Diminished trapezohedron

Summary

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
Facesn kites
n triangles
1 n-gon
Edges4n
Vertices2n + 1
Symmetry groupCnv, [n], (*nn)
Rotation groupCn, [n]+, (nn)
Dual polyhedronself-dual
Propertiesconvex

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

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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

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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

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References

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  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,C3v:[1] 9,C4v:[2] 11,C5v:[3], 13,C6v:[4], 15,C7v:[5].