Tetracontagon

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Regular tetracontagon
Regular polygon 40.svg
A regular tetracontagon
Type Regular polygon
Edges and vertices 40
Schläfli symbol {40}
t{20}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel 0x.pngCDel node.png
CDel node 1.pngCDel 20.pngCDel node 1.png
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.[1][2] A regular tetracontagon is represented by Schläfli symbol {40} and can be constructed as a quasiregular truncated icosagon, t{20}, which alternates two types of edges.

The sum of any tetracontagon's interior angles is 6840 degrees.

Regular tetracontagon properties[edit]

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}

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})}

Tetracontagram[edit]

A tetracontagram is a 40-sided star polygon. There are 7 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}
Star polygon 40-3.svg
{40/3}
Star polygon 40-7.svg
{40/7}
Star polygon 40-9.svg
{40/9}
Star polygon 40-11.svg
{40/11}
Star polygon 40-13.svg
{40/13}
Star polygon 40-17.svg
{40/17}
Star polygon 40-19.svg
{40/19}
Regular compound polygons
Star polygon 40-2.png
{40/2}=2{20}
Star polygon 40-4.png
{40/4}=4{10}
Star polygon 40-5.png
{40/5}=5{8}
Star polygon 40-6.png
{40/6}=2{20/3}
Star polygon 40-8.png
{40/8}=8{5}
Star polygon 40-10.png
{40/10}=10{4}
Star polygon 40-12.png
{40/12}=4{10/3}
Star polygon 40-14.png
{40/14}=2{20/7}
Star polygon 40-15.png
{40/15}=5{8/3}
Star polygon 40-16.png
{40/16}=8{5/2}
Star polygon 40-18.png
{40/18}=2{20/9}
Star polygon 40-20.png
{40/20}=20{2}

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}.[4]

Regular polygon truncation 20 1.svg
t{20}={40}
CDel node 1.pngCDel 20.pngCDel node 1.png
Regular polygon truncation 20 2.svg Regular polygon truncation 20 3.svg Regular polygon truncation 20 4.svg Regular polygon truncation 20 5.svg Regular polygon truncation 20 6.svg Regular polygon truncation 20 7.svg Regular polygon truncation 20 8.svg Regular polygon truncation 20 9.svg Regular polygon truncation 20 10.svg Regular polygon truncation 20 11.svg
t{20/19}={40/19}
CDel node 1.pngCDel 20.pngCDel rat.pngCDel 19.pngCDel node 1.png

References[edit]

  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 Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum