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 truncateddodecagon, 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).
Symmetries of a regular icositetragon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center.
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.
Dissection
24-gon with 264 rhombs
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.
Examples
12-cube
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}.
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
3 regular skew zig-zag icositetragons
{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.
^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
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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 Regular icositetragonA regular icositetragonTypeRegular polygonEdges and vertices24Schlafli symbol 24 t 12 tt 6 ttt 3 Coxeter Dynkin diagramsSymmetry groupDihedral D24 order 2 24Internal angle degrees 165 PropertiesConvex cyclic equilateral isogonal isotoxal Contents 1 Regular icositetragon 1 1 Construction 2 Symmetry 3 Dissection 4 Related polygons 5 Skew icositetragon 5 1 Petrie polygons 6 ReferencesRegular icositetragon EditThe regular icositetragon is represented by Schlafli 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 Further information Trigonometric constants expressed in real radicals 7 5 regular icositetragon 24 sided polygon 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 6 t 2 cot p 24 6 t 2 2 2 3 6 displaystyle 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 tetracontaoctagon 48 gon and enneacontahexagon 96 gon Construction Edit 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 Symmetry Edit Symmetries of a regular icositetragon Vertices are colored by their symmetry positions Blue mirrors are drawn through vertices and purple mirrors are drawn through edge Gyration orders are given in the center 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 Dissection Edit24 gon with 264 rhombs regular IsotoxalCoxeter 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 Examples 12 cube Related polygons Edit 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 Schlafli 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 figuresForm Convex polygon Compounds Star polygon CompoundImage 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 CompoundImage 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 dodecagramQuasiregular Isogonal Quasiregular t 12 24 t 12 11 24 11 t 12 5 24 5 t 12 7 24 7 Skew icositetragon Edit3 regular skew zig zag icositetragons 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 Petrie polygons Edit 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 cellE8 421 241 142References Edit Constructible Polygon John H Conway Heidi Burgiel Chaim Goodman Strauss 2008 The Symmetries of Things ISBN 978 1 56881 220 5 Chapter 20 Generalized Schaefli symbols Types of symmetry of a polygon pp 275 278 Coxeter Mathematical recreations and Essays Thirteenth edition p 141 The Lighter Side of Mathematics Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and its History 1994 Metamorphoses of polygons Branko Grunbaum Weisstein Eric W Icositetragon MathWorld Naming Polygons and Polyhedra simple polygon icosatetragon Retrieved from https en wikipedia org w index php title Icositetragon amp oldid 1095639107, wikipedia, wiki, book, books, library,
