Tetracontagon

Regular tetracontagon
A regular tetracontagon
Type	Regular polygon
Edges and vertices	40
Schläfli symbol	{40}, t{20}, tt{10}, ttt{5}
Coxeter diagram
Symmetry group	Dihedral (D40), order 2×40
Internal angle (degrees)	171°
Dual polygon	self
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a tetracontagon or tessaracontagon is a forty-sided polygon or 40-gon.^[1][2] The sum of any tetracontagon's interior angles is 6840 degrees.

Regular tetracontagon

A regular tetracontagon is represented by Schläfli symbol {40} and can also be constructed as a truncated icosagon, t{20}, which alternates two types of edges. Furthermore, it can also be constructed as a twice-truncated decagon, tt{10}, or a thrice-truncated pentagon, ttt{5}.

One interior angle in a regular tetracontagon is 171°, meaning that one exterior angle would be 9°.

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

$A = 10t^2 \cot \frac{\pi}{40}$

and its inradius is

$r = \frac{1}{2}t \cot \frac{\pi}{40}$

The factor $\cot \frac{\pi}{40}$ is a root of the octic equation $x^{8} - 8x^{7} - 60x^{6} - 8x^{5} + 134x^{4} + 8x^{3} - 60x^{2} + 8x + 1$.

The circumradius of a regular tetracontagon is

$R = \frac{1}{2}t \csc \frac{\pi}{40}$

As 40 = 2³ × 5, a regular tetracontagon is constructible using a compass and straightedge.^[3] As a truncated icosagon, it can be constructed by an edge-bisection of a regular icosagon. This means that the values of $\sin \frac{\pi}{40}$ and $\cos \frac{\pi}{40}$ may be expressed in radicals as follows:

$\sin \frac{\pi}{40} = \frac{1}{4}(\sqrt{2}-1)\sqrt{\frac{1}{2}(2+\sqrt{2})(5+\sqrt{5})}-\frac{1}{8}\sqrt{2-\sqrt{2}}(1+\sqrt{2})(\sqrt{5}-1)$

$\cos \frac{\pi}{40} = \frac{1}{8}(\sqrt{2}-1)\sqrt{2+\sqrt{2}}(\sqrt{5}-1)+\frac{1}{4}(1+\sqrt{2})\sqrt{\frac{1}{2}(2-\sqrt{2})(5+\sqrt{5})}$

Symmetry

The symmetries of a regular tetracontagon. Light blue lines show subgroups of index 2. The left and right subgraphs are positionally related by index 5 subgroups.

The regular tetracontagon has Dih₄₀ dihedral symmetry, order 80, represented by 40 lines of reflection. Dih₄₀ has 7 dihedral subgroups: (Dih₂₀, Dih₁₀, Dih₅), and (Dih₈, Dih₄, Dih₂, Dih₁). It also has eight more cyclic symmetries as subgroups: (Z₄₀, Z₂₀, Z₁₀, Z₅), and (Z₈, Z₄, Z₂, Z₁), with Z_n representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^[4] He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.

These lower symmetries allows degrees of freedoms in defining irregular tetracontagons. Only the g40 subgroup has no degrees of freedom but can seen as directed edges.

Tetracontagram

A tetracontagram is a 40-sided star polygon. There are seven regular forms given by Schläfli symbols {40/3}, {40/7}, {40/9}, {40/11}, {40/13}, {40/17}, and {40/19}, and 12 compound star figures with the same vertex configuration.

Regular star polygons {40/k}

Picture	{40/3}	{40/7}	{40/9}	{40/11}	{40/13}	{40/17}	{40/19}
Interior angle	153°	117°	99°	81°	63°	27°	9°

Regular compound polygons

Picture	{40/2}=2{20}	{40/4}=4{10}	{40/5}=5{8}	{40/6}=2{20/3}	{40/8}=8{5}	{40/10}=10{4}
Interior angle	162°	144°	135°	126°	108°	90°
Picture	{40/12}=4{10/3}	{40/14}=2{20/7}	{40/15}=5{8/3}	{40/16}=8{5/2}	{40/18}=2{20/9}	{40/20}=20{2}
Interior angle	72°	54°	45°	36°	18°	0°

Many isogonal tetracontagrams can also be constructed as deeper truncations of the regular icosagon {20} and icosagrams {20/3}, {20/7}, and {20/9}. These also create four quasitruncations: t{20/11}={40/11}, t{20/13}={40/13}, t{20/17}={40/17}, and t{20/19}={40/19}. Some of the isogonal tetracontagrams are depicted below, as a truncation sequence with endpoints t{20}={40} and t{20/19}={40/19}.^[5]

t{20}={40}
				t{20/19}={40/19}

References

1. Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 165, ISBN 9781438109572.
2. The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
3. Constructible Polygon
4. The Symmetries of Things, Chapter 20
5. The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

• Weisstein, Eric W., "Tetracontagon", MathWorld.
• Naming Polygons and Polyhedra
• tessaracontagon