Hectogon

Regular hectogon
A regular hectogon
Type	Regular polygon
Edges and vertices	100
Schläfli symbol	{100}, t{50}, tt{25}
Coxeter diagram
Symmetry group	Dihedral (D100), order 2×100
Internal angle (degrees)	176.4°
Dual polygon	self
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a hectogon or hecatontagon is a hundred-sided polygon or 100-gon. 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 176.4°, meaning that one exterior angle would be 3.6°.

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

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

and its inradius is

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

The circumradius of a regular hectogon is

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

Because 100 = 2² × 5², the number of sides is neither a product of distinct Fermat primes nor a power of two. Thus the regular hectogon is not a constructible polygon. Indeed, it is not even constructible with the use of neusis or 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.

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 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. 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.

Hectogram

A hectogram is an 100-sided star polygon. There are 19 regular forms 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.

Regular star polygons {100/k}

Picture	{100/3}	{100/7}	{100/9}	{100/11}	{100/13}	{100/17}	{100/19}
Interior angle	169.2°	154.8°	147.6°	140.4°	133.2°	118.8°	111.6°
Picture	{100/21}	{100/23}	{100/27}	{100/29}	{100/31}	{100/33}	{100/37}
Interior angle	104.4°	97.2°	82.8°	75.6°	68.4°	61.2°	46.8°
Picture	{100/39}	{100/41}	{100/43}	{100/47}	{100/49}
Interior angle	39.6°	32.4°	25.2°	10.8°	3.6°

References

• Weisstein, Eric W., "Hectogon", MathWorld.
• Naming Polygons and Polyhedra
• hecatonagon