Icositetragon

Regular icositetragon

A regular icositetragon
Type Regular polygon
Edges and vertices 24
Schläfli symbol {24}
t{12}
Coxeter diagram
Symmetry group Dihedral (D24), order 2×24
Internal angle (degrees) 165°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, an icositetragon (or icosikaitetragon or tetracosagon) 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 quasiregular truncated icosagon, t{12}, which alternates two types of edges.

Regular icositetragon

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)

A = 6t^2 \cot \frac{\pi}{24} = {6}t^2(2+\sqrt{2}+\sqrt{3}+\sqrt{6}).

The icositetragon appeared in Archimedes' polygon approximation of pi, along with the hexagon (6-gon), dodecagon (12-gon), 48-gon, and 96-gon.

Construction

A regular icositetragon is constructible using a compass and straightedge.[1] As a truncated dodecagon, it can be constructed by an edge-bisection of a regular dodecagon.

Related polygons


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}. [2]

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}

Petrie polygons

The regular icositetragon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes, including:

E8

421
241
142

References

  1. ↑ Constructible Polygon
  2. ↑ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum