# Characteristics of Regular 4-Dimensional Polytopes

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## Theorem

The $4$-dimensional regular polytopes have the following characteristics:

Name | No. of cells | No. of faces | No. of edges | No. of vertices | Dual |
---|---|---|---|---|---|

Pentatope | $5$ | $10$ | $10$ | $5$ | Self-dual |

Tesseract | $8$ | $24$ | $32$ | $16$ | $16$-cell |

$16$-cell | $16$ | $32$ | $24$ | $8$ | Tesseract |

$24$-cell | $24$ | $96$ | $96$ | $24$ | Self-dual |

$120$-cell | $120$ | $720$ | $1200$ | $600$ | $600$-cell |

$600$-cell | $600$ | $1200$ | $720$ | $120$ | $120$-cell |

## Proof

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $6$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $6$