Hexagonal trapezohedron

The hexagonal trapezohedron or deltohedron is the fourth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has twelve faces which are congruent kites.

Hexagonal trapezohedron
Typetrapezohedra
ConwaydA6
Coxeter diagram
Faces12 kites
Edges24
Vertices14
Face configurationV6.3.3.3
Symmetry groupD6d, [2+,12], (2*6), order 24
Rotation groupD6, [2,6]+, (66), order 12
Dual polyhedronhexagonal antiprism
Propertiesconvex, face-transitive

Variations

One degree of freedom within D6 symmetry changes the kites into congruent quadrilaterals with 3 edges lengths. In the limit, one edge of each quadrilateral goes to zero length, and these become bipyramids.

Crystal arrangements of atoms can repeat in space with hexagonal trapezohedral cells.[1]

If the kites surrounding the two peaks are of different shapes, it can only have C6v symmetry, order 12. These can be called unequal trapezohedra. The dual is an unequal antiprism, with the top and bottom polygons of different radii. If it twisted and unequal its symmetry is reduced to cyclic symmetry, C6 symmetry, order 6.

Example variations
Type Twisted trapezohedra (isohedral) Unequal trapezohedra Unequal and twisted
Symmetry D6, (662), [6,2]+, order 12 C6v, (*66), [6], order 12 C6, (66), [6]+, order 6
Image
(n=6)
Net
Family of trapezohedra Vn.3.3.3
Polyhedron
Tiling
Config. V2.3.3.3 V3.3.3.3 V4.3.3.3 V5.3.3.3 V6.3.3.3 V7.3.3.3 V8.3.3.3 V10.3.3.3 V12.3.3.3 ... V∞.3.3.3
gollark: Then set the x^4/x^3/x^2/x^1 coefficients and constant terms on each side to be equal and work out a/b/c/d.
gollark: Set it equal to `(x-1)(ax^3+bx^2+cx+d)` (the thing you know it's divisible by times the generalized cubic thingy), and expand that out/simplify.
gollark: It would be annoying and inconsistent if it was 0. It's 1.
gollark: It's 1, or the nice neat recursive factorial calculation algorithms would stop working.
gollark: It's not an example, this seems to be true in all cases.
  1. 3 2 and Hexagonal-trapezohedric Class, 6 2 2
  • Weisstein, Eric W. "Trapezohedron". MathWorld.
  • Virtual Reality Polyhedra The Encyclopedia of Polyhedra
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