Icosihexagon

In geometry, an icosihexagon (or icosikaihexagon) or 26-gon is a twenty-six-sided polygon. The sum of any icosihexagon's interior angles are 4320° .

Regular icosihexagon
A regular icosihexagon
TypeRegular polygon
Edges and vertices26
Schläfli symbol{26}, t{13}
Coxeter diagram
Symmetry groupDihedral (D26), order 2×26
Internal angle (degrees)≈166.154°
Dual polygonSelf
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal

Regular icosihexagon

The regular icosihexagon is represented by Schläfli symbol {26} and can also be constructed as a truncated tridecagon, t{13}.

The area of a regular icosihexagon is: (with t = edge length)

Construction

As 26 = 2 × 13, the icosihexagon can be constructed by truncating a regular tridecagon. However, the icosihexagon is not constructible with a compass and straightedge, since 13 is not a Fermat prime. It can be constructed with an angle trisector, since 13 is a Pierpont prime.

Symmetry

The regular icosihexagon has Dih26 symmetry, order 52. There are 3 subgroup dihedral symmetries: Dih11, Dih2, and Dih1, and 4 cyclic group symmetries: Z26, Z13, Z2, and Z1.

These 8 symmetries can be seen in 10 distinct symmetries on the icosihexagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order.[1] The full symmetry of the regular form is r52 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries n are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g26 subgroup has no degrees of freedom but can seen as directed edges.

The highest symmetry irregular icosihexagons are d26, an isogonal icosihexagon constructed by thirteen mirrors which can alternate long and short edges, and p26, an isotoxal icosihexagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular icosihexagon.

Dissection
26-gon with 312 rhombs

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular icosihexagon, m=13, and it can be divided into 78: 6 sets of 13 rhombs. This decomposition is based on a Petrie polygon projection of a 13-cube.[2]

Examples

An icosihexagram is a 26-sided star polygon. There are 5 regular forms given by Schläfli symbols: {26/3}, {26/5}, {26/7}, {26/9}, and {26/11}.


{26/3}
{26/5}
{26/7}
{26/9}
{26/11}

There are also isogonal icosihexagrams constructed as deeper truncations of the regular tridecagon {13} and tridecagrams {13/2}, {13/3}, {13/4}, {13/5} and {13/6}.[3]

References
  1. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
  2. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  3. The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum
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