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The bisymmetric hendecahedron contains 11 faces and can be arranged in 3D without gaps.

A hendecahedron (or undecahedron) is a polyhedron with 11 faces. There are numerous topologically distinct forms of a hendecahedron, for example the decagonal pyramid, and enneagonal prism.

Three forms are Johnson solids: augmented hexagonal prism, biaugmented triangular prism, and elongated pentagonal pyramid.

Two classes, the bisymmetric and the sphenoid hendecahedra, are space-filling.[1]

Name of hendecahedron[edit]

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 Hendecahedron[edit]

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.

Bisymmetric Hendecahedron[edit]

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]

Net of 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.  

Hendecahedron in Chemistry[edit]

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]  


There are 440,564 topologically distinct convex hendecahedra, excluding mirror images, having at least 8 vertices.[2] (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)


  1. ^ Inchbald (1996)
  2. ^ Counting polyhedra
  • 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

External links[edit]