# Reeve tetrahedra

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Reeve tetrahedra | |
---|---|

Rank | 3 |

Dimension | 3 |

Type | Integral |

Elements | |

Faces | 1+1+1+1 |

Edges | 1+1+1+1+1+1 |

Vertices | 1+1+1+1 |

Measures (edge length 1) | |

Volume | r/6 |

Central density | 1 |

Abstract & topological properties | |

Flag count | 24 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | I×I×I, order 1 |

Flag orbits | 24 |

Convex | Yes |

Nature | Tame |

The **Reeve tetrahedra** are a family of integral polyhedra. Their existence demonstrates that there is no direct generalization of Pick's theorem to 3 dimensions.

## Vertex coordinates[edit | edit source]

A Reeve tetrahedron is a tetrahedron with vertex coordinates:

- ,
- ,
- ,
- ,

where r is a positive integer.

## Pick's theorem[edit | edit source]

Although no Reeve tetrahedron contains no interior integral points, and no boundary points other than its 4 vertices, its volume *r*/6 is unbounded. Thus there is no way to directly generalize Pick's theorem so that it correctly gives the volume of the Reeve tetrahedra.

However generalizations of Pick's theorem involving Ehrhart polynomials do correctly calculate the volumes of the Reeve tetrahedra.

## External links[edit | edit source]

- Wikipedia contributors. "Reeve tetrahedra".

## Bibliography[edit | edit source]

- Kiradjiev, Kristian (December 2018), "Connecting the Dots with Pick's Theorem" (PDF),
*Mathematics Today*, Institute of mathematics and its applications, retrieved January 6, 2023