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 base face, triangle faces around the base, and kites meeting on top. The kites can also be replaced by rhombi with specific proportions.
Along with the set of pyramids and elongated pyramids, these figures are topologically self-dual.
It can also be seen as an augmented antiprism, with a pyramid augmented onto one of the 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 . This sequence has sets of two triangles instead of kite faces.
Examples + Diminished trapezohedraSymmetryC3vC4vC5vC6vC7v C8v ...Image Rhombicform Net Faces 3 trapezoids3+1 triangles 4 trapezoids4 triangles1 square 5 trapezoids5 triangles1 pentagon 6 trapezoids6 triangles1 hexagon 7 trapezoids7 triangles1 heptagon 8 trapezoids7 triangles1 octagonEdges121620242832Vertices7911131517TrapezohedraSymmetryD3dD4dD5dD6dD7dD8dImage 3 4 5 6Faces 3+3 rhombi(Or squares) 4+4 kites 5+5 kites 6+6 kites 7+7 kitesEdges1216202428Vertices810121416Gyroelongated pyramid or (augmented antiprisms)SymmetryC3vC4vC5vC6vC7vC8vImage34 56Faces9+1 triangles12 triangles1 squares15 triangles1 pentagon 18 triangles1 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. 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 variationsHeptahedron topology #31Diminished cubeChestahedron(Equal area faces)Augmented octahedron(Equilateral faces)3 squares3 45-45-90 triangles1 equilateral triangle face3 kite faces3+1 equilateral triangle faces3 60 degree rhombic faces3+1 equilateral triangle faces
Gallery
See also Elongated pyramid Gyroelongated bipyramid Elongated bipyramid Gyroelongated pyramid
References
Symmetries of Canonical Self-Dual Polyhedra 7F,C3v: 9,C4v: 11,C5v:, 13,C6v:, 15,C7v:.