
  - [Data Fields](#data-fields)
  - [Data Format](#data-format)
- [Usage](#usage)
  - [Getting Started](#getting-started)
  - [Machine Learning Applications](#machine-learning-applications)
  - [Citations](#citations)

## Dataset Description

### General Information

Calabi-Yau threefolds are a special class of smooth, compact three-dimensional spaces that have become fundamental objects in both mathematics and theoretical physics. In the context of string theory, they serve as the internal dimensions over which strings compactify, leading to a four-dimensional effective theory. The geometry of these threefolds is closely related to many physical phenomena, including the number of particle generations, gauge symmetries, and the cosmological constant. This dataset encompasses all 4319 reflexive polyhedra in 3 dimensions, offering a comprehensive view of potential Calabi-Yau geometries. The reflexive polyhedra serve as dual representations of these threefolds and are crucial in understanding their topological and geometric properties.

### Dataset Origin

The dataset is derived from the original work documented in [hep-th/9805190](https://arxiv.org/abs/hep-th/9805190). While the original dataset was in a PALP-compatible structure, this version has been converted to a nested JSON format to better accommodate machine learning applications. The PALP-compatible version of the dataset can be accessed at [CYk3](http://hep.itp.tuwien.ac.at/~kreuzer/CY/CYk3.html).

## Dataset Characteristics

### Schema

The dataset is presented in a nested JSON format, with each entry containing both metadata and a matrix representing the vertices of the corresponding polyhedron.

### Data Fields

- `M1`, `M2`: These are point and vertex numbers in the M lattice, which is a mathematical lattice in the context of toric geometry. This lattice serves as the foundational geometric space from which the polyhedron is constructed.
  
- `N1`, `N2`: Similar to the M lattice, these are point and vertex numbers in the N lattice, which is dual to the M lattice. The N lattice provides a different but equally important geometric perspective for understanding the polyhedron.
  
- `Pic`: The Picard number is a topological invariant that measures the rank of the Néron-Severi group of a manifold. In the context of Calabi-Yau threefolds, it helps to determine the number of independent 2-cycles, which has physical implications like the number of U(1) gauge fields in the effective theory.

- `Cor`: The correction term is a specific mathematical entity that adjusts the Picard number to account for certain topological peculiarities. The Picard numbers of a polyhedron and its dual add up to \( 20 + \text{correction} \).

- `Matrix`: A 3xN matrix containing the coordinates of the vertices of the polyhedron. Each row represents a dimension in 3D space, and each column represents a vertex.

### Data Format

Each entry in the dataset is structured as follows:

```json
{
  "M1": ...,
  "M2": ...,
  "N1": ...,
  "N2": ...,
  "Pic": ...,
  "Cor": ...,
  "Matrix": [
    [...],
    [...],
    [...]
  ]
}
```

## Usage

### Getting Started

To access the dataset using the Hugging Face `datasets` library, the following Python code can be used:

```python
from datasets import load_dataset

dataset = load_dataset("calabi-yau-threefolds")
```

### Machine Learning Applications

This dataset provides rich opportunities for various machine learning tasks:

- Geometric deep learning for topological invariant prediction.
- Unsupervised learning techniques for polyhedra clustering.
- Graph neural networks to model vertex connections.

### Citations

### Citations

For dataset usage, please cite the original paper using the following BibTeX entry:

```bibtex
@misc{kreuzer1998classification,
      title={Classification of Reflexive Polyhedra in Three Dimensions}, 
      author={M. Kreuzer and H. Skarke},
      year={1998},
      eprint={hep-th/9805190},
      archivePrefix={arXiv},
      primaryClass={hep-th}
}
```
