User:Tomruen/Higher polygons

Enneadecagon

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Regular enneadecagon
 
A regular enneadecagon
TypeRegular polygon
Edges and vertices19
Schläfli symbol{19}
Coxeter–Dynkin diagrams   
Symmetry groupDihedral (D19), order 2×19
Internal angle (degrees)≈161.052°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, an enneadecagon, enneakaidecagon, nonadecagon or 19-gon is a polygon with nineteen sides.

Regular form

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A regular enneadecagon is represented by Schläfli symbol {19}.

The radius of the circumcircle of the regular enneadecagon with side length t is   (angle in degrees). The area, where t is the edge length, is  

Construction

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As 19 is a Pierpont prime but not a Fermat prime, the regular enneadecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisector.

 
Approximated enneadecagon, inscribed in a circle

Symmetry

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Symmetries of a regular enneadecagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.

The regular enneadecagon has Dih19 symmetry, order 38. Since 19 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z19, and Z1.

These 4 symmetries can be seen in 4 distinct symmetries on the enneadecagon. John Conway labels these by a letter and group order.[1] Full symmetry of the regular form is r38 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 g19 subgroup has no degrees of freedom but can seen as directed edges.

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A enneadecagram is a 19-sided star polygon. There are eight regular forms given by Schläfli symbols: {19/2}, {19/3}, {19/4}, {19/5}, {19/6}, {19/7}, {19/8}, and {19/9}. Since 19 is prime, all enneadecagrams are regular stars and not compound figures.

Picture  
{19/2}
 
{19/3}
 
{19/4}
 
{19/5}
Interior angle ≈142.105° ≈123.158° ≈104.211° ≈85.2632°
Picture  
{19/6}
 
{19/7}
 
{19/8}
 
{19/9}
Interior angle ≈66.3158° ≈47.3684° ≈28.4211° ≈9.47368°

Petrie polygons

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The regular enneadecagon is the Petrie polygon for one higher-dimensional polytope, projected in a skew orthogonal projection:

 
18-simplex (18D)

References

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  1. ^ 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)
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Category:Polygons by the number of sides

Icosidigon

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Regular icosidigon
 
A regular icosidigon
TypeRegular polygon
Edges and vertices22
Schläfli symbol{22}, t{11}
Coxeter–Dynkin diagrams    
   
Symmetry groupDihedral (D22), order 2×22
Internal angle (degrees)≈163.636°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, an icosidigon (or icosikaidigon) or 22-gon is a twenty-two-sided polygon. The sum of any icosidigon's interior angles is 3600 degrees.

Regular icosidigon

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The regular icosidigon is represented by Schläfli symbol {22} and can also be constructed as a truncated hendecagon, t{11}.

The area of a regular icosidigon is: (with t = edge length)

 

Construction

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As 22 = 2 × 11, the icosidigon can be constructed by truncating a regular hendecagon. However, the icosidigon is not constructible with a compass and straightedge, since 11 is not a Fermat prime. Consequently, the icosidigon cannot be constructed even with an angle trisector, because 11 is not a Pierpont prime. It can, however, be constructed with the neusis method.

Symmetry

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The regular icosidigon has Dih22 symmetry, order 44. There are 3 subgroup dihedral symmetries: Dih11, Dih2, and Dih1, and 4 cyclic group symmetries: Z22, Z11, Z2, and Z1.

These 8 symmetries can be seen in 10 distinct symmetries on the icosidigon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order.[1] The full symmetry of the regular form is r44 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 n are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g22 subgroup has no degrees of freedom but can seen as directed edges.

The highest symmetry irregular icosidigons are d22, an isogonal icosidigon constructed by eleven mirrors which can alternate long and short edges, and p22, an isotoxal icosidigon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular icosidigon.

Dissection

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22-gon with 220 rhombs

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. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular icosidigon, m=11, and it can be divided into 55: 5 sets of 11 rhombs. This decomposition is based on a Petrie polygon projection of a 11-cube.[2]

Examples
 
11-cube
       
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An icosidigram is a 22-sided star polygon. There are 4 regular forms given by Schläfli symbols: {22/3}, {22/5}, {22/7}, and {22/9}. There are also 7 regular star figures using the same vertex arrangement: 2{11}, 11{2}.

There are also isogonal icosidigrams constructed as deeper truncations of the regular hendecagon {11} and hendecagrams {11/2}, {11/3}, {11/4} and {11/5}.[3]

References

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  1. ^ 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)
  2. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  3. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

Icosihexagon

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Regular icosihexagon
 
A regular icosihexagon
TypeRegular polygon
Edges and vertices26
Schläfli symbol{26}, t{13}
Coxeter–Dynkin diagrams    
   
Symmetry groupDihedral (D26), order 2×26
Internal angle (degrees)≈166.154°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, an icosihexagon (or icosikaihexagon) or 26-gon is a twenty-six-sided polygon. The sum of any icosihexagon's interior angles are 4320° .

Regular icosihexagon

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The regular icosihexagon is represented by Schläfli symbol {26} and can also be constructed as a truncated tridecagon, t{13}.

The area of a regular icosihexagon is: (with t = edge length)

 

Construction

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As 26 = 2 × 13, the icosihexagon can be constructed by truncating a regular tridecagon. However, the icosihexagon is not constructible with a compass and straightedge, since 13 is not a Fermat prime. It can be constructed with an angle trisector, since 13 is a Pierpont prime.

Symmetry

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The regular icosihexagon has Dih26 symmetry, order 52. There are 3 subgroup dihedral symmetries: Dih11, Dih2, and Dih1, and 4 cyclic group symmetries: Z26, Z13, Z2, and Z1.

These 8 symmetries can be seen in 10 distinct symmetries on the icosihexagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order.[1] The full symmetry of the regular form is r52 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 n are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g26 subgroup has no degrees of freedom but can seen as directed edges.

The highest symmetry irregular icosihexagons are d26, an isogonal icosihexagon constructed by thirteen mirrors which can alternate long and short edges, and p26, an isotoxal icosihexagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular icosihexagon.

Dissection

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26-gon with 312 rhombs

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. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular icosihexagon, m=13, and it can be divided into 78: 6 sets of 13 rhombs. This decomposition is based on a Petrie polygon projection of a 13-cube.[2]

Examples
       
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An icosihexagram is a 26-sided star polygon. There are 5 regular forms given by Schläfli symbols: {26/3}, {26/5}, {26/7}, {26/9}, and {26/11}.

 
{26/3}
 
{26/5}
 
{26/7}
 
{26/9}
 
{26/11}

There are also isogonal icosihexagrams constructed as deeper truncations of the regular tridecagon {13} and tridecagrams {13/2}, {13/3}, {13/4}, {13/5} and {13/6}.[3]

References

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  1. ^ 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)
  2. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  3. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

Icosioctagon

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Regular icosioctagon
 
A regular icosioctagon
TypeRegular polygon
Edges and vertices28
Schläfli symbol{28}, t{14}
Coxeter–Dynkin diagrams    
   
Symmetry groupDihedral (D28), order 2×28
Internal angle (degrees)≈167.143°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, an icosioctagon (or icosikaioctagon) or 28-gon is a twenty eight sided polygon. The sum of any icosioctagon's interior angles is 4680 degrees.

Regular icosioctagon

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The regular icosioctagon is represented by Schläfli symbol {28} and can also be constructed as a truncated tetradecagon, t{14}, or a twice-truncated heptagon, tt{7}.

The area of a regular icosioctagon(28 sided polygon) is: (with t = edge length)

 

Construction

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As 28 = 22 × 7, the icosioctagon is not constructible with a compass and straightedge, since 7 is not a Fermat prime. However, it can be constructed with an angle trisector, because 7 is a Pierpont prime.

Symmetry

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The regular icosioctagon has Dih28 symmetry, order 56. There are 5 subgroup dihedral symmetries: (Dih14, Dih7), and (Dih4, Dih2, and Dih1), and 6 cyclic group symmetries: (Z28, Z14, Z7), and (Z4, Z2, Z1).

These 10 symmetries can be seen in 16 distinct symmetries on the icosioctagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order.[1] The full symmetry of the regular form is r56 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 g28 subgroup has no degrees of freedom but can seen as directed edges.

The highest symmetry irregular icosioctagons are d28, an isogonal icosioctagon constructed by ten mirrors which can alternate long and short edges, and p28, an isotoxal icosioctagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular icosioctagon.

Dissection

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28-gon with 364 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. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular icosioctagon, m = 14, and it can be divided into 91: 7 squares and 6 sets of 14 rhombs. This decomposition is based on a Petrie polygon projection of a 14-cube.[2]

Examples
     
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An icosioctagram is a 28-sided star polygon. There are 5 regular forms given by Schläfli symbols: {28/3}, {28/5}, {28/9}, {28/11} and {28/13}.

 
{28/3}
 
{28/5}
 
{28/9}
 
{28/11}
 
{28/13}

There are also isogonal icosioctagrams constructed as deeper truncations of the regular tetradecagon {14} and tetradecagrams {28/3}, {28/5}, {28/9}, and {28/11}.[3]

References

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  1. ^ 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)
  2. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p. 141
  3. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum


Triacontadigon

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Regular triacontadigon
 
A regular triacontadigon
TypeRegular polygon
Edges and vertices32
Schläfli symbol{32}, t{16}, tt{8}, ttt{4}
Coxeter–Dynkin diagrams    
   
Symmetry groupDihedral (D32), order 2×32
Internal angle (degrees)168.75°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, a triacontadigon (or triacontakaidigon) or 32-gon is a thirty-two-sided polygon. In Greek, the prefix triaconta- means 30 and di- means 2. The sum of any triacontadigon's interior angles is 5400 degrees.

An older name is tricontadoagon.[1] Another name is icosidodecagon, suggesting a (20 and 12)-gon, in parallel to the 32-faced icosidodecahedron, which has 20 triangles and 12 pentagons.[2]

Regular triacontadigon

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The regular triacontadigon can be constructed as a truncated hexadecagon, t{16}, a twice-truncated octagon, tt{8}, and a thrice-truncated square. A truncated triacontadigon, t{32}, is a hexacontatetragon, {64}.

One interior angle in a regular triacontadigon is 16834°, meaning that one exterior angle would be 1114°.

The area of a regular triacontadigon is (with t = edge length)

 

and its inradius is

 

The circumradius of a regular triacontadigon is

 

Construction

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As 32 = 25 (a power of two), the regular triacontadigon is a constructible polygon. It can be constructed by an edge-bisection of a regular hexadecagon.[3]

Symmetry

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  The symmetries of a regular triacontadigon. Lines of reflections are blue through vertices, and purple through edges. Gyrations are given as numbers in the center. Vertices are colored by their symmetry positions.

The regular triacontadigon has Dih32 dihedral symmetry, order 64, represented by 32 lines of reflection. Dih32 has 5 dihedral subgroups: Dih16, Dih8, Dih4, Dih2 and Dih1 and 6 more cyclic symmetries: Z32, Z16, Z8, Z4, Z2, and Z1, with Zn representing π/n radian rotational symmetry.

On the regular triacontadigon, there are 17 distinct symmetries. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[4] He gives r64 for the full reflective symmetry, Dih16, 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 triacontadigons. Only the g32 subgroup has no degrees of freedom but can seen as directed edges.

Dissection

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32-gon with 480 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.[5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular triacontadigon, m=16, and it can be divided into 120: 8 squares and 7 sets of 16 rhombs. This decomposition is based on a Petrie polygon projection of a 16-cube.

Examples
       

Triacontadigram

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A triacontadigram is a 32-sided star polygon. There are seven regular forms given by Schläfli symbols {32/3}, {32/5}, {32/7}, {32/9}, {32/11}, {32/13}, and {32/15}, and eight compound star figures with the same vertex configuration.

Many isogonal triacontadigrams can also be constructed as deeper truncations of the regular hexadecagon {16} and hexadecagrams {16/3}, {16/5}, and {16/7}. These also create four quasitruncations: t{16/9} = {32/9}, t{16/11} = {32/11}, t{16/13} = {32/13}, and t{16/15} = {32/15}. Some of the isogonal triacontadigrams are depicted below as part of the aforementioned truncation sequences.[6]

References

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  1. ^ A Mathematical Solution Book Containing Systematic Solutions to Many of the Most Difficult Problems by Benjamin Franklin Finkel
  2. ^ Weisstein, Eric W. "Icosidodecagon". MathWorld.
  3. ^ Constructible Polygon
  4. ^ 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)
  5. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  6. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

Triacontatetragon

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Regular triacontatetragon
 
A regular triacontatetragon
TypeRegular polygon
Edges and vertices34
Schläfli symbol{34}, t{17}
Coxeter–Dynkin diagrams    
   
Symmetry groupDihedral (D34), order 2×34
Internal angle (degrees)169.412°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, a triacontatetragon or triacontakaitetragon is a thirty-four-sided polygon or 34-gon.[1] The sum of any triacontatetragon's interior angles is 5760 degrees.

Regular triacontatetragon

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A regular triacontatetragon is represented by Schläfli symbol {34} and can also be constructed as a truncated 17-gon, t{17}, which alternates two types of edges.

One interior angle in a regular triacontatetragon is (2880/17)°, meaning that one exterior angle would be (180/17)°.

The area of a regular triacontatetragon is (with t = edge length)

 

and its inradius is

 

The factor   is a root of the equation  .

The circumradius of a regular triacontatetragon is

 

As 34 = 2 × 17 and 17 is a Fermat prime, a regular triacontatetragon is constructible using a compass and straightedge.[2][3][4] As a truncated 17-gon, it can be constructed by an edge-bisection of a regular 17-gon. This means that the values of   and   may be expressed in terms of nested radicals.

Symmetry

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The regular triacontatetragon has Dih34 symmetry, order 68. There are 3 subgroup dihedral symmetries: Dih17, Dih2, and Dih1, and 4 cyclic group symmetries: Z34, Z17, Z2, and Z1.

These 8 symmetries can be seen in 10 distinct symmetries on the icosidigon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order.[5] The full symmetry of the regular form is labeled r68 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 n are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g34 subgroup has no degrees of freedom but can seen as directed edges.

The highest symmetry irregular triacontatetragons are d34, an isogonal triacontatetragon constructed by seventeen mirrors which can alternate long and short edges, and p34, an isotoxal triacontatetragon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular triacontatetragon.

Dissection

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34-gon with 544 rhombs

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.[6] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular triacontatetragon, m=17, it can be divided into 136: 8 sets of 17 rhombs. This decomposition is based on a Petrie polygon projection of a 17-cube.

Examples
         

Triacontatetragram

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A triacontatetragram is a 34-sided star polygon. There are seven regular forms given by Schläfli symbols {34/3}, {34/5}, {34/7}, {34/9}, {34/11}, {34/13}, and {34/15}, and nine compound star figures with the same vertex configuration.

