# Pentagonal trapezohedron

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Pentagonal trapezohedron

The pentagonal trapezohedron or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedron 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.

As a die

Some role-playing games and miniature wargames use ten-sided dice, typically pentagonal trapezohedra, to get random decimal numbers, such as percentages. 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 be 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.

Regular icosahedra with two sides each marked 0 to 9 are also referred to as ten-sided dice, and sometimes preferred due to their more regular shape (see platonic solid) that improves rolling.

References

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

*
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] www.georgehart.com: The Encyclopedia of Polyhedra
** VRML [http://www.georgehart.com/virtual-polyhedra/vrml/pentagonal_trapezohedron.wrl model]
** [http://www.georgehart.com/virtual-polyhedra/conway_notation.html Conway Notation for Polyhedra] Try: "dA5"
* [http://www.wizards.com/default.asp?x=dnd/dnd/20040517a Dungeons & Dragons Dice Roller]

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