Pentagonal trapezohedron
Type	trapezohedra Trapezohedron The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kites . The faces are symmetrically staggered.The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry...
Faces	10 kites Kite (geometry) In Euclidean geometry a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are next to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite each other rather than next to each other...
Edges	20
Vertices	12
Face configuration Face configuration In geometry, a face configuration is notational description of a face-transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face....	5,3,3,3
Symmetry group	D5d, [2+,10], (2*5)
Dual polyhedron Dual polyhedron In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...	pentagonal antiprism Pentagonal antiprism In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for a total of 12 faces...
Properties	convex, face-transitive

The pentagonal trapezohedron

Trapezohedron

The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kites . The faces are symmetrically staggered.The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry...

or deltohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprism

Antiprism

In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...

s. It has ten faces (i.e., it is a decahedron

Decahedron

In geometry, a decahedron is a polyhedron with 10 faces. There are 32300 topologically distinct decahedra and none are regular, so this name is ambiguous.With regular faces:* Octagonal prism * Square antiprism...

) which are congruent

Congruence (geometry)

In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...

 kites

Kite (geometry)

In Euclidean geometry a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are next to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite each other rather than next to each other...

.

It can be decomposed into two pentagonal pyramid

Pentagonal pyramid

In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point . Like any pyramid, it is self-dual....

s and a pentagonal antiprism

Pentagonal antiprism

In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for a total of 12 faces...

 in the middle. It can also be decomposed into two pentagonal pyramids and a Dodecahedron in the middle.

Die

During the 1970s

1970s

File:1970s decade montage.png|From left, clockwise: US President Richard Nixon doing the V for Victory sign after his resignation from office after the Watergate scandal in 1974; Refugees aboard a US naval boat after the Fall of Saigon, leading to the end of the Vietnam War in 1975; The 1973 oil...

 the first role-playing game

Role-playing game

A role-playing game is a game in which players assume the roles of characters in a fictional setting. Players take responsibility for acting out these roles within a narrative, either through literal acting, or through a process of structured decision-making or character development...

s and miniature wargames that used percentile-based skills (RuneQuest

RuneQuest

RuneQuest is a fantasy role-playing game first published in 1978 by Chaosium, created by Steve Perrin and set in Greg Stafford's mythical world of Glorantha. RuneQuest was notable for its original gaming system and for its verisimilitude in adhering to an original fantasy world...

was the first of them) first used a couple of twenty-sided dice to get random decimal numbers, such as percentages. To do that the players used to roll two twenty-sided dice and then divided the results as if the dice were ten-sided. On the other hand, 20-sided dice labeled with single digits -twice in a row- are still nowadays commercialized, as they were back in the seventies. Although it is claimed the typically pentagonal-trapezohedrical ten-sided dice were invented in the late seventies (their invention was announced during the 1980 Gen Con

Gen Con

Gen Con is one of the largest and most prominent annual gaming conventions in North America. It features traditional pen-and-paper, board, and card-style games, including role-playing games, miniatures wargames, board games, live action role-playing games, collectible card games, non-collectible...

) the US patent office has a patent showing a similar die dated 1906.

To improve rolling, the edges are usually rounded or sub-faces introduced by truncation

Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...

. Each face has two long edges and two short edges. The five odd-numbered faces meet at the common vertex

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

 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

Zigzag

A zigzag is a pattern made up of small corners at variable angles, though constant within the zigzag, tracing a path between two parallel lines; it can be described as both jagged and fairly regular....

 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 sometimes read as 0 and sometimes as 100, depending on the game system being used, just as "0" a single die is sometimes treated as a roll of 10.

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

Platonic solid

In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

) that improves rolling.

Sources

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

External links

• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
  • VRML
    VRML
    VRML is a standard file format for representing 3-dimensional interactive vector graphics, designed particularly with the World Wide Web in mind...
    
    model
  • Conway Notation for Polyhedra Try: "dA5"
• Dungeons & Dragons Dice Roller