 
{34/3}
 
{34/5}
 
{34/7}
 
{34/9}
 
{34/11}
 
{34/13}
 
{34/15}

Many isogonal triacontatetragrams can also be constructed as deeper truncations of the regular heptadecagon {17} and heptadecagrams {17/2}, {17/3}, {17/4}, {17/5}, {17/6}, {17/7}, and {17/8}. These also create eight quasitruncations: t{17/9} = {34/9}, t{17/10} = {34/10}, t{17/11} = {34/11}, t{17/12} = {34/12}, t{17/13} = {34/13}, t{17/14} = {34/14}, t{17/15} = {34/15}, and t{17/16} = {34/16}. Some of the isogonal triacontatetragrams are depicted below, as a truncation sequence with endpoints t{17}={34} and t{17/16}={34/16}.[7]

 
t{17}={34}
   
                 
t{17/16}={34/16}
     
 
t{17/3}={34/3}
                 
t{17/14}={34/14}
 
t{17/5}={34/5}
               
t{17/12}={34/12}
 
t{17/12}={34/12}
 
t{17/7}={34/7}
                 
t{17/10}={34/5}
 
t{17/9}={34/9}
                 
t{17/8}={34/8}

References

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  1. ^ "Ask Dr. Math: Naming Polygons and Polyhedra". mathforum.org. Retrieved 2017-09-05.
  2. ^ W., Weisstein, Eric. "Constructible Polygon". mathworld.wolfram.com. Retrieved 2017-09-01.{{cite web}}: CS1 maint: multiple names: authors list (link)
  3. ^ Chepmell, C. H. (1913-03-01). "A construction of the regular polygon of 34 sides" (PDF). Mathematische Annalen. 74 (1): 150–151. doi:10.1007/bf01455349. ISSN 0025-5831.
  4. ^ White, Charles Edgar (1913). Theory of Irreducible Cases of Equations and Its Applications in Algebra, Geometry, and Trigonometry. p. 79.
  5. ^ 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)
  6. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  7. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

Tetracontagon

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Regular tetracontagon
 
A regular tetracontagon
TypeRegular polygon
Edges and vertices40
Schläfli symbol{40}, t{20}, tt{10}, ttt{5}
Coxeter–Dynkin diagrams    
   
Symmetry groupDihedral (D40), order 2×40
Internal angle (degrees)171°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, a tetracontagon or tessaracontagon is a forty-sided polygon or 40-gon.[1][2] The sum of any tetracontagon's interior angles is 6840 degrees.

Regular tetracontagon

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A regular tetracontagon is represented by Schlafli symbol {40} and can also be constructed as a truncated icosagon, t{20}, which alternates 2 types of edges. Furthermore, it can also be constructed as a twice-truncated decagon, tt{10}, or a thrice-truncated pentagon, ttt{5}.

One interior angle in a regular tetracontagon is 171°, meaning that one exterior angle would be 9°.

The area of a regular tetracontagon is (with t = edge length)

 

and its inradius is

 

The factor   is a root of the octic equation  .

The circumradius of a regular tetracontagon is

 

As 40 = 23 × 5, a regular tetracontagon is constructible using a compass and straightedge.[3] As a truncated icosagon, it can be constructed by an edge-bisection of a regular icosagon. This means that the values of   and   may be expressed in radicals as follows:

 
 

Construction of a regular tetracontagon

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Regular tetracontagon with given circumcircle

Circumcircle is given

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  1. Construct first the side length JE1 of a pentagon.
  2. Transfer this on the circumcircle, there arises the intersection E39.
  3. Connect the point E39 with the central point M, there arises the angle E39ME1 with 72°.
  4. Halve the angle E39ME1, there arise the intersection E40 and the angle E40ME1 with 9°.
  5. Connect the point E1 with the point E40, there arises the first side length a of the tetracontagon.
  6. Finally you transfer the segment E1E40 (side length a) repeatedly counterclockwise on the circumcircle until arises a regular tetracontagon.

The golden ratio

 

Side length is given

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Regular tetracontagon with given side length
(The construction is very similar to that of icosagon with given side length)
  1. Draw a segment E40E1 whose length is the given side length a of the tetracontagon.
  2. Extend the segment E40E1 by more than two times.
  3. Draw each a circular arc about the points E1 and E40, there arise the intersections A and B.
  4. Draw a vertical straight line from point B through point A.
  5. Draw a parallel line too the segment AB from the point E1 to the circular arc, there arises the intersection D.
  6. Draw a circle arc about the point C with the radius CD till to the extension of the side length, there arises the intersection F.
  7. Draw a circle arc about the point E40 with the radius E40F till to the vertical straight line, there arises the intersection G and the angle E40GE1 with 36°.
  8. Draw a circle arc about the point G with radius E40G till to the vertical straight line, there arises the intersection H and the angle E40HE1 with 18°.
  9. Draw a circle arc about the point H with radius E40H till to the vertical straight line, there arises the central point M of the circumcircle and the angle E40ME1 with 9°.
  10. Draw around the central point M with radius E40M the circumcircle of the tetracontagon.
  11. Finally transfer the segment E40E1 (side length a) repeatedly counterclockwise on the circumcircle until to arises a regular tetracontagon.

The golden ratio

 

Symmetry

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The symmetries of a regular tetracontagon. Light blue lines show subgroups of index 2. The left and right subgraphs are positionally related by index 5 subgroups.

The regular tetracontagon has Dih40 dihedral symmetry, order 80, represented by 40 lines of reflection. Dih40 has 7 dihedral subgroups: (Dih20, Dih10, Dih5), and (Dih8, Dih4, Dih2, Dih1). It also has eight more cyclic symmetries as subgroups: (Z40, Z20, Z10, Z5), and (Z8, Z4, Z2, Z1), with Zn representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[4] 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 tetracontagons. Only the g40 subgroup has no degrees of freedom but can seen as directed edges.

Dissection

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40-gon with 560 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. These tilings are contained as subsets of vertices, edges and faces in orthogonal projections m-cubes[5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular tetracontagon, m=20, and it can be divided into 190: 10 squares and 9 sets of 20 rhombs. This decomposition is based on a Petrie polygon projection of a 20-cube.

Examples
     

Tetracontagram

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A tetracontagram is a 40-sided star polygon. There are seven regular forms given by Schläfli symbols {40/3}, {40/7}, {40/9}, {40/11}, {40/13}, {40/17}, and {40/19}, and 12 compound star figures with the same vertex configuration.

Regular star polygons {40/k}
Picture  
{40/3}
 
{40/7}
 
{40/9}
 
{40/11}
 
{40/13}
 
{40/17}
 
{40/19}
Interior angle 153° 117° 99° 81° 63° 27°
Regular compound polygons
Picture  
{40/2}=2{20}
 
{40/4}=4{10}
 
{40/5}=5{8}
 
{40/6}=2{20/3}
 
{40/8}=8{5}
 
{40/10}=10{4}
Interior angle 162° 144° 135° 126° 108° 90°
Picture  
{40/12}=4{10/3}
 
{40/14}=2{20/7}
 
{40/15}=5{8/3}
 
{40/16}=8{5/2}
 
{40/18}=2{20/9}
 
{40/20}=20{2}
Interior angle 72° 54° 45° 36° 18°

Many isogonal tetracontagrams can also be constructed as deeper truncations of the regular icosagon {20} and icosagrams {20/3}, {20/7}, and {20/9}. These also create four quasitruncations: t{20/11}={40/11}, t{20/13}={40/13}, t{20/17}={40/17}, and t{20/19}={40/19}. Some of the isogonal tetracontagrams are depicted below, as a truncation sequence with endpoints t{20}={40} and t{20/19}={40/19}.[6]

 
t{20}={40}
   
         
         
t{20/19}={40/19}
     

References

edit
  1. ^ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 165, ISBN 9781438109572.
  2. ^ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
  3. ^ Constructible Polygon
  4. ^ The Symmetries of Things, Chapter 20
  5. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  6. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum
edit

Tetracontadigon

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Regular tetracontadigon
 
A regular tetracontadigon
TypeRegular polygon
Edges and vertices42
Schläfli symbol{42}, t{21}
Coxeter–Dynkin diagrams    
    
Symmetry groupDihedral (D42), order 2×42
Internal angle (degrees)≈171.429°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, a tetracontadigon (or tetracontakaidigon) or 42-gon is a forty-two-sided polygon. (In Greek, the prefix tetraconta- means 40 and di- means 2.) The sum of any tetracontadigon's interior angles is 7200 degrees.

Regular tetracontadigon

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The regular tetracontadigon can be constructed as a truncated icosihenagon, t{21}.

One interior angle in a regular tetracontadigon is 17137°, meaning that one exterior angle would be 847°.

