Diminished trapezohedron

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
Example square form
Faces	n kites n triangles 1 n-gon
Edges	4n
Vertices	2n + 1
Symmetry group	Cnv, [n], (*nn)
Rotation group	Cn, [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.

Examples

Diminished trapezohedra

Symmetry	C3v	C4v	C5v	C6v	C7v	C8v ...
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	D3d	D4d	D5d	D6d	D7d	D8d
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	C3v	C4v	C5v	C6v	C7v	C8v
Image	3	4	5	6
Faces	9+1 triangles	12 triangles 1 squares	15 triangles 1 pentagon	18 triangles 1 hexagon

Special cases

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.

Diminished trigonal trapezohedron variations

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

See also

• Elongated pyramid
• Gyroelongated bipyramid
• Elongated bipyramid
• Gyroelongated pyramid
• Tetrahedrally diminished dodecahedron

References

1. "Chestahedron Geometry". The Art & Science of Frank Chester. Retrieved 2020-01-22.
2. Donke, Hans-Joakim (March 2011). "Transforming a Tetrahedron into a Chestahedron". Wolfram Alpha. Archived from the original on 2014-10-07. Retrieved 22 January 2020.

• Symmetries of Canonical Self-Dual Polyhedra 7F,C_3v: 9,C_4v: 11,C_5v:, 13,C_6v:, 15,C_7v:.