Pentagonal trapezohedron
A ten-faced polyhedron used as a gaming die.
The pentagonal trapezohedron is the third shape in an infinite family of face-transitive polyhedra called trapezohedra. It has ten faces, each a congruent kite, making it a decahedron. Its dual is the pentagonal antiprism. One version of this polyhedron can be split into two pentagonal pyramids with a regular dodecahedron between them.
This shape has ten quadrilateral faces, twenty edges, and twelve vertices. In one form, each kite face has three interior angles of 108° and one of 36°. Truncating the top and bottom vertices of this polyhedron produces a regular dodecahedron. The pentagonal trapezohedron shares its symmetry group with its dual, the pentagonal antiprism, which has five-fold antiprismatic symmetry of order 20.
Ten-sided dice in the shape of a pentagonal trapezohedron are used in role-playing games with percentile-based skills, though a twenty-sided die labeled 0–9 twice can also serve this purpose. When all vertices lie on a sphere, two internal angles of each kite become 90°. Later patents made minor refinements, such as rounding or truncating edges, to improve tumbling and randomness. Ten-sided dice are typically numbered 0–9, allowing two dice to be rolled for a percentile result: one die gives the tens digit, the other the units, with double-zero read as 100. Some sets include one die numbered 0–9 and another numbered 00–90 to avoid confusion. Dice may also be marked 1–10.
The pentagonal trapezohedron also exists as a spherical tiling, with two vertices at the poles and alternating vertices equally spaced above and below the equator.
- type
- Polyhedron
- family
- Trapezohedra
- faces
- 10 congruent kites
- edges
- 20
- vertices
- 12
- symmetry_group
- D5d of order 20
- dual
- Pentagonal antiprism
Quick Facts
- Faces
- 10 kites
- Edges
- 20
- Vertices
- 12
- Face Config
- V5.3.3.3
- Symmetry
- D_{5\mathrm{d
Facts from the source article.
Lore & Background
The pentagonal trapezohedron has ten quadrilaterals, twenty edges, and twelve vertices. Each face is a kite, with one version having its three interior angles at 108° and one interior angle at 36°. Truncating both top and bottom vertices of this polyhedron achieves the regular dodecahedron. The dual of a pentagonal trapezohedron is the pentagonal antiprism, an antiprism with pentagonal bases connected by alternating band of triangles. It has the same three-dimensional symmetry group as the pentagonal antiprism, the five-fold antiprismatic symmetry D5d of order 20.
Reader's Guide
The pentagonal trapezohedron was patented for use as a gaming die in 1906. These dice are used for role-playing games that use percentile-based skills. This version of the pentagonal trapezohedron has an additional constraint, namely that all vertices are on the surface of a sphere. One consequence of this is that the kite faces have two internal angles equal to 90°. Subsequent patents on ten-sided dice have made minor refinements to the basic design by rounding or truncating the edges, enabling 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. 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. Some ten-sided 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. 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.
Did You Know?
- The pentagonal trapezohedron can be decomposed into two pentagonal pyramids and a regular dodecahedron in the middle.
- When all vertices are on the surface of a sphere, the kite faces have two internal angles equal to 90°.
Frequently Asked Questions
What is a pentagonal trapezohedron?
It is a ten-faced, face-transitive polyhedron and the third member of the infinite trapezohedron family. Every one of its ten faces is a congruent kite-shaped quadrilateral, which also earns it the label of decahedron.
How many faces, edges, and vertices does a pentagonal trapezohedron have?
The solid has exactly ten kite faces, twenty edges, and twelve vertices. In one standard form, each kite face carries three interior angles of 108° and a single angle of 36°.
What polyhedron is the dual of the pentagonal trapezohedron?
Its dual is the pentagonal antiprism, so swapping the roles of faces and vertices between the two solids maps one onto the other. This dual pairing is a standard relationship in the catalog of well-known polyhedra.
How does a pentagonal trapezohedron relate to a regular dodecahedron?
Truncating the top and bottom vertices of the trapezohedron yields a regular dodecahedron. One particular version of the shape can also be split into two pentagonal pyramids with a regular dodecahedron sitting between them.
What symmetry group does the pentagonal trapezohedron belong to?
It carries the D5d symmetry group, whose order is twenty, meaning twenty distinct rotations and reflections leave the shape looking unchanged. This moderate level of symmetry is what lets it work as a fair ten-sided gaming die.
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