The area of a regular tetracontadigon is (with t = edge length)

 

and its inradius is

 

The circumradius of a regular tetracontadigon is

 

Since 42 = 2 × 3 × 7, a regular tetracontadigon is not constructible using a compass and straightedge,[1] but is constructible if the use of an angle trisector is allowed.[2]

Symmetry

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  The symmetries of a regular tetracontadigon, related as subgroups of index 2, 3, and 7. Lines of reflections are blue through vertices, and purple through edges. Gyrations are given as numbers in the center. Vertices are colored by their symmetry positions.

The regular tetracontadigon has Dih42 dihedral symmetry, order 84, represented by 42 lines of reflection. Dih42 has 7 dihedral subgroups: Dih21, (Dih14, Dih7), (Dih6, Dih3), and (Dih2, Dih1) and 8 more cyclic symmetries: (Z42, Z21), (Z14, Z7), (Z6, Z3), and (Z2, Z1), with Zn representing π/n radian rotational symmetry.

These 16 symmetries generate 20 unique symmetries on the regular tetracontadigon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[3] He gives r84 for the full reflective symmetry, Dih42, 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 tetracontadigons. Only the g42 subgroup has no degrees of freedom but can seen as directed edges.

Dissection

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42-gon with 840 rhombs

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.[4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular tetracontadigon, m=21, it can be divided into 210: 10 sets of 21 rhombs. This decomposition is based on a Petrie polygon projection of a 21-cube.

Examples
       
edit

 
An equilateral triangle, a regular heptagon, and a regular tetracontadigon can completely fill a plane vertex, one of 17 different combinations of regular polygons with this property.[5] However, the entire plane cannot be tiled with regular polygons while including this vertex figure,[6] though it can be used in a tiling with equilateral polygons and rhombi.[7]

Tetracontadigram

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A tetracontadigram is a 42-sided star polygon. There are five regular forms given by Schläfli symbols {42/5}, {42/11}, {42/13}, {42/17}, and {42/19}, as well as 15 compound star figures with the same vertex configuration.

Regular star polygons {42/k}
Picture  
{42/5}
 
{42/11}
 
{42/13}
 
{42/17}
 
{42/19}
Interior angle ≈137.143° ≈85.7143° ≈68.5714° ≈34.2857° ≈17.1429°

References

edit
  1. ^ Constructible Polygon
  2. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2015-07-14. Retrieved 2015-02-19.{{cite web}}: CS1 maint: archived copy as title (link)
  3. ^ 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)
  4. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  5. ^ Dallas, Elmslie William (1855), The Elements of Plane Practical Geometry, Etc, John W. Parker & Son, p. 134.
  6. ^ [1] Topics in Mathematics for Elementary Teachers: A Technology-enhanced ... By Sergei Abramovich
  7. ^ Shield - a 3.7.42 tiling

Tetracontaoctagon

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Regular tetracontaoctagon
 
A regular tetracontaoctagon
TypeRegular polygon
Edges and vertices48
Schläfli symbol{48}, t{24}, tt{12}, ttt{6}, tttt{3}
Coxeter–Dynkin diagrams    
    
Symmetry groupDihedral (D48), order 2×48
Internal angle (degrees)172.5°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, a tetracontaoctagon (or tetracontakaioctagon) or 48-gon is a forty-eight sided polygon. The sum of any tetracontaoctagon's interior angles is 8280 degrees.

Regular tetracontaoctagon

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The regular tetracontaoctagon is represented by Schläfli symbol {48} and can also be constructed as a truncated icositetragon, t{24}, or a twice-truncated dodecagon, tt{12}, or a thrice-truncated hexagon, ttt{6}, or a fourfold-truncated triangle, tttt{3}.

One interior angle in a regular tetracontaoctagon is 17212°, meaning that one exterior angle would be 712°.

The area of a regular tetracontaoctagon is: (with t = edge length)

 

The tetracontaoctagon appeared in Archimedes' polygon approximation of pi, along with the hexagon (6-gon), dodecagon (12-gon), icositetragon (24-gon), and enneacontahexagon (96-gon).

Construction

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Since 48 = 24 × 3, a regular tetracontaoctagon is constructible using a compass and straightedge.[1] As a truncated icositetragon, it can be constructed by an edge-bisection of a regular icositetragon.

Symmetry

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Symmetries of a regular tetracontaoctagon

The regular tetracontaoctagon has Dih48 symmetry, order 96. There are nine subgroup dihedral symmetries: (Dih24, Dih12, Dih6, Dih3), and (Dih16, Dih8, Dih4, Dih2 Dih1), and 10 cyclic group symmetries: (Z48, Z24, Z12, Z6, Z3), and (Z16, Z8, Z4, Z2, Z1).

These 20 symmetries can be seen in 28 distinct symmetries on the tetracontaoctagon. John Conway labels these by a letter and group order.[2] The full symmetry of the regular form is r96 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 g48 subgroup has no degrees of freedom but can seen as directed edges.

Dissection

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48-gon with 1104 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 tetracontaoctagon, m=24, and it can be divided into 276: 12 squares and 11 sets of 24 rhombs. This decomposition is based on a Petrie polygon projection of a 24-cube.

Examples
     

Tetracontaoctagram

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A tetracontaoctagram is a 48-sided star polygon. There are seven regular forms given by Schläfli symbols {48/5}, {48/7}, {48/11}, {48/13}, {48/17}, {48/19}, and {48/23}, as well as 16 compound star figures with the same vertex configuration.

Regular star polygons {48/k}
Picture  
{48/5}
 
{48/7}
 
{48/11}
 
{48/13}
 
{48/17}
 
{48/19}
 
{48/23}
Interior angle 142.5° 127.5° 97.5° 82.5° 52.5° 37.5° 7.5°

References

edit
  1. ^ Constructible Polygon
  2. ^ 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)
  3. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

Pentacontagon

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Regular pentacontagon
 
A regular pentacontagon
TypeRegular polygon
Edges and vertices50
Schläfli symbol{50}, t{25}
Coxeter–Dynkin diagrams    
    
Symmetry groupDihedral (D50), order 2×50
Internal angle (degrees)172.8°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, a pentacontagon or pentecontagon or 50-gon is a fifty-sided polygon.[1][2] The sum of any pentacontagon's interior angles is 8640 degrees.

A regular pentacontagon is represented by Schläfli symbol {50} and can be constructed as a quasiregular truncated icosipentagon, t{25}, which alternates two types of edges.

Regular pentacontagon properties

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One interior angle in a regular pentacontagon is 17245°, meaning that one exterior angle would be 715°.

The area of a regular pentacontagon is (with t = edge length)

 

and its inradius is

 

The circumradius of a regular pentacontagon is

 

Since 50 = 2 × 52, a regular pentacontagon is not constructible using a compass and straightedge,[3] and is not constructible even if the use of an angle trisector is allowed.[4] However, it is constructible using an auxiliary curve (such as the quadratrix of Hippias or an Archimedean spiral), as such curves can be used to divide angles into any number of equal parts. For example, one can construct a 36° angle using compass and straightedge and proceed to quintisect it (divide it into five equal parts) using an Archimedean spiral, giving the 7.2° angle required to construct a pentacontagon.

It is not known if the pentacontagon is neusis-constructible.

Symmetry

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The symmetries of a regular pentacontagon. Light blue lines show subgroups of index 2. The 3 boxed subgraphs are positionally related by index 5 subgroups.

The regular pentacontagon has Dih50 dihedral symmetry, order 100, represented by 50 lines of reflection. Dih50 has 5 dihedral subgroups: Dih25, (Dih10, Dih5), and (Dih2, Dih1). It also has 6 more cyclic symmetries as subgroups: (Z50, Z25), (Z10, Z5), and (Z2, Z1), with Zn representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[5] 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 freedom in defining irregular pentacontagons. Only the g50 subgroup has no degrees of freedom but can seen as directed edges.

Dissection

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50-gon with 1200 rhombs

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.[6] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular pentacontagon, m=25, it can be divided into 300: 12 sets of 25 rhombs. This decomposition is based on a Petrie polygon projection of a 25-cube.

