# Hectogon

Regular hectogon
A regular hectogon
TypeRegular polygon
Edges and vertices100
Schläfli symbol{100}, t{50}, tt{25}
Coxeter diagram
Symmetry groupDihedral (D100), order 2×100
Internal angle (degrees)176.4°
Dual polygonSelf
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal

In geometry, a hectogon or hecatontagon or 100-gon[1][2] is a hundred-sided polygon.[3][4] The sum of any hectogon's interior angles is 17640 degrees.

## Regular hectogon

A regular hectogon is represented by Schläfli symbol {100} and can be constructed as a truncated pentacontagon, t{50}, or a twice-truncated icosipentagon, tt{25}.

One interior angle in a regular hectogon is 176​25°, meaning that one exterior angle would be 3​35°.

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

${\displaystyle A=25t^{2}\cot {\frac {\pi }{100}}}$

${\displaystyle r={\frac {1}{2}}t\cot {\frac {\pi }{100}}}$

The circumradius of a regular hectogon is

${\displaystyle R={\frac {1}{2}}t\csc {\frac {\pi }{100}}}$

Because 100 = 22 × 52, the number of sides contains a repeated Fermat prime (the number 5). Thus the regular hectogon is not a constructible polygon.[5] Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three.[6] It is not known if the regular hectogon is neusis constructible.

### Exact construction with help the quadratrix of Hippias

Hectogon, exact construction using the quadratrix of Hippias as an additional aid

## Symmetry

The symmetries of a regular hectogon. Light blue lines show subgroups of index 2. The 3 boxed subgraphs are positionally related by index 5 subgroups.

The regular hectogon has Dih100 dihedral symmetry, order 200, represented by 100 lines of reflection. Dih100 has 8 dihedral subgroups: (Dih50, Dih25), (Dih20, Dih10, Dih5), (Dih4, Dih2, and Dih1). It also has 9 more cyclic symmetries as subgroups: (Z100, Z50, Z25), (Z20, Z10, Z5), and (Z4, 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.[7] r200 represents full symmetry and a1 labels no symmetry. 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.

These lower symmetries allows degrees of freedom in defining irregular hectogons. Only the g100 subgroup has no degrees of freedom but can seen as directed edges.

## Dissection

100-gon with 2900 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. [8] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular hectogon, m=50, it can be divided into 1225: 25 squares and 24 sets of 50 rhombs. This decomposition is based on a Petrie polygon projection of a 50-cube.

## Hectogram

A hectogram is a 100-sided star polygon. There are 19 regular forms[9] given by Schläfli symbols {100/3}, {100/7}, {100/9}, {100/11}, {100/13}, {100/17}, {100/19}, {100/21}, {100/23}, {100/27}, {100/29}, {100/31}, {100/33}, {100/37}, {100/39}, {100/41}, {100/43}, {100/47}, and {100/49}, as well as 30 regular star figures with the same vertex configuration.

Picture Interior angle Picture Interior angle Picture Interior angle {100/3} {100/7} {100/11} {100/13} {100/17} {100/19} 169.2° 154.8° 140.4° 133.2° 118.8° 111.6° {100/21} {100/23} {100/27} {100/29} {100/31} {100/37} 104.4° 97.2° 82.8° 75.6° 68.4° 46.8° {100/39} {100/41} {100/43} {100/47} {100/49} 39.6° 32.4° 25.2° 10.8° 3.6°