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Regular enneacontagon  

A regular enneacontagon


Type  Regular polygon 
Edges and vertices  90 
Schläfli symbol  {90}, t{45} 
Coxeter diagram  
Symmetry group  Dihedral (D_{90}), order 2×90 
Internal angle (degrees)  176° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, an enneacontagon or enenecontagon or 90gon (from Ancient Greek ἑννενήκοντα, ninety^{[1]}) is a ninetysided polygon.^{[2]}^{[3]} The sum of any enneacontagon's interior angles is 15840 degrees.
A regular enneacontagon is represented by Schläfli symbol {90} and can be constructed as a truncated tetracontapentagon, t{45}, which alternates two types of edges.
One interior angle in a regular enneacontagon is 176°, meaning that one exterior angle would be 4°.
The area of a regular enneacontagon is (with t = edge length)
and its inradius is
The circumradius of a regular enneacontagon is
Since 90 = 2 × 3^{2} × 5, a regular enneacontagon is not constructible using a compass and straightedge,^{[4]} but is constructible if the use of an angle trisector is allowed.^{[5]}
The regular enneacontagon has Dih_{90} dihedral symmetry, order 180, represented by 90 lines of reflection. Dih_{90} has 11 dihedral subgroups: Dih_{45}, (Dih_{30}, Dih_{15}), (Dih_{18}, Dih_{9}), (Dih_{10}, Dih_{5}), (Dih_{6}, Dih_{3}), and (Dih_{2}, Dih_{1}). And 12 more cyclic symmetries: (Z_{90}, Z_{45}), (Z_{30}, Z_{15}), (Z_{18}, Z_{9}), (Z_{10}, Z_{5}), (Z_{6}, Z_{3}), and (Z_{2}, Z_{1}), with Z_{n} representing π/n radian rotational symmetry.
These 24 symmetries are related to 30 distinct symmetries on the enneacontagon. 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 freedom in defining irregular enneacontagons. Only the g90 symmetry has no degrees of freedom but can seen as directed edges.
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 enneacontagon, m=45, it can be divided into 990: 22 sets of 45 rhombs. This decomposition is based on a Petrie polygon projection of a 45cube.
An enneacontagram is a 90sided star polygon. There are 11 regular forms given by Schläfli symbols {90/7}, {90/11}, {90/13}, {90/17}, {90/19}, {90/23}, {90/29}, {90/31}, {90/37}, {90/41}, and {90/43}, as well as 33 regular star figures with the same vertex configuration.
Pictures  {90/7} 
{90/11} 
{90/13} 
{90/17} 
{90/19} 
{90/23} 

Interior angle  152°  136°  128°  112°  104°  88° 
Pictures  {90/29} 
{90/31} 
{90/37} 
{90/41} 
{90/43} 

Interior angle  64°  56°  32°  16°  8° 