Examples
       

Pentacontagram

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A pentacontagram is a 50-sided star polygon. There are 9 regular forms given by Schläfli symbols {50/3}, {50/7}, {50/9}, {50/11}, {50/13}, {50/17}, {50/19}, {50/21}, and {50/23}, as well as 16 compound star figures with the same vertex configuration.

Regular star polygons {50/k}
Picture  
{503}
 
{507}
 
{509}
 
{5011}
 
5013
Interior angle 158.4° 129.6° 115.2° 100.8° 86.4°
Picture  
{5017 }
 
{5019 }
 
{5021 }
 
{5023 }
 
Interior angle 57.6° 43.2° 28.8° 14.4°  

References

edit
  1. ^ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 120, ISBN 9781438109572.
  2. ^ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
  3. ^ Constructible Polygon
  4. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2015-07-14. Retrieved 2015-02-19.{{cite web}}: CS1 maint: archived copy as title (link)
  5. ^ The Symmetries of Things, Chapter 20
  6. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

Hexacontagon

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Regular hexacontagon
 
A regular hexacontagon
TypeRegular polygon
Edges and vertices60
Schläfli symbol{60}, t{30}, tt{15}
Coxeter–Dynkin diagrams    
    
Symmetry groupDihedral (D60), order 2×60
Internal angle (degrees)174°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, a hexacontagon or hexecontagon or 60-gon is a sixty-sided polygon.[1][2] The sum of any hexacontagon's interior angles is 10440 degrees.

Regular hexacontagon properties

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A regular hexacontagon is represented by Schläfli symbol {60} and also can be constructed as a truncated triacontagon, t{30}, or a twice-truncated pentadecagon, tt{15}. A truncated hexacontagon, t{60}, is a 120-gon, {120}.

One interior angle in a regular hexacontagon is 174°, meaning that one exterior angle would be 6°.

The area of a regular hexacontagon is (with t = edge length)

 

and its inradius is

 

The circumradius of a regular hexacontagon is

 

This means that the trigonometric functions of π/60 can be expressed in radicals.

Constructible

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Since 60 = 22 × 3 × 5, a regular hexacontagon is constructible using a compass and straightedge.[3] As a truncated triacontagon, it can be constructed by an edge-bisection of a regular triacontagon.

Symmetry

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The symmetries of a regular hexacontagon, divided into 4 subgraphs containing index 2 subgroups. Each symmetry within a subgraph is related to the lower connected subgraphs.

The regular hexacontagon has Dih60 dihedral symmetry, order 120, represented by 60 lines of reflection. Dih60 has 11 dihedral subgroups: (Dih30, Dih15), (Dih20, Dih10, Dih5), (Dih12, Dih6, Dih3), and (Dih4, Dih2, Dih1). And 12 more cyclic symmetries: (Z60, Z30, Z15), (Z20, Z10, Z5), (Z12, Z6, Z3), and (Z4, Z2, Z1), with Zn representing π/n radian rotational symmetry.

These 24 symmetries are related to 32 distinct symmetries on the hexacontagon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[4] 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 freedom in defining irregular hexacontagons. Only the g60 symmetry has no degrees of freedom but can seen as directed edges.

Dissection

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60-gon with 1740 rhombs

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. [5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular hexacontagon, m=30, and it can be divided into 435: 15 squares and 14 sets of 30 rhombs. This decomposition is based on a Petrie polygon projection of a 30-cube.

Examples
 
 
 

Hexacontagram

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A hexacontagram is a 60-sided star polygon. There are 7 regular forms given by Schläfli symbols {60/7}, {60/11}, {60/13}, {60/17}, {60/19}, {60/23}, and {60/29}, as well as 22 compound star figures with the same vertex configuration.

Regular star polygons {60/k}
Picture  
{60/7}
 
{60/11}
 
{60/13}
 
{60/17}
 
{60/19}
 
{60/23}
 
{60/29}
Interior angle 138° 114° 102° 78° 66° 42°

References

edit
  1. ^ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 78, ISBN 9781438109572.
  2. ^ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
  3. ^ Constructible Polygon
  4. ^ 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)
  5. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

Hexacontatetragon

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Regular hexacontatetragon
 
A regular hexacontatetragon
TypeRegular polygon
Edges and vertices64
Schläfli symbol{64}, t{32}, tt{16}, ttt{8}, tttt{4}
Coxeter–Dynkin diagrams    
    
Symmetry groupDihedral (D64), order 2×64
Internal angle (degrees)174.375°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, a hexacontatetragon (or hexacontakaitetragon) or 64-gon is a sixty-four-sided polygon. (In Greek, the prefix hexaconta- means 60 and tetra- means 4.) The sum of any hexacontatetragon's interior angles is 11160 degrees.

Regular hexacontatetragon

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The regular hexacontatetragon can be constructed as a truncated triacontadigon, t{32}, a twice-truncated hexadecagon, tt{16}, a thrice-truncated octagon, ttt{8}, a fourfold-truncated square, tttt{4}, and a fivefold-truncated digon, ttttt{2}.

One interior angle in a regular hexacontatetragon is 17438°, meaning that one exterior angle would be 558°.

The area of a regular hexacontatetragon is (with t = edge length)

 

and its inradius is

 

The circumradius of a regular hexacontatetragon is

 

Construction

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Since 64 = 26 (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 edge-bisection of a regular triacontadigon.

Symmetry

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Symmetries of hexacontatetragons

The regular hexacontatetragon has Dih64 dihedral symmetry, order 128, represented by 64 lines of reflection. Dih64 has 6 dihedral subgroups: Dih32, Dih16, Dih8, Dih4, Dih2 and Dih1 and 7 more cyclic symmetries: Z64, Z32, Z16, Z8, Z4, Z2, and Z1, with Zn 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, Dih64, 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.

Dissection

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64-gon with 1740 rhombs

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 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 32-cube.

Examples
     

Hexacontatetragram

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A hexacontatetragram is a 64-sided 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.

Regular star polygons {64/k}
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°  

References

edit
  1. ^ Constructible Polygon
  2. ^ 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)
  3. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

Heptacontagon

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Regular heptacontagon
 
A regular heptacontagon
TypeRegular polygon
Edges and vertices70
Schläfli symbol{70}, t{35}
Coxeter–Dynkin diagrams    
    
Symmetry groupDihedral (D70), order 2×70
Internal angle (degrees)≈174.857°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, a heptacontagon (or hebdomecontagon from Ancient Greek ἑβδομήκοντα, seventy[1]) or 70-gon is a seventy-sided polygon.[2][3] The sum of any heptacontagon's interior angles is 12240 degrees.

A regular heptacontagon is represented by Schläfli symbol {70} and can also be constructed as a truncated triacontapentagon, t{35}, which alternates two types of edges.

Regular heptacontagon properties

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One interior angle in a regular heptacontagon is 17467°, meaning that one exterior angle would be 517°.

The area of a regular heptacontagon is (with t = edge length)

 

and its inradius is

 

The circumradius of a regular heptacontagon is

 

Since 70 = 2 × 5 × 7, a regular heptacontagon is not constructible using a compass and straightedge,[4] but is constructible if the use of an angle trisector is allowed.[5]

Symmetry

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The symmetries of a regular heptacontagon. Light blue lines show subgroups of index 2. The four subgraphs are positionally related by index 5 and index 7 subgroups.

The regular heptacontagon has Dih70 dihedral symmetry, order 140, represented by 70 lines of reflection. Dih70 has 7 dihedral subgroups: Dih35, (Dih14, Dih7), (Dih10, Dih5), and (Dih2, Dih1). It also has 8 more cyclic symmetries as subgroups: (Z70, Z35), (Z14, Z7), (Z10, Z5), and (Z2, Z1), with Zn representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[6] 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 heptacontagons. Only the g70 subgroup has no degrees of freedom but can seen as directed edges.

