Pentagonal trapezohedron

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.

One particular pentagonal trapezohedron can be decomposed into two pentagonal pyramids and a regular dodecahedron in the middle. ^[1]

Properties

The pentagonal trapezohedron has ten quadrilaterals, twenty edges, and 12 vertices. Each face is 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. ^[2]

10-sided dice

Ten-sided dice

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. 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. 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.

Spherical tiling

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.

See also

Family of n-gonal trapezohedra

Trapezohedron name	Digonal trapezohedron (Tetrahedron)	Trigonal trapezohedron	Tetragonal trapezohedron	Pentagonal trapezohedron	Hexagonal trapezohedron	...	Apeirogonal trapezohedron
Polyhedron image						...
Spherical tiling image						Plane tiling image
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

References

1. Alsina & Nelsen (2020), p.  75.
2. Montroll, John (2002). A Plethora of Polyhedra in Origami. Dover Publications. p. 105. ISBN   978-0-486-42271-8.
3. U.S. patent 809,293
4. "Greg Peterson about Gen Con 1980: The big news of the year was that someone had 'invented' the ten-sided die". Archived from the original on 2016-08-14.

Sources

• Cundy, H. M.; Rollett, A. P. (1981). Mathematical models (3rd ed.). Tarquin. p. 117.
• Alsina, Claudi; Nelsen, Roger B. (2020). "Section 3.4: Kites". A Cornucopia of Quadrilaterals. The Dolciani Mathematical Expositions. Vol. 55. Providence, Rhode Island: MAA Press and American Mathematical Society. pp. 73–78. ISBN   978-1-4704-5312-1. MR   4286138.

External links

• Generalized formula of uniform polyhedron (trapezohedron) having 2n congruent right kite faces from Academia.edu
• Weisstein, Eric W. "Trapezohedron". MathWorld .
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
  • VRML model Archived 2018-02-24 at the Wayback Machine
  • Conway Notation for Polyhedra Try: "dA5"