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In geometry, a **diminished trapezohedron** is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular n-gonal base face, n triangle faces around the base, and n kites meeting on top. The kites can also be replaced by rhombi with specific proportions.

Diminished trapezohedron | |
---|---|

Faces | n kites n triangles 1 n-gon |

Edges | 4n |

Vertices | 2n + 1 |

Symmetry group | C_{nv}, [n], (*nn) |

Rotation group | C_{n}, [n]^{+}, (nn) |

Dual polyhedron | self-dual |

Properties | convex |

Along with the set of pyramids and elongated pyramids, these figures are topologically self-dual.

It can also be seen as an augmented n-gonal antiprism, with a n-gonal pyramid augmented onto one of the n-gonal faces, and whose height is adjusted so the upper antiprism triangle faces can be made coparallel to the pyramid faces and merged into kite-shaped faces.

They're also related to the gyroelongated pyramids, as augmented antiprisms and which are Johnson solids for *n* = 4, 5. This sequence has sets of two triangles instead of kite faces.

Symmetry | C_{3v} |
C_{4v} |
C_{5v} |
C_{6v} |
C_{7v} |
C_{8v} ...
| |
---|---|---|---|---|---|---|---|

Image | |||||||

Rhombic form |
|||||||

Net | |||||||

Faces | 3 trapezoids 3+1 triangles |
4 trapezoids 4 triangles 1 square |
5 trapezoids 5 triangles 1 pentagon |
6 trapezoids 6 triangles 1 hexagon |
7 trapezoids 7 triangles 1 heptagon |
8 trapezoids 7 triangles 1 octagon | |

Edges | 12 | 16 | 20 | 24 | 28 | 32 | |

Vertices | 7 | 9 | 11 | 13 | 15 | 17 | |

Trapezohedra | |||||||

Symmetry | D_{3d} |
D_{4d} |
D_{5d} |
D_{6d} |
D_{7d} |
D_{8d} | |

Image | 3 |
4 |
5 |
6 |
|||

Faces | 3+3 rhombi (Or squares) |
4+4 kites | 5+5 kites | 6+6 kites | 7+7 kites | ||

Edges | 12 | 16 | 20 | 24 | 28 | ||

Vertices | 8 | 10 | 12 | 14 | 16 | ||

Gyroelongated pyramid or (augmented antiprisms) | |||||||

Symmetry | C_{3v} |
C_{4v} |
C_{5v} |
C_{6v} |
C_{7v} |
C_{8v} | |

Image | 3 |
4 |
5 |
6 | |||

Faces | 9+1 triangles | 12 triangles 1 squares |
15 triangles 1 pentagon |
18 triangles 1 hexagon |

There are three special case geometries of the *diminished trigonal trapezohedron*. The simplest is a **diminished cube**. The **Chestahedron**, named after artist Frank Chester, is constructed with equilateral triangles around the base, and the geometry adjusted so the kite faces have the same area as the equilateral triangles.^{[1]}^{[2]} The last can be seen by augmenting a regular tetrahedron and an octahedron, leaving 10 equilateral triangle faces, and then merging 3 sets of coparallel equilateral triangular faces into 3 (60 degree) rhombic faces. It can also be seen as a tetrahedron with 3 of 4 of its vertices rectified. The three rhombic faces fold out flat to form half of a hexagram.

Heptahedron topology #31 Diminished cube |
Chestahedron (Equal area faces) |
Augmented octahedron (Equilateral faces) |
---|---|---|

3 squares 3 45-45-90 triangles 1 equilateral triangle face |
3 kite faces 3+1 equilateral triangle faces |
3 60 degree rhombic faces 3+1 equilateral triangle faces |

- Symmetries of Canonical Self-Dual Polyhedra 7F,C
_{3v}:[1] 9,C_{4v}:[2] 11,C_{5v}:[3], 13,C_{6v}:[4], 15,C_{7v}:[5].