| Regular icositetragon | |
|---|---|
 A regular icositetragon | |
| Type | Regular polygon |
| Edges and vertices | 24 |
| Schläfli symbol | {24}, t{12}, tt{6}, ttt{3} |
| Coxeter–Dynkin diagrams |      |
| Symmetry group | Dihedral (D24), order 2×24 |
| Internal angle (degrees) | 165° |
| Properties | Convex, cyclic, equilateral, isogonal, isotoxal |
In geometry, an icositetragon (or icosikaitetragon) or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.
The regular icositetragon is represented by Schläfli symbol {24} and can also be constructed as a truncated dodecagon, t{12}, or a twice-truncated hexagon, tt{6}, or thrice-truncated 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 (6-gon), dodecagon (12-gon), tetracontaoctagon (48-gon), and enneacontahexagon (96-gon).
As 24 = 23 × 3, a regular icositetragon is constructible using an angle trisector. [1] As a truncated dodecagon, it can be constructed by an edge-bisection of a regular dodecagon.
The regular icositetragon has Dih24 symmetry, order 48. There are 7 subgroup dihedral symmetries: (Dih12, Dih6, Dih3), and (Dih8, Dih4, Dih2 Dih1), and 8 cyclic group symmetries: (Z24, Z12, Z6, Z3), and (Z8, Z4, Z2, Z1).
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.
 regular |  Isotoxal |
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. [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 12-cube.
 12-cube |  |  |  |  |
A regular triangle, octagon, and icositetragon can completely fill a plane vertex.
An icositetragram is a 24-sided 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} | |||||
| {12}#{ } | {12/5}#{ } | {12/7}#{ } |
|---|---|---|
 |  |  |
| A regular skew icositetragon is seen as zig-zagging edges of a dodecagonal antiprism, a dodecagrammic antiprism, and a dodecagrammic crossed-antiprism. | ||
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 zig-zag icositetragon has vertices alternating between two parallel planes.
A regular skew icositetragon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew icositetragon and can be seen in the vertices and side edges of a dodecagonal antiprism with the same D12d, [2+,24] symmetry, order 48. The dodecagrammic antiprism, s{2,24/5} and dodecagrammic crossed-antiprism, s{2,24/7} also have regular skew dodecagons.
The regular icositetragon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes, including:
| 2F4 | ||
|---|---|---|
 Bitruncated 24-cell |  Runcinated 24-cell |  Omnitruncated 24-cell |
| E8 | ||
|---|---|---|
 421 |  241 |  142 |
In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An.
In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.
In geometry, an octagon is an eight-sided polygon or 8-gon.
In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex or star. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.
In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells.
In geometry, a dodecagon or 12-gon is any twelve-sided polygon.
In geometry, a myriagon or 10000-gon is a polygon with 10,000 sides. Several philosophers have used the regular myriagon to illustrate issues regarding thought.
In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.
In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (n+m)-polytope, where n and m are dimensions of 2 (polygon) or higher.
In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon.
In mathematics, a hexadecagon is a sixteen-sided polygon.
In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least four vertices. The interior surface of such a polygon is not uniquely defined.
In geometry, an octadecagon or 18-gon is an eighteen-sided polygon.
In geometry, a dodecagram is a star polygon or compound with 12 vertices. There is one regular dodecagram polygon. There are also 4 regular compounds {12/2},{12/3},{12/4}, and {12/6}.
In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lines. Infinite helical polygons are 3-dimensional infinite skew polygons with vertices on the surface of a cylinder.
