Diminished trapezohedron

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Set of diminished trapezohedra
Diminished square trapezohedron.png
Example square form
Faces n kites
n triangles
1 n-gon
Edges 4n
Vertices 2n+1
Symmetry group Cnv, [n], (*nn)
Rotational group Cn, [n]+, (nn)
Dual polyhedron self-dual
Properties convex

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 triangles faces around the base, and n kite faces meeting on top.

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.

Diminished trapezohedra
Symmetry C3v C4v C5v C6v C7v...
Image Diminished trigonal trapezohedron.png Diminished square trapezohedron.png Diminished pentagonal trapezohedron.png Diminished hexagonal trapezohedron.png Diminished heptagonal trapezohedron.png
Faces 3 kites
3+1 triangles
4 kites
4 triangles
1 square
5 kites
5 triangles
1 pentagon
6 kites
6 triangles
1 hexagon
7 kites
7 triangles
1 heptagon
Edges 12 16 20 24 28
Vertices 7 9 11 13 15
Symmetry D3d D4d D5d D6d D7d...
Image Trigonal trapezohedron.png
Tetragonal trapezohedron.png
Pentagonal trapezohedron.png
Hexagonal trapezohedron.png
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 C3v C4v C5v C6v C7v...
Image Augmented octahedron.png
Gyroelongated square pyramid.png
Gyroelongated pentagonal pyramid.png
Augmented hexagonal antiprism flat.png
Faces 9+1 triangles 12 triangles
1 squares
15 triangles
1 pentagon
18 triangles
1 hexagon

Special cases[edit]

There are three special case geometries of the diminished trigonal trapezohedron. The simplest is a diminished cube. The Chestahedron is named after artist Frank Chester, 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.

Diminished trigonal trapezohedron variations
Heptahedron topology #31
Diminished cube
(Equal area faces)
Augmented octahedron
(Equilateral faces)
Heptahedron31.GIF Chesahedron transparent.png Augmented octahedron.png
Diminished cube net.png Chestahedron net.png Augmented octahedgon net.png
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

See also[edit]