Dissection

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70-gon with 2380 rhombs

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.[7] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular heptacontagon, m=35, it can be divided into 595: 17 sets of 35 rhombs. This decomposition is based on a Petrie polygon projection of a 35-cube.

Examples
       

Heptacontagram

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A heptacontagram is a 70-sided star polygon. There are 11 regular forms given by Schläfli symbols {70/3}, {70/9}, {70/11}, {70/13}, {70/17}, {70/19}, {70/23}, {70/27}, {70/29}, {70/31}, and {70/33}, as well as 23 regular star figures with the same vertex configuration.

Regular star polygons {70/k}
Picture  
{70/3}
 
{70/9}
 
{70/11}
 
{70/13}
 
{70/17}
 
{70/19}
Interior angle ≈164.571° ≈133.714° ≈123.429° ≈113.143° ≈92.5714° ≈82.2857°
Picture  
{70/23}
 
{70/27}
 
{70/29}
 
{70/31}
 
{70/33}
 
Interior angle ≈61.7143° ≈41.1429° ≈30.8571° ≈20.5714° ≈10.2857°  

References

edit
  1. ^ Greek Numbers and Numerals (Ancient and Modern) by Harry Foundalis
  2. ^ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 77, ISBN 9781438109572.
  3. ^ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
  4. ^ Constructible Polygon
  5. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2015-07-14. Retrieved 2015-02-19.{{cite web}}: CS1 maint: archived copy as title (link)
  6. ^ The Symmetries of Things, Chapter 20
  7. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

Octacontagon

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Regular octacontagon
 
A regular octacontagon
TypeRegular polygon
Edges and vertices80
Schläfli symbol{80}, t{40}, tt{20}, ttt{10}, tttt{5}
Coxeter–Dynkin diagrams    
    
Symmetry groupDihedral (D80), order 2×80
Internal angle (degrees)175.5°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, an octacontagon (or ogdoëcontagon or 80-gon from Ancient Greek ὁγδοήκοντα, eighty[1]) is an eighty-sided polygon.[2][3] The sum of any octacontagon's interior angles is 14040 degrees.

Regular octacontagon

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A regular octacontagon is represented by Schläfli symbol {80} and can also be constructed as a truncated tetracontagon, t{40}, or a twice-truncated icosagon, tt{20}, or a thrice-truncated decagon, ttt{10}, or a four-fold-truncated pentagon, tttt{5}.

One interior angle in a regular octacontagon is 17512°, meaning that one exterior angle would be 412°.

The area of a regular octacontagon is (with t = edge length)

 

and its inradius is

 

The circumradius of a regular octacontagon is

 

Construction

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Since 80 = 24 × 5, a regular octacontagon is constructible using a compass and straightedge.[4] As a truncated tetracontagon, it can be constructed by an edge-bisection of a regular tetracontagon. This means that the trigonometric functions of π/80 can be expressed in radicals:

 
 
 

Symmetry

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The symmetries of a regular octacontagon. Light blue lines show subgroups of index 2. The left and right subgraphs are positionally related by index 5 subgroups.

The regular octacontagon has Dih80 dihedral symmetry, order 80, represented by 80 lines of reflection. Dih40 has 9 dihedral subgroups: (Dih40, Dih20, Dih10, Dih5), and (Dih16, Dih8, Dih4, and Dih2, Dih1). It also has 10 more cyclic symmetries as subgroups: (Z80, Z40, Z20, Z10, Z5), and (Z16, Z8, Z4, Z2, Z1), with Zn representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[5] r160 represents full symmetry and a1 labels 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.

These lower symmetries allows degrees of freedoms in defining irregular octacontagons. Only the g80 subgroup has no degrees of freedom but can seen as directed edges.

Dissection

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80-gon with 3120 rhombs

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. [6] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular octacontagon, m=40, and it can be divided into 780: 20 squares and 19 sets of 40 rhombs. This decomposition is based on a Petrie polygon projection of a 40-cube.

Examples
     

Octacontagram

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An octacontagram is an 80-sided star polygon. There are 15 regular forms given by Schläfli symbols {80/3}, {80/7}, {80/9}, {80/11}, {80/13}, {80/17}, {80/19}, {80/21}, {80/23}, {80/27}, {80/29}, {80/31}, {80/33}, {80/37}, and {80/39}, as well as 24 regular star figures with the same vertex configuration.

Regular star polygons {80/k}
Picture  
{80/3}
 
{80/7}
 
{80/9}
 
{80/11}
 
{80/13}
 
{80/17}
 
{80/19}
 
{80/21}
Interior angle 166.5° 148.5° 139.5° 130.5° 121.5° 103.5° 94.5° 85.5°
Picture  
{80/23}
 
{80/27}
 
{80/29}
 
{80/31}
 
{80/33}
 
{80/37}
 
{80/39}
 
Interior angle 76.5° 58.5° 49.5° 40.5° 31.5° 13.5° 4.5°  

References

edit
  1. ^ Greek Numbers and Numerals (Ancient and Modern) by Harry Foundalis
  2. ^ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 110, ISBN 9781438109572.
  3. ^ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
  4. ^ Constructible Polygon
  5. ^ The Symmetries of Things, Chapter 20
  6. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

Enneacontagon

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Regular enneacontagon
 
A regular enneacontagon
TypeRegular polygon
Edges and vertices90
Schläfli symbol{90}, t{45}
Coxeter–Dynkin diagrams    
    
Symmetry groupDihedral (D90), order 2×90
Internal angle (degrees)176°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, an enneacontagon or enenecontagon or 90-gon (from Ancient Greek ἑννενήκοντα, ninety[1]) is a ninety-sided polygon.[2][3] The sum of any enneacontagon's interior angles is 15840 degrees.

A regular enneacontagon is represented by Schläfli symbol {90} and can be constructed as a truncated tetracontapentagon, t{45}, which alternates two types of edges.

Regular enneacontagon properties

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One interior angle in a regular enneacontagon is 176°, meaning that one exterior angle would be 4°.

The area of a regular enneacontagon is (with t = edge length)

 

and its inradius is

 

The circumradius of a regular enneacontagon is

 

Since 90 = 2 × 32 × 5, a regular enneacontagon is not constructible using a compass and straightedge,[4] but is constructible if the use of an angle trisector is allowed.[5]

Symmetry

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The symmetries of a regular enneacontagon, divided into 6 subgraphs containing index 2 subgroups. Each symmetry within a subgraph is related to the lower connected subgraphs by index 3 or 5.

The regular enneacontagon has Dih90 dihedral symmetry, order 180, represented by 90 lines of reflection. Dih90 has 11 dihedral subgroups: Dih45, (Dih30, Dih15), (Dih18, Dih9), (Dih10, Dih5), (Dih6, Dih3), and (Dih2, Dih1). And 12 more cyclic symmetries: (Z90, Z45), (Z30, Z15), (Z18, Z9), (Z10, Z5), (Z6, Z3), and (Z2, Z1), with Zn representing π/n radian rotational symmetry.

These 24 symmetries are related to 30 distinct symmetries on the enneacontagon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[6] 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 freedom in defining irregular enneacontagons. Only the g90 symmetry has no degrees of freedom but can seen as directed edges.

Dissection

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90-gon with 3960 rhombs

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. [7] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular enneacontagon, m=45, it can be divided into 990: 22 sets of 45 rhombs. This decomposition is based on a Petrie polygon projection of a 45-cube.

Examples
       

Enneacontagram

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An enneacontagram is a 90-sided star polygon. There are 11 regular forms given by Schläfli symbols {90/7}, {90/11}, {90/13}, {90/17}, {90/19}, {90/23}, {90/29}, {90/31}, {90/37}, {90/41}, and {90/43}, as well as 33 regular star figures with the same vertex configuration.

