Pentagonal trapezohedron

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Pentagonal trapezohedron
Pentagonal trapezohedron
Type trapezohedra
Coxeter diagram CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 10.pngCDel node.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 5.pngCDel node fh.png
Faces 10 kites
Edges 20
Vertices 12
Face configuration V5.3.3.3
Symmetry group D5d, [2+,10], (2*5), order 20
Rotation group D5, [2,5]+, (225), order 10
Dual polyhedron pentagonal antiprism
Properties convex, face-transitive

The pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms. It has ten faces (i.e., it is a decahedron) which are congruent kites.

It can be decomposed into two pentagonal pyramids and a pentagonal antiprism in the middle. It can also be decomposed into two pentagonal pyramids and a Dodecahedron in the middle.

10-sided dice[edit]

Ten ten-sided dice

The pentagonal trapezohedron was patented for use as a gaming die in 1906.[1] It is convenient for role-playing games that use percentile-based skills; however, it is not strictly necessary since the outcome of a twenty-sided die can be divided ten ways and is sometimes preferred due to its regular shape. When a ten-sided die is rolled for a random digit, the outcome can be interpreted as 0-9 or (more commonly in role-playing) 1-10. Similarly two rolls provide 100 equally probable outcomes ranging 0-99 or 1-100.

Subsequent patents on ten-sided dice have made minor refinements to the basic design by rounding or truncating the edges. This enables the die to tumble so that the outcome is less predictable. One such refinement became notorious at the 1980 Gen Con[2] when the patent was incorrectly thought to cover ten-sided dice in general.

A fairly consistent arrangement of the faces on ten-digit dice has been observed. If one holds such a die between one's fingers at two of the vertices such that the even numbers are on top, and reads the numbers from left to right in a zigzag pattern, the sequence obtained is 0, 7, 4, 1, 6, 9, 2, 5, 8, 3, and back to 0. The even and odd digits are divided among the two opposing "caps" of the die, and each pair of opposite faces adds to nine.

See also[edit]

Family of trapezohedra V.n.3.3.3
Polyhedron Digonal trapezohedron.png TrigonalTrapezohedron.svg Tetragonal trapezohedron.png Pentagonal trapezohedron.svg Hexagonal trapezohedron.png Octagonal trapezohedron.png Decagonal trapezohedron.png
Tiling Spherical digonal antiprism.png Spherical trigonal trapezohedron.png Spherical tetragonal trapezohedron.png Spherical pentagonal trapezohedron.png Spherical hexagonal trapezohedron.png Spherical heptagonal trapezohedron.png Spherical octagonal trapezohedron.png Spherical decagonal trapezohedron.png Spherical dodecagonal trapezohedron.png E2 tiling 22i-8 dual.png
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


  1. ^ U.S. Patent 809,293
  2. ^ Greg Peterson about Gen Con 1980: The big news of the year was that someone had 'invented' the ten-sided die.


  • Cundy H.M and Rollett, A.P. Mathematical models, 2nd Edn. Oxford University Press (1961), (3rd edition 1989) p. 117

External links[edit]