| Pentagonal trapezohedron | |
|---|---|
 | |
| Type | Trapezohedra |
| Faces | 10 kites |
| Edges | 20 |
| Vertices | 12 |
| Symmetry group | D5d |
| Dual polyhedron | pentagonal antiprism |
| Properties | convex, face-transitive |
In geometry, a pentagonal trapezohedron is the third in the infinite family of trapezohedra, face-transitive polyhedra. Its dual polyhedron is the pentagonal antiprism. As a decahedron it has ten faces which are congruent kites.
It can be decomposed into two pentagonal pyramids and a regular dodecahedron in the middle.[1]
The pentagonal trapezohedron has ten quadrilaterals, twenty edges, and 12 vertices. Each face is kite, where its three interior angles are 108° and one interior angle is 36°. Truncating both top and bottom apices of the polyhedron achieves the regular dodecahedron.[2]
The pentagonal trapezohedron was patented for use as a gaming die (i.e. "game apparatus") in 1906.[3] These dice are used for role-playing games that use percentile-based skills; however, a twenty-sided die can be labeled with the numbers 0-9 twice to use for percentages instead.
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 when the patent was incorrectly thought to cover ten-sided dice in general.[4]
Ten-sided dice are commonly numbered from 0 to 9, as this allows two to be rolled in order to easily obtain a percentile result. Where one die represents the 'tens', the other represents 'units' therefore a result of 7 on the former and 0 on the latter would be combined to produce 70. A result of double-zero is commonly interpreted as 100. Some ten-sided dice (often called 'Percentile Dice') are sold in sets of two where one is numbered from 0 to 9 and the other from 00 to 90 in increments of 10, thus making it impossible to misinterpret which one is the tens and which the units die. Ten-sided dice may also be marked 1 to 10 when a random number in this range is desirable.
The pentagonal trapezohedron also exists as a spherical tiling, with 2 vertices on the poles, and alternating vertices equally spaced above and below the equator.
| Trapezohedron name | Digonal trapezohedron (Tetrahedron) |
Trigonal trapezohedron | Tetragonal trapezohedron | Pentagonal trapezohedron | Hexagonal trapezohedron | ... | Apeirogonal trapezohedron |
|---|---|---|---|---|---|---|---|
| Polyhedron image | 
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| Spherical tiling image | 
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Plane tiling image | 
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| Face configuration | V2.3.3.3 | V3.3.3.3 | V4.3.3.3 | V5.3.3.3 | V6.3.3.3 | ... | V∞.3.3.3 |
