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Regular hexacontatetragon  

A regular hexacontatetragon


Type  Regular polygon 
Edges and vertices  64 
Schläfli symbol  {64}, t{32}, tt{16}, ttt{8}, tttt{4} 
Coxeter diagram  
Symmetry group  Dihedral (D_{64}), order 2×64 
Internal angle (degrees)  174.375° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, a hexacontatetragon (or hexacontakaitetragon) or 64gon is a sixtyfoursided polygon. (In Greek, the prefix hexaconta means 60 and tetra means 4.) The sum of any hexacontatetragon's interior angles is 11160 degrees.
The regular hexacontatetragon can be constructed as a truncated triacontadigon, t{32}, a twicetruncated hexadecagon, tt{16}, a thricetruncated octagon, ttt{8}, a fourfoldtruncated square, tttt{4}, and a fivefoldtruncated digon, ttttt{2}.
One interior angle in a regular hexacontatetragon is 174^{3}⁄_{8}°, meaning that one exterior angle would be 5^{5}⁄_{8}°.
The area of a regular hexacontatetragon is (with t = edge length)
and its inradius is
The circumradius of a regular hexacontatetragon is
Since 64 = 2^{6} (a power of two), a regular hexacontatetragon is constructible using a compass and straightedge.^{[1]} As a truncated triacontadigon, it can be constructed by an edgebisection of a regular triacontadigon.
The regular hexacontatetragon has Dih_{64} dihedral symmetry, order 128, represented by 64 lines of reflection. Dih_{64} has 6 dihedral subgroups: Dih_{32}, Dih_{16}, Dih_{8}, Dih_{4}, Dih_{2} and Dih_{1} and 7 more cyclic symmetries: Z_{64}, Z_{32}, Z_{16}, Z_{8}, Z_{4}, Z_{2}, and Z_{1}, with Z_{n} representing π/n radian rotational symmetry.
These 13 symmetries generate 20 unique symmetries on the regular hexacontatetragon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^{[2]} He gives r128 for the full reflective symmetry, Dih_{64}, and a1 for 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. a1 labels no symmetry.
These lower symmetries allows degrees of freedoms in defining irregular hexacontatetragons. Only the g64 subgroup 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. ^{[3]} In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular hexacontatetragon, m=32, and it can be divided into 496: 16 squares and 15 sets of 32 rhombs. This decomposition is based on a Petrie polygon projection of a 32cube.
A hexacontatetragram is a 64sided star polygon. There are 15 regular forms given by Schläfli symbols {64/3}, {64/5}, {64/7}, {64/9}, {64/11}, {64/13}, {64/15}, {64/17}, {64/19}, {64/21}, {64/23}, {64/25}, {64/27}, {64/29}, {64/31}, as well as 16 compound star figures with the same vertex configuration.
Picture  {64/3} 
{64/5} 
{64/7} 
{64/9} 
{64/11} 
{64/13} 
{64/15} 
{64/17} 

Interior angle  163.125°  151.875°  140.625°  129.375°  118.125°  106.875°  95.625°  84.375° 
Picture  {64/19} 
{64/21} 
{64/23} 
{64/25} 
{64/27} 
{64/29} 
{64/31} 

Interior angle  73.125°  61.875°  50.625°  39.375°  28.125°  16.875°  5.625° 