Pentagonal trapezohedron

Pentagonal trapezohedron

Type	trapezohedra
Coxeter diagram
Faces	10 kites
Edges	20
Vertices	12
Face configuration	V5.3.3.3
Symmetry group	D_5d, [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

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.