Heptacontagon

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

Regular heptacontagon
A regular heptacontagon
TypeRegular polygon
Edges and vertices70
Schläfli symbol{70}, t{35}
Coxeter diagram
Symmetry groupDihedral (D70), order 2×70
Internal angle (degrees)≈174.857°
Dual polygonSelf
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal

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 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 Zn 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
70-gon with 2380 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.[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 35-cube.

Examples

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. "Archived copy" (PDF). Archived from the original (PDF) on 2015-07-14. Retrieved 2015-02-19.CS1 maint: archived copy as title (link)
  6. The Symmetries of Things, Chapter 20
  7. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
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