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​²⁄₅°, meaning that one exterior angle would be 3​³⁄₅°.

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

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

and its inradius is

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

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.

However, a hectogon is constructible using an auxiliary curve such as an Archimedean spiral. A 72° angle is constructible with compass and straightedge, so a possible approach to constructing one side of a hectogon is to construct a 72° angle using compass and straightedge, use an Archimedean spiral to construct a 14.4° angle, and bisect one of the 14.4° angles twice.

Exact construction with help the quadratrix of Hippias

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

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.[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.

100-gon with 4900 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.

Examples

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.

Regular star polygons {100/k}

Picture	{100/3}	{100/7}	{100/11}	{100/13}	{100/17}	{100/19}
Interior angle	169.2°	154.8°	140.4°	133.2°	118.8°	111.6°
Picture	{100/21}	{100/23}	{100/27}	{100/29}	{100/31}	{100/37}
Interior angle	104.4°	97.2°	82.8°	75.6°	68.4°	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°

• Decagon (10)
• Chiliagon (1000)
• Myriagon (10000)
• Megagon (1000000)

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