Heptacontagon
Regular heptacontagon  

A regular heptacontagon  
Type  Regular polygon 
Edges and vertices  70 
Schläfli symbol  {70}, t{35} 
Coxeter diagram 

Symmetry group  Dihedral (D_{70}), order 2×70 
Internal angle (degrees)  ≈174.857° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, a heptacontagon (or hebdomecontagon from Ancient Greek ἑβδομήκοντα, seventy^{[1]}) or 70gon is a seventysided polygon.^{[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 174^{6}⁄_{7}°, meaning that one exterior angle would be 5^{1}⁄_{7}°.
The area of a regular heptacontagon is (with t = edge length)
and its inradius is
The circumradius of a regular heptacontagon is
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 regular heptacontagon has Dih_{70} dihedral symmetry, order 140, represented by 70 lines of reflection. Dih_{70} has 7 dihedral subgroups: Dih_{35}, (Dih_{14}, Dih_{7}), (Dih_{10}, Dih_{5}), and (Dih_{2}, Dih_{1}). It also has 8 more cyclic symmetries as subgroups: (Z_{70}, Z_{35}), (Z_{14}, Z_{7}), (Z_{10}, Z_{5}), and (Z_{2}, Z_{1}), 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.
Dissection
Coxeter states that every zonogon (a 2mgon whose opposite sides are parallel and of equal length) can be dissected into m(m1)/2 parallelograms.^{[7]} In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular heptacontagon, m=35, it can be divided into 595: 17 sets of 35 rhombs. This decomposition is based on a Petrie polygon projection of a 35cube.
Heptacontagram
A heptacontagram is a 70sided 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.
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
 ↑ Greek Numbers and Numerals (Ancient and Modern) by Harry Foundalis
 ↑ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 77, ISBN 9781438109572 .
 ↑ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
 ↑ Constructible Polygon
 ↑ "Archived copy" (PDF). Archived from the original (PDF) on 20150714. Retrieved 20150219.
 ↑ The Symmetries of Things, Chapter 20
 ↑ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141