Heptacontagon

Regular heptacontagon
A regular heptacontagon
Type	Regular polygon
Edges and vertices	70
Schläfli symbol	{70}, t{35}
Coxeter diagram
Symmetry group	Dihedral (D70), order 2×70
Internal angle (degrees)	≈174.9°
Dual polygon	self
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a heptacontagon (or hebdomecontagon from Ancient Greek ἑβδομήκοντα, seventy^[1]) is a seventy-sided polygon or 70-gon.^[2][3] The sum of any heptacontagon's interior angles is 12240 degrees.

A regular heptacontagon is represented by Schläfli symbol {70} and can also be constructed as a truncated triacontapentagon, t{35}, which alternates two types of edges.

Regular heptacontagon properties

One interior angle in a regular heptacontagon is 174⁶⁄₇°, meaning that one exterior angle would be 5¹⁄₇°.

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

$A = \frac{35}{2}t^2 \cot \frac{\pi}{70}$

and its inradius is

$r = \frac{1}{2}t \cot \frac{\pi}{70}$

The circumradius of a regular heptacontagon is

$R = \frac{1}{2}t \csc \frac{\pi}{70}$

Since 70 = 2 × 5 × 7, a regular heptacontagon is not constructible using a compass and straightedge,^[4] but is constructible if the use of an angle trisector is allowed.^[5]

Symmetry

The symmetries of a regular heptacontagon. Light blue lines show subgroups of index 2. The four subgraphs are positionally related by index 5 and index 7 subgroups.

The regular heptacontagon has Dih₇₀ dihedral symmetry, order 140, represented by 70 lines of reflection. Dih₇₀ has 7 dihedral subgroups: Dih₃₅, (Dih₁₄, Dih₇), (Dih₁₀, Dih₅), and (Dih₂, Dih₁). It also has 8 more cyclic symmetries as subgroups: (Z₇₀, Z₃₅), (Z₁₄, Z₇), (Z₁₀, Z₅), and (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.^[6] 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 heptacontagons. Only the g70 subgroup has no degrees of freedom but can seen as directed edges.

Heptacontagram

A heptacontagram is a 70-sided star polygon. There are 11 regular forms given by Schläfli symbols {70/3}, {70/9}, {70/11}, {70/13}, {70/17}, {70/19}, {70/23}, {70/27}, {70/29}, {70/31}, and {70/33}, as well as 23 regular star figures with the same vertex configuration.

Regular star polygons {70/k}

Picture	{70/3}	{70/9}	{70/11}	{70/13}	{70/17}	{70/19}
Interior angle	≈164.571°	≈133.714°	≈123.429°	≈113.143°	≈92.5714°	≈82.2857°
Picture	{70/23}	{70/27}	{70/29}	{70/31}	{70/33}
Interior angle	≈61.7143°	≈41.1429°	≈30.8571°	≈20.5714°	≈10.2857°

References

1. Greek Numbers and Numerals (Ancient and Modern) by Harry Foundalis
2. Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 77, ISBN 9781438109572.
3. The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
4. Constructible Polygon
5. http://www.math.iastate.edu/thesisarchive/MSM/EekhoffMSMSS07.pdf
6. The Symmetries of Things, Chapter 20

• Weisstein, Eric W., "Heptacontagon", MathWorld.
• Naming Polygons and Polyhedra
• hebdomacontagon