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Hendecahedron

In geometry, Hendecahedron refers to a polyhedron with eleven faces.[1] The common hendecahedrons are cones, cylinders, some Jason polyhedrons, and the semi-regular polyhedron. The semi-regular polyhedron here is not the Archimedean solid, but Enneagonal prism.

In the eleven faces of a hendecahedron, four of them are the Jansen polyhedron, which are: Elongated pentagonal pyramid, Gyroelongated pentagonal pyramid, Augmented pentagonal prism, and Augmented sphenocorona.

In the Hendecahedron, there are also some of the farrists, such as the truncated triangulum and the truncated triangular-cone.

Other Hendecahedrons include the enneagonal prism, spherical octagonal pyramid, two side taper triangular prism of a duality of six, and bisymmetric Hendecahedron, which can close shop space.

Content:

Name of hendecahedron 

The name of hendecahedron is based on its meaning. Hen- represents one. Deca represents ten, and then combined with the polyhedron suf-hedron, it is Hendecahedron.

Common Hendecahedrons

In all the convex hendecahedrons, there are a total of 440,564 convex ones with distinct differences in topology.[2] [3]There are significant differences in the structure of topology, which means two types of polyhedrons cannot be transformed by moving vertex positions, twisting, or scaling, such as a pentagonal pyramid and a nine diagonal column. They can't change with each other, so their topology structure is different. But the pentagonal prism and enneagonal prism can interchange by stretching out or drawing back one of the nine sides of the scale, so the triangulum prism and the triangulum pyramid have no obvious difference in topology.

The common hendecahedrons are cones, cylinders, some Jason polyhedrons, and the semi-regular polyhedron. The semi-regular polyhedron here is not the Archimedean solid, but the enneagonal prism.

Other hendecahedrons include enneagonal prism, Spherical octagonal pyramid, two side taper triangular prism of the duality of six, side cone Angle and bisymmetric hendecahedron, which can close shop space. [4]

Truncated triangular prism

In geometry, truncated triangular prism refers to the triangular prism after cut half, which is a kind of eleven side body. Its side is square, the bottom is a triangle, and others faces are six isosceles triangle surface. Truncated triangular prism can be constructed by cutting the halfway point of the triangular prism. The triangular prism will edge as a new vertex, and old vertex disappear. In other words, that is to cut a triangular prism from an edge to the midpoint of another edge (namely cut the triangular prism vertex, rather than cut Angle). It has the symmetry of D3h dihedral group.

Truncated triangular prism with one side

Jason polyhedron

Three of the eleven sides are Jansen polyhedron, which are respectively: the positive pentagonal pyramidal column, the two-sided cone, and the lateral cone.

Enneagonal prism

A nine-sided column is a nine-sided cylinder, consisting of 11 sides, 27 edges and 18 vertices. The regular Enneagonal prism represents that each surface is a regular polygon nine Angle. Every vertex is the public point of two squares and one enneagon, so it has the nature of equiangular, which can be categorized as hendecahedron.

Enneagonal Prism

Enneagonal Prism vertigo

A nine-sided column is a nine-sided cylinder, consisting of 11 sides, 27 edges and 18 vertices. The regular Enneagonal prism represents that each surface is a regular polygon nine Angle. Every vertex is the public point of two squares and one enneagon, so it has the nature of equiangular, which can be categorized as hendecahedron.

Spherical octagonal pyramid

Ten pyramid is a cone with a base of ten sides, with 11 faces, 20 edges and 11 vertices. And its dual polyhedron is itself. A regular ten pyramid is a ten - angle pyramid with a regular decagon.

Spherical octagonal pyramid

Bisymmetric Hendecahedron

Bisymmetric Hendecahedron is a polyhedron of hendecahedron. According to Plato and Archimedes, there are only a few that can be placed in space, that is to say, stacking together and leaving no gaps to fill the space. Guy Inchbald describes an interesting polyhedron that can complete the tesselation with hendecahedron in a surprising way. [4] [5] [6]

Bisymmetric Hendecahedron

Someone has proposed a hendecahedron, which has the same number of faces and vertices. After being distorted, it will get different characteristics. The most symmetric hendecahedron is the double symmetric hendecahedron. Because it has two symmetric faces, it is named as the double symmetric hendecahedron.   

Elongated pentagonal pyramid

Elongated pentagonal pyramid

It belongs to one of the 92 types of Johnson polyhedron, which can be composed of a positive pentagonal cone in the Johnson polyhedron and a pentagonal plane of the same size in the Platonic solid. These 92 types of Johnson Spaces were first named and described by Johnson Norman in 1996.

Another type of the Johnson polyhedron (J11) is the gyroelongated pentagonal pyramid. As the name implies, it can be synthesized from the normal pentagonal cone of the Johnson polyhedron and the pentagonal plane of the same size. At the same time, it is also the solid body of the regular polyhedron, which is the result of removing a normal pentagonal cone.

Augmented pentagonal prism

Augmented pentagonal prism is also one kind of Johnson Polyhedra (J52), which is composed of a regular quadrangle (J1) and a regular pentagonal column by combing the bottom and sides. It has a similar structure to the biaugmented pentagonal prism. These 92 kinds of Johnson polyhedrons were first named and observed by Norman Johnson in 1996.

Augmented sphenocorona

Augmented sphenocorona is one of the Johnson polyhedra. Although it can be gotten by adding an elongated square pyramid on the side of Sphenocorona, it cannot get by cutting or providing a supplement on Platonic solid (regular polyhedron) and Archimedes solid (semi-regular polyhedron). These 92 kinds of Johnson polyhedrons were first named and observed by Norman Johnson in 1996.

Hendecahedron in Chemistry

In the chemistry, after removing all 18 sides in borane hydrogen ions ([B11H11]), it is an Octadecahedron. If making a perpendicular to the center of gravity to the surface of a boron atom, a new polyhedron is constructed, which is 18 surface structures of the dual polyhedron, also one of hendecahedrons. [7]  

References

• Thomas H. Sidebotham. The A to Z of Mathematics: A Basic Guide. John Wiley & Sons.

2003: 237. ISBN 9780471461630

• Steven Dutch: How Many Polyhedra are There? (http://www.uwgb.edu/dutchs/symmetry/POLYHOW M.HTM)

• Counting polyhedra (http://www.numericana.com/data/polycount.htm) numericana.com [2016-1-10]

• Inchbald, Guy. "Five Space-Filling Polyhedra." The Mathematical Gazette 80, no. 489 (November 1996): 466-475

• Space-Filling Bisymmetric Hendecahedron. [2013-04-11]

• Anderson, Ian. "Constructing Tournament Designs." The Mathematical Gazette 73, no. 466 (December 1989): 284-292

• Holleman, A. F.; Wiberg, E., Inorganic Chemistry, San Diego: Academic Press: 1165, 2001, ISBN 0-12-352651-5