Regular star polygons {90/k}
Pictures  
{90/7}
 
{90/11}
 
{90/13}
 
{90/17}
 
{90/19}
 
{90/23}
Interior angle 152° 136° 128° 112° 104° 88°
Pictures  
{90/29}
 
{90/31}
 
{90/37}
 
{90/41}
 
{90/43}
 
Interior angle 64° 56° 32° 16°  

References

edit
  1. ^ Greek Numbers and Numerals (Ancient and Modern) by Harry Foundalis
  2. ^ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 57, ISBN 9781438109572.
  3. ^ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
  4. ^ Constructible Polygon
  5. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2015-07-14. Retrieved 2015-02-19.{{cite web}}: CS1 maint: archived copy as title (link)
  6. ^ 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)
  7. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

Enneacontahexagon

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Regular enneacontahexagon
 
A regular enneacontahexagon
TypeRegular polygon
Edges and vertices96
Schläfli symbol{96}, t{48}, tt{24}, ttt{12}, tttt{6}, ttttt{3}
Coxeter–Dynkin diagrams    
    
Symmetry groupDihedral (D96), order 2×96
Internal angle (degrees)176.25°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, an enneacontahexagon or enneacontakaihexagon or 96-gon is a ninety-six-sided polygon. The sum of any enneacontahexagon's interior angles is 16920 degrees.

Regular enneacontahexagon

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The regular enneacontahexagon is represented by Schläfli symbol {96} and can also be constructed as a truncated tetracontaoctagon, t{48}, or a twice-truncated icositetragon, tt{24}, or a thrice-truncated dodecagon, ttt{12}, or a fourfold-truncated hexagon, tttt{6}, or a fivefold-truncated triangle, ttttt{3}.

One interior angle in a regular enneacontahexagon is 17614°, meaning that one exterior angle would be 334°.

The area of a regular enneacontahexagon is: (with t = edge length)

 

The enneacontahexagon appeared in Archimedes' polygon approximation of pi, along with the hexagon (6-gon), dodecagon (12-gon), icositetragon (24-gon), and tetracontaoctagon (48-gon).

Construction

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Since 96 = 25 × 3, a regular enneacontahexagon is constructible using a compass and straightedge.[1] As a truncated tetracontaoctagon, it can be constructed by an edge-bisection of a regular tetracontaoctagon.

Symmetry

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Symmetries of enneacontahexagon. The symmetries in each box are related as index 2 subgroups. The right box groups are related to the left box as index 3 subgroups.

The regular enneacontahexagon has Dih96 symmetry, order 192. There are 11 subgroup dihedral symmetries: (Dih48, Dih24, Dih12, Dih6, Dih3), (Dih32, Dih16, Dih8, Dih4, Dih2 and Dih1), and 12 cyclic group symmetries: (Z96, Z48, Z24, Z12, Z6, Z3), (Z32, Z16, Z8, Z4, Z2, and Z1).

These 24 symmetries can be seen in 34 distinct symmetries on the enneacontahexagon. John Conway labels these by a letter and group order.[2] The full symmetry of the regular form is r192 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 g96 subgroup has no degrees of freedom but can seen as directed edges.

Dissection

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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 enneacontahexagon, m=48, and it can be divided into 1128: 24 squares and 23 sets of 48 rhombs. This decomposition is based on a Petrie polygon projection of a 48-cube.

Examples
   

Enneacontahexagram

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An enneacontahexagram is a 96-sided star polygon. There are 15 regular forms given by Schläfli symbols {96/5}, {96/7}, {96/11}, {96/13}, {96/17}, {96/19}, {96/23}, {96/25}, {96/29}, {96/31}, {96/35}, {96/37}, {96/41}, {96/43}, and {96/47}, as well as 32 compound star figures with the same vertex configuration.

Regular star polygons {96/k}
Picture  
{96/5}
 
{96/7}
 
{96/11}
 
{96/13}
 
{96/17}
 
{96/19}
 
{96/23}
 
{96/25}
Interior angle 161.25° 153.75° 138.75° 131.25° 116.25° 108.75° 93.75° 86.25°
Picture  
{96/29}
 
{96/31}
 
{96/35}
 
{96/37}
 
{96/41}
 
{96/43}
 
{96/47}
 
Interior angle 71.25° 63.75° 48.75° 41.25° 26.25° 18.75° 3.75°  

References

edit
  1. ^ Constructible Polygon
  2. ^ 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)
  3. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

Hectogon

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Regular hectogon
 
A regular hectogon
TypeRegular polygon
Edges and vertices100
Schläfli symbol{100}, t{50}, tt{25}
Coxeter–Dynkin diagrams    
    
Symmetry groupDihedral (D100), order 2×100
Internal angle (degrees)176.4°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, a hectogon or hecatontagon or 100-gon[1][2] is a hundred-sided polygon.[3][4] The sum of a hectogon's interior angles are 17640 degrees.

Regular hectogon

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A regular hectogon is represented by Schläfli symbol {100} and can be constructed as a truncated pentacontagon, t{50}, or a twice-truncated icosipentagon, tt{25}.

One interior angle in a regular hectogon is 17625°, meaning that one exterior angle would be 335°.

The area of a regular hectogon is (with t = edge length)

 

and its inradius is

 

The circumradius of a regular hectogon is

 

Because 100 = 22 × 52, the number of sides contains a repeated Fermat prime (the number 5). Thus the regular hectogon is not a constructible polygon.[5] Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three.[6]

It is not known if the regular hectogon is neusis constructible. Its neusis constructibility is equivalent to that of the 25-gon, which is an open problem.[7]

However, a hectogon is constructible using an auxiliary curve such as an Archimedean spiral. A 72° angle is constructible with compass and straightedge, so a possible approach to constructing one side of a hectogon is to construct a 72° angle using compass and straightedge, use an Archimedean spiral to construct a 14.4° angle, and bisect one of the 14.4° angles twice.

Exact construction with help the quadratrix of Hippias

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Hectogon, exact construction using the quadratrix of Hippias as an additional aid

Symmetry

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The symmetries of a regular hectogon. Light blue lines show subgroups of index 2. The 3 boxed subgraphs are positionally related by index 5 subgroups.

The regular hectogon has Dih100 dihedral symmetry, order 200, represented by 100 lines of reflection. Dih100 has 8 dihedral subgroups: (Dih50, Dih25), (Dih20, Dih10, Dih5), (Dih4, Dih2, and Dih1). It also has 9 more cyclic symmetries as subgroups: (Z100, Z50, Z25), (Z20, Z10, Z5), and (Z4, Z2, Z1), with Zn representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[8] r200 represents full symmetry and a1 labels 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.

These lower symmetries allows degrees of freedom in defining irregular hectogons. Only the g100 subgroup has no degrees of freedom but can seen as directed edges.

Dissection

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100-gon with 4900 rhombs

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. [9] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular hectogon, m=50, it can be divided into 1225: 25 squares and 24 sets of 50 rhombs. This decomposition is based on a Petrie polygon projection of a 50-cube.

Examples
   

Hectogram

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A hectogram is a 100-sided star polygon. There are 19 regular forms[10] given by Schläfli symbols {100/3}, {100/7}, {100/9}, {100/11}, {100/13}, {100/17}, {100/19}, {100/21}, {100/23}, {100/27}, {100/29}, {100/31}, {100/33}, {100/37}, {100/39}, {100/41}, {100/43}, {100/47}, and {100/49}, as well as 30 regular star figures with the same vertex configuration.

