# Pentagonal trapezohedron

Pentagonal trapezohedron
Type trapezohedra
Coxeter diagram
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.

## Die

Ten ten-sided dice

RuneQuest, published in 1978, was one of the first of many role-playing games and miniature wargames that use percentile-based skills. Before trapezohedral dice became available, players used twenty-sided dice to get random digits, either by dividing by two or by discarding the tens digit. Regular icosahedra with two sides each marked 0 to 9 are also referred to as ten-sided dice, and sometimes preferred because their more regular shape (see platonic solid) improves rolling.

A new pentagonal-trapezohedral ten-sided die was announced during the 1980 Gen Con,[1] although the US patent office has a patent[2] showing a similar die dated 1906.

To improve rolling, the edges are usually rounded or sub-faces introduced by truncation. Each face has two long edges and two short edges. The five odd-numbered faces meet at the common vertex of their long edges. The five even-numbered faces meet at the common vertex of their long edges.

There seems to be a standard configuration for the numbers on 10-sided dice. 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. (In this position, odd numbers appear upside-down.) Opposite sides on such a die total nine.

These dice are often sold in pairs for use as a percentile die. One die will signify tens from 00 through 90, and the other units from 0 to 9. The use of such markings is to generate random numbers from 00 to 99, also known as percentile. "00" is read as 100 for most games but is read as 0 for some games, just as "0" on a single die is most often treated as a roll of 10.

Family of trapezohedra
2 3 4 5 6 7 8 9 10 11 12 ...

As spherical polyhedra

## Sources

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