# Convex triamond polyhedron

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A **convex triamond polyhedron** is a strictly convex polyhedron with all regular faces of edge length 1 except for at least one "triamond", defined as a trapezoid with edge lengths 2-1-1-1 (a blend of three coplanar equilateral triangles).^{[1]} In such polyhedra, triamonds always come in pairs joined at their length-2 edges. There are 40 known convex triamond polyhedra, found by Roger Kaufman and others. All are achiral. It is not known if there are others.

## List of convex triamond polyhedra[edit | edit source]

The following are all of the convex triamond polyhedra that are currently known.^{[1]} The categories are those given by Roger Kaufman.

Name | Image | Element counts | Discoverer | Category | |||
---|---|---|---|---|---|---|---|

Vertices | Edges | Faces | Triamond faces | ||||

Triamond stretched octahedron | 8 | 14 | 8 | 4 | Roger Kaufman | Stretched platonics | |

Triamond stretched icosahedron | 16 | 36 | 22 | 4 | Alex Doskey | Stretched platonics | |

Triamond triangular bifrustum | 9 | 15 | 8 | 6 | Roger Kaufman | Bifrusta | |

Triamond square bifrustum | 12 | 20 | 10 | 8 | Roger Kaufman | Bifrusta | |

Triamond pentagonal bifrustum | 15 | 25 | 12 | 10 | Robert Austin | Bifrusta | |

Triamond triangular bicupola | 18 | 30 | 14 | 6 | Roger Kaufman | Bicupolae | |

Triamond square bicupola | 24 | 40 | 18 | 8 | Roger Kaufman | Bicupolae | |

Triamond pentagonal bicupola | 30 | 50 | 22 | 10 | Roger Kaufman | Bicupolae | |

Triamond bilateral cupola + stretched J1 wedge | 10 | 17 | 9 | 4 | Roger Kaufman | Triamond bilateral cupola augmentations | |

Triamond bilateral cupola + stretched J1 wedge + meta J1 | 11 | 21 | 12 | 4 | Jim McNeil | Triamond bilateral cupola augmentations | |

Triamond bilateral cupola + stretched J1 wedge + para J1 | 11 | 21 | 12 | 4 | Roger Kaufman | Triamond bilateral cupola augmentations | |

Triamond bilateral cupola + stretched J1 wedge + 2 J1 | 12 | 25 | 15 | 4 | Roger Kaufman | Triamond bilateral cupola augmentations | |

Triamond pentagonal birotunda | 50 | 80 | 32 | 10 | Jim McNeil | Rotunda augmentations | |

Triamond pentagonal rotunda 1 + triamond pentagonal cupola | 40 | 65 | 27 | 10 | Jim McNeil | Rotunda augmentations | |

Triamond pentagonal rotunda 2 + triamond pentagonal cupola | 50 | 80 | 32 | 10 | Jim McNeil | Rotunda augmentations | |

Triamond gyroelongated square bicupola | 32 | 56 | 26 | 16 | Roger Kaufman | Augmentations of triamond antiprisms, order 1 | |

Triamond gyroelongated pentagonal bicupola | 40 | 70 | 32 | 20 | Roger Kaufman | Augmentations of triamond antiprisms, order 1 | |

Triamond antiprism 2,2 | 6 | 9 | 5 | 2 | Roger Kaufman | Augmentations of the triamond antiprism 2,2 | |

Triamond antiprism 2,2 + triamond antiprism 2,2 | 8 | 14 | 8 | 2 | Roger Kaufman | Augmentations of the triamond antiprism 2,2 | |

Triamond antiprism 4,2 + triamond square frustrum | 16 | 28 | 14 | 8 | Roger Kaufman | Augmentations of the triamond antiprism 4,2 | |

Triamond antiprism 4,2 + triamond square frustrum + ortho J4 | 20 | 40 | 22 | 8 | Roger Kaufman | Augmentations of the triamond antiprism 4,2 | |

Triamond antiprism 4,2 + triamond square frustrum + gyro J4 | 20 | 40 | 22 | 8 | Roger Kaufman | Augmentations of the triamond antiprism 4,2 | |

Triamond antiprism 5,2 + triamond pentagonal cupola | 20 | 35 | 17 | 10 | Roger Kaufman | Augmentations of the triamond antiprism 5,2 | |

Triamond antiprism 5,2 + triamond pentagonal cupola + gyro J5 | 25 | 50 | 27 | 10 | Roger Kaufman | Augmentations of the triamond antiprism 5,2 | |

Triamond antiprism 5,2 + triamond pentagonal cupola + ortho J5 | 25 | 50 | 27 | 10 | Roger Kaufman | Augmentations of the triamond antiprism 5,2 | |

Triamond antiprism 5,2 + triamond pentagonal cupola + J6 | 30 | 60 | 32 | 10 | Roger Kaufman | Augmentations of the triamond antiprism 5,2 | |

Triamond stretched J13 | 10 | 20 | 12 | 2 | Roger Kaufman | Stretched Johnson solids | |

Triamond stretched J17 | 14 | 30 | 18 | 4 | Jim McNeil | Stretched Johnson solids | |

Triamond stretched J84 | 12 | 25 | 15 | 2 | Jim McNeil | Stretched Johnson solids | |

Triamond stretched J86 | 14 | 29 | 17 | 2 | Jim McNeil | Stretched Johnson solids | |

Triamond stretched J88 | 18 | 39 | 23 | 2 | Jim McNeil | Stretched Johnson solids | |

Triamond stretched J51 | 12 | 24 | 14 | 6 | Jim McNeil | Stretched Johnson solids | |

Triamond J91 + 2 J2 | 15 | 31 | 18 | 2 | Roger Kaufman | J91 augmentations | |

Triamond J91 + 4 J2 - J2 | 15 | 31 | 18 | 4 | Roger Kaufman | J91 augmentations | |

Triamond J91 + 4 J2 - 2 J2 (cis) | 14 | 26 | 14 | 4 | Roger Kaufman | J91 augmentations | |

Triamond J91 + 4 J2 - 2 J2 (trans) | 14 | 26 | 14 | 4 | Roger Kaufman | J91 augmentations | |

Triamond J3 section | 7 | 11 | 6 | 2 | Roger Kaufman | Sectioned Johnson solids | |

Triamond J92 section + triamond J92 section |
16 | 32 | 18 | 2 | Roger Kaufman | Sectioned Johnson solids | |

Triamond J92 section + stretched J2 wedge | 13 | 26 | 15 | 2 | Roger Kaufman | Sectioned Johnson solids | |

Misc1 | 10 | 22 | 14 | 2 | Roger Kaufman | Miscellaneous |

## References[edit | edit source]

- ↑
^{1.0}^{1.1}Kaufman, Roger. "Convex Triamond Regular Polyhedra."