Icositetragon
Regular icositetragon  

A regular icositetragon  
Type  Regular polygon 
Edges and vertices  24 
Schläfli symbol  {24}, t{12}, tt{6}, ttt{3} 
Coxeter diagram 

Symmetry group  Dihedral (D_{24}), order 2×24 
Internal angle (degrees)  165° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, an icositetragon (or icosikaitetragon or tetracosagon) or 24gon is a twentyfoursided polygon. The sum of any icositetragon's interior angles is 3960 degrees.
Regular icositetragon
The regular icositetragon is represented by Schläfli symbol {24} and can also be constructed as a truncated dodecagon, t{12}, or a twicetruncated hexagon, tt{6}, or thricetruncated triangle, ttt{3}.
One interior angle in a regular icositetragon is 165°, meaning that one exterior angle would be 15°.
The area of a regular icositetragon is: (with t = edge length)
The icositetragon appeared in Archimedes' polygon approximation of pi, along with the hexagon (6gon), dodecagon (12gon), tetracontaoctagon (48gon), and enneacontahexagon (96gon).
Construction
As 24 = 2^{3} × 3, a regular icositetragon is constructible using a compass and straightedge.^{[1]} As a truncated dodecagon, it can be constructed by an edgebisection of a regular dodecagon.
Symmetry
The regular icositetragon has Dih_{24} symmetry, order 48. There are 7 subgroup dihedral symmetries: (Dih_{12}, Dih_{6}, Dih_{3}), and (Dih_{8}, Dih_{4}, Dih_{2} Dih_{1}), and 8 cyclic group symmetries: (Z_{24}, Z_{12}, Z_{6}, Z_{3}), and (Z_{8}, Z_{4}, Z_{2}, Z_{1}).
These 16 symmetries can be seen in 22 distinct symmetries on the icositetragon. John Conway labels these by a letter and group order.^{[2]} The full symmetry of the regular form is r48 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g24 subgroup has no degrees of freedom but can seen as directed edges.
Dissection
regular 
Isotoxal 
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 icositetragon, m=12, and it can be divided into 66: 6 squares and 5 sets of 12 rhombs. This decomposition is based on a Petrie polygon projection of a 12cube.
12cube 
Related polygons
A regular triangle, octagon, and icositetragon can completely fill a plane vertex.
An icositetragram is a 24sided star polygon. There are 3 regular forms given by Schläfli symbols: {24/5}, {24/7}, and {24/11}. There are also 7 regular star figures using the same vertex arrangement: 2{12}, 3{8}, 4{6}, 6{4}, 8{3}, 3{8/3}, and 2{12/5}.
Icositetragrams as star polygons and star figures  

Form  Convex polygon  Compounds  Star polygon  Compound  
Image  {24/1}={24} 
{24/2}=2{12} 
{24/3}=3{8} 
{24/4}=4{6} 
{24/5} 
{24/6}=6{4}  
Interior angle  165°  150°  135°  120°  105°  90°  
Form  Star polygon  Compounds  Star polygon  Compound  
Image  {24/7} 
{24/8}=8{3} 
{24/9}=3{8/3} 
{24/10}=2{12/5} 
{24/11} 
{24/12}=12{2}  
Interior angle  75°  60°  45°  30°  15°  0° 
There are also isogonal icositetragrams constructed as deeper truncations of the regular dodecagon {12} and dodecagram {12/5}. These also generate two quasitruncations: t{12/11}={24/11}, and t{12/7}={24/7}. ^{[4]}
Isogonal truncations of regular dodecagon and dodecagram  

Quasiregular  Isogonal  Quasiregular  
t{12}={24} 
t{12/11}={24/11}  
t{12/5}={24/5} 
t{12/7}={24/7} 
Skew icositetragon
{12}#{ }  {12/5}#{ }  {12/7}#{ } 

A regular skew icositetragon is seen as zigzagging edges of a dodecagonal antiprism, a dodecagrammic antiprism, and a dodecagrammic crossedantiprism. 
A skew icositetragon is a skew polygon with 24 vertices and edges but not existing on the same plane. The interior of such an icositetragon is not generally defined. A skew zigzag icositetragon has vertices alternating between two parallel planes.
A regular skew icositetragon is vertextransitive with equal edge lengths. In 3dimensions it will be a zigzag skew icositetragon and can be seen in the vertices and side edges of a dodecagonal antiprism with the same D_{12d}, [2^{+},24] symmetry, order 48. The dodecagrammic antiprism, s{2,24/5} and dodecagrammic crossedantiprism, s{2,24/7} also have regular skew dodecagons.
Petrie polygons
The regular icositetragon is the Petrie polygon for many higherdimensional polytopes, seen as orthogonal projections in Coxeter planes, including:
E_{8}  

4_{21} 
2_{41} 
1_{42} 
References
 ↑ Constructible Polygon
 ↑ John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, (2008) The Symmetries of Things, ISBN 9781568812205 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275278)
 ↑ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
 ↑ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum