User:Tomruen/Higher polygons

Enneadecagon

Regular enneadecagon
A regular enneadecagon
Type	Regular polygon
Edges and vertices	19
Schläfli symbol	{19}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D19), order 2×19
Internal angle (degrees)	≈161.052°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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

Regular form

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

The radius of the circumcircle of the regular enneadecagon with side length t is $R={\frac {t}{2}}\csc {\frac {180}{19}}$ (angle in degrees). The area, where t is the edge length, is ${\frac {19}{4}}t^{2}\cot {\frac {\pi }{19}}\simeq 28.4652\,t^{2}.$

Construction

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

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. 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.

Related polygons

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

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

18-simplex (18D)

References

• enneadecagon

External links

• Weisstein, Eric W. "Enneadecagon". MathWorld.

Category:Polygons by the number of sides

Icosidigon

Regular icosidigon
A regular icosidigon
Type	Regular polygon
Edges and vertices	22
Schläfli symbol	{22}, t{11}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D22), order 2×22
Internal angle (degrees)	≈163.636°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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

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)

$A={\frac {11t^{2}}{2}}\cot {\frac {\pi }{22}}.$

Construction

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

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. 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

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.

Examples

11-cube

Related polygons

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}.

Isogonal truncations of regular hendecagon and hendecagrams
Quasiregular	Isogonal					Quasiregular
t{11}={22}						t{11/10}={22/10}
t{11/2}={22/2}	{11/9}:t6	{11/9}:t5	{11/9}:t4	{11/9}:t3	{11/9}:t2	t{11/9}={22/9}
t{11/3} = {22/3}	{11/3}:t2	{11/3}:t3	{11/3}:t4	{11/3}:t5	{11/3}:t6	t{11/8}={22/8}
t{11/4}={22/4}	{11/7}:t6	{11/7}:t5	{11/7}:t4	{11/7}:t3	{11/7}:t2	t{11/7}={22/7}
t{11/5} = {22/5}	{11/5}:t2	{11/5}:t3	{11/5}:t4	{11/5}:t5	{11/5}:t6	t{11/6}={22/6}

References

• Naming Polygons and Polyhedra
• (simple) polygon

Icosihexagon

Regular icosihexagon
A regular icosihexagon
Type	Regular polygon
Edges and vertices	26
Schläfli symbol	{26}, t{13}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D26), order 2×26
Internal angle (degrees)	≈166.154°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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

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)

$A=6.5t^{2}\cot {\frac {\pi }{26}}.$

Construction

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

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. 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

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.

Examples

Related polygons

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}.

Isogonal truncations of regular tridecagon and tridecagrams
Quasiregular	Isogonal						Quasiregular
t{13} = {26}							t{13/12}={26/12}
t{13/2}={26/2}							t{13/11}={26/11}
t{13/3} = {26/3}							t{13/10}={26/10}
t{13/4}={26/4}							t{13/9}={26/9}
t{13/5} = {26/5}							t{13/8}={26/8}
t{13/6}={26/6}							t{13/7}={26/7}

References

• Naming Polygons and Polyhedra
• (simple) polygon

Icosioctagon

Regular icosioctagon
A regular icosioctagon
Type	Regular polygon
Edges and vertices	28
Schläfli symbol	{28}, t{14}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D28), order 2×28
Internal angle (degrees)	≈167.143°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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

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)

$A=7t^{2}\cot {\frac {\pi }{28}}.$

Construction

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

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. 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

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.

Examples

Related polygons

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}.

Isogonal truncations of regular tetradecagon and tetradecagrams
Quasiregular	Isogonal						Quasiregular
t{14} = {28}							t{14/13}={28/13}
t{14/3} = {28/3}							t{14/11}={28/11}
t{14/5} = {28/5}							t{14/9}={28/9}

References

• Naming Polygons and Polyhedra
• (simple) polygon

Triacontadigon

Regular triacontadigon
A regular triacontadigon
Type	Regular polygon
Edges and vertices	32
Schläfli symbol	{32}, t{16}, tt{8}, ttt{4}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D32), order 2×32
Internal angle (degrees)	168.75°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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. Another name is icosidodecagon, suggesting a (20 and 12)-gon, in parallel to the 32-faced icosidodecahedron, which has 20 triangles and 12 pentagons.

Regular triacontadigon

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 1683⁄4°, meaning that one exterior angle would be 111⁄4°.

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

${\begin{aligned}A=&8t^{2}\cot {\frac {\pi }{32}}\\=&8t^{2}\left(1+{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}+{\sqrt {8+4{\sqrt {2}}+2{\sqrt {20+14{\sqrt {2}}}}}}\right)\end{aligned}}$

and its inradius is

$r={\frac {1}{2}}t\cot {\frac {\pi }{32}}$

The circumradius of a regular triacontadigon is

${\begin{aligned}R=&{\frac {1}{2}}t\csc {\frac {\pi }{32}}\\=&{\frac {1}{2}}t\left({\sqrt {16+8{\sqrt {2}}+4{\sqrt {20+14{\sqrt {2}}}}+2{\sqrt {168+116{\sqrt {2}}+2{\sqrt {13780+9742{\sqrt {2}}}}}}}}\right)\end{aligned}}$

Further information: Trigonometric constants expressed in real radicals § 5.625°: regular triacontadigon (32-sided polygon)

Construction

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.

Symmetry

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 Z_n 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. 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

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. 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

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.

Regular star polygons {32/k}
Picture	{32/3}	{32/5}	{32/7}	{32/9}	{32/11}	{32/13}	{32/15}
Interior angle	146.25°	123.75°	101.25°	78.75°	56.25°	33.75°	11.25°

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.

isogonal triacontadigrams
t{16} = {32}								t{16/15}={32/15}
t{16/3} = {32/3}								t{16/13}={32/13}
t{16/5} = {32/5}								t{16/11}={32/11}
t{16/7} = {32/7}								t{16/9}={32/9}

References

• Naming Polygons and Polyhedra
• CRC Concise Encyclopedia of Mathematics, Second Edition, Eric W. Weisstein icosidodecagon

Triacontatetragon

Regular triacontatetragon
A regular triacontatetragon
Type	Regular polygon
Edges and vertices	34
Schläfli symbol	{34}, t{17}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D34), order 2×34
Internal angle (degrees)	169.412°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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

Regular triacontatetragon

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)

$A={\frac {17}{2}}t^{2}\cot {\frac {\pi }{34}}$

and its inradius is

$r={\frac {1}{2}}t\cot {\frac {\pi }{34}}$

The factor $\cot {\frac {\pi }{34}}$ is a root of the equation $x^{16}-136x^{14}+2380x^{12}-12376x^{10}+24310x^{8}-19448x^{6}+6188x^{4}-680x^{2}+17=0$.

The circumradius of a regular triacontatetragon is

$R={\frac {1}{2}}t\csc {\frac {\pi }{34}}$

As 34 = 2 × 17 and 17 is a Fermat prime, a regular triacontatetragon is constructible using a compass and straightedge. As a truncated 17-gon, it can be constructed by an edge-bisection of a regular 17-gon. This means that the values of $\sin {\frac {\pi }{34}}$ and $\cos {\frac {\pi }{34}}$ may be expressed in terms of nested radicals.

Symmetry

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. 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

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. 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

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}.

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}

Tetracontagon

Regular tetracontagon
A regular tetracontagon
Type	Regular polygon
Edges and vertices	40
Schläfli symbol	{40}, t{20}, tt{10}, ttt{5}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D40), order 2×40
Internal angle (degrees)	171°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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

Regular tetracontagon

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)

${\begin{aligned}A=10t^{2}\cot {\frac {\pi }{40}}=&10\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)^{2}+1}}\right)t^{2}\\=&10\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {\left(1+{\sqrt {5}}\right)^{2}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}+\left({\sqrt {5+2{\sqrt {5}}}}\right)^{2}+1}}\right)t^{2}\\=&10\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {\left(6+{\binom {2}{1}}{\sqrt {5}}\right)^{}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}+\left(5+2{\sqrt {5}}\right)^{}+1}}\right)t^{2}\\=&10\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {\left(11+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}\right)+1}}\right)t^{2}\\=&10\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)t^{2}\end{aligned}}$

and its inradius is

$r={\frac {1}{2}}t\cot {\frac {\pi }{40}}$

The factor $\cot {\frac {\pi }{40}}$ is a root of the octic equation $x^{8}-8x^{7}-60x^{6}-8x^{5}+134x^{4}+8x^{3}-60x^{2}+8x+1$.

The circumradius of a regular tetracontagon is

${\begin{aligned}R={\frac {1}{2}}t\csc {\frac {\pi }{40}}\end{aligned}}$

As 40 = 2³ × 5, a regular tetracontagon is constructible using a compass and straightedge. As a truncated icosagon, it can be constructed by an edge-bisection of a regular icosagon. This means that the values of $\sin {\frac {\pi }{40}}$ and $\cos {\frac {\pi }{40}}$ may be expressed in radicals as follows:

$\sin