Regular star polygons {100/k}
Picture  
{100/3}
 
{100/7}
 
{100/11}
 
{100/13}
 
{100/17}
 
{100/19}
Interior angle 169.2° 154.8° 140.4° 133.2° 118.8° 111.6°
Picture  
{100/21}
 
{100/23}
 
{100/27}
 
{100/29}
 
{100/31}
 
{100/37}
Interior angle 104.4° 97.2° 82.8° 75.6° 68.4° 46.8°
Picture  
{100/39}
 
{100/41}
 
{100/43}
 
{100/47}
 
{100/49}
 
Interior angle 39.6° 32.4° 25.2° 10.8° 3.6°  

See also

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References

edit
  1. ^ Barnes-Svarney, Patricia; Svarney, Thomas E. (May 2012). The Handy Math Answer Book. ISBN 9781578593866.
  2. ^ Salomon, David (18 September 2011). The Computer Graphics Manual. ISBN 9780857298867.
  3. ^ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 110, ISBN 9781438109572.
  4. ^ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
  5. ^ Constructible Polygon
  6. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2015-07-14. Retrieved 2015-02-19.{{cite web}}: CS1 maint: archived copy as title (link)
  7. ^ Arthur Baragar (2002) Constructions Using a Compass and Twice-Notched Straightedge, The American Mathematical Monthly, 109:2, 151-164, doi:10.1080/00029890.2002.11919848
  8. ^ The Symmetries of Things, Chapter 20
  9. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  10. ^ 19 = 50 cases - 1 (convex) - 10 (multiples of 5) - 25 (multiples of 2)+ 5 (multiples of 2 and 5)

120-gon

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Regular 120-gon
 
A regular 120-gon
TypeRegular polygon
Edges and vertices120
Schläfli symbol{120}, t{60}, tt{30}, ttt{15}
Coxeter–Dynkin diagrams    
    
Symmetry groupDihedral (D120), order 2×120
Internal angle (degrees)177°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, a 120-gon is a polygon with 120 sides. The sum of any 120-gon's interior angles is 21240 degrees.

Alternative names include dodecacontagon and hecatonicosagon.[1]

Regular 120-gon properties

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A regular 120-gon is represented by Schläfli symbol {120} and also can be constructed as a truncated hexacontagon, t{60}, or a twice-truncated triacontagon, tt{30}, or a thrice-truncated pentadecagon, ttt{15}.

One interior angle in a regular 120-gon is 177°, meaning that one exterior angle would be 3°.

The area of a regular 120-gon is (with t = edge length)

 

and its inradius is

 

The circumradius of a regular 120-gon is

 

This means that the trigonometric functions of π/120 can be expressed in radicals.

Constructible

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Since 120 = 23 × 3 × 5, a regular 120-gon is constructible using a compass and straightedge.[2] As a truncated hexacontagon, it can be constructed by an edge-bisection of a regular hexacontagon.

Symmetry

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The symmetries of a regular 120-gon. Symmetries are related as index 2 subgroups in each box. The 4 boxes are related as 3 and 5 index subgroups.

The regular 120-gon has Dih120 dihedral symmetry, order 240, represented by 120 lines of reflection. Dih120 has 15 dihedral subgroups: (Dih60, Dih30, Dih15), (Dih40, Dih20, Dih10, Dih5), (Dih24, Dih12, Dih6, Dih3), and (Dih8, Dih4, Dih2, Dih1). And 16 more cyclic symmetries: (Z120, Z60, Z30, Z15), (Z40, Z20, Z10, Z5), (Z24, Z12, Z6, Z3), and (Z8, Z4, Z2, Z1), with Zn representing π/n radian rotational symmetry.

These 32 symmetries are related to 44 distinct symmetries on the 120-gon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[3] 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 allow degrees of freedom in defining irregular 120-gons. Only the g120 symmetry has no degrees of freedom but can be seen as directed edges.

Dissection

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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.[4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular 120-gon, m=60, and it can be divided into 1770: 30 squares and 29 sets of 60 rhombs. This decomposition is based on a Petrie polygon projection of a 60-cube.

Examples
   

120-gram

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A 120-gram is a 120-sided star polygon. There are 15 regular forms given by Schläfli symbols {120/7}, {120/11}, {120/13}, {120/17}, {120/19}, {120/23}, {120/29}, {120/31}, {120/37}, {120/41}, {120/43}, {120/47}, {120/49}, {120/53}, and {120/59}, as well as 44 compound star figures with the same vertex configuration.

Regular star polygons {120/k}
Picture  
{120}
 
{120/7}
 
{120/11}
 
{120/13}
 
{120/17}
 
{120/19}
 
{120/23}
 
{120/29}
Interior angle 177° 159° 147° 141° 129° 123° 111° 93°
Picture  
{120/31}
 
{120/37}
 
{120/41}
 
{120/43}
 
{120/47}
 
{120/49}
 
{120/53}
 
{120/59}
Interior angle 87° 69° 57° 51° 39° 33° 21°

References

edit
  1. ^ Norman Johnson, Geometries and Transformations (2018), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5 full polychoric groups
  2. ^ Constructible Polygon
  3. ^ 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)
  4. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

360-gon

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In geometry, a 360-gon (triacosiahexecontagon or triacosiahexeacontagon) is a polygon with 360 sides. The sum of any 360-gon's interior angles is 64440 degrees.

Regular 360-gon

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A regular 360-gon is represented by Schläfli symbol {360} and also can be constructed as a truncated 180-gon, t{180}, or a twice-truncated enneacontagon, tt{90}, or a thrice-truncated tetracontapentagon, ttt{45}.

One interior angle in a regular 360-gon is 179°, meaning that one exterior angle would be 1°.

The area of a regular 360-gon is (with t = edge length)

 

and its inradius is

 

The circumradius of a regular 360-gon is

 

Since 360 = 23 × 32 × 5, a regular 360-gon is not constructible using a compass and straightedge,[1] but is constructible if the use of an angle trisector is allowed.[2]

Symmetry

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The symmetries of a regular 360-gon. Symmetries are related as index 2 subgroups in each box. The 6 boxes are related as 3 and 5 index subgroups.

The regular 360-gon has Dih360 dihedral symmetry, order 720, represented by 360 lines of reflection. Dih360 has 23 dihedral subgroups: (Dih180, Dih90, Dih45), (Dih120, Dih60, Dih30, Dih15), (Dih72, Dih36, Dih18, Dih9), (Dih40, Dih20, Dih10, Dih5), (Dih24, Dih12, Dih6, Dih3), and (Dih8, Dih4, Dih2, Dih1). And 24 more cyclic symmetries: (Z360, Z180, Z90, Z45), (Z120, Z60, Z30, Z15), (Z72, Z36, Z18, Z9), (Z40, Z20, Z10, Z5), (Z24, Z12, Z6, Z3), and (Z8, Z4, Z2, Z1), with Zn representing π/n radian rotational symmetry.

These 48 symmetries are related to 66 distinct symmetries on the 360-gon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[3] Full symmetry is r720 and a1 labels 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.

These lower symmetries allows degrees of freedom in defining irregular 360-gons. Only the g360 symmetry has no degrees of freedom but can seen as directed edges.

Dissection

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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.[4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular 360-gon, m=180, and it can be divided into 16110: 90 squares and 89 sets of 180 rhombs. This decomposition is based on a Petrie polygon projection of a 180-cube.

Examples
   

360-gram

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A 360-gram is a 360-sided star polygon. There are 47 regular forms given by Schläfli symbols {360/7}, {360/11}, {360/13}, {360/17}, {360/19}, {360/23}, {360/29}, {360/31}, {360/37}, {360/41}, {360/43}, {360/47}, {360/49}, {360/53}, {360/59}, {360/61}, {360/67}, {360/71}, {360/73}, {360/77}, {360/79}, {360/83}, {360/89}, {360/91}, {360/97}, {360/101}, {360/103}, {360/107}, {360/109}, {360/113}, {360/119}, {360/121}, {360/127}, {360/131}, {360/133}, {360/137}, {360/139}, {360/143}, {360/149}, {360/151}, {360/157}, {360/161}, {360/163}, {360/167}, {360/169}, {360/173}, and {360/179}, as well as 132 compound star figures with the same vertex configuration. Many of the more intricate 360-grams show moiré patterns.

The regular convex and star polygons whose interior angles are some integer number of degrees are precisely those whose numbers of sides are integer divisors of 360 that are not unity, i.e. {2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360}.

References

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  1. ^ Constructible Polygon
  2. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2015-07-14. Retrieved 2015-02-19.{{cite web}}: CS1 maint: archived copy as title (link)
  3. ^ 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)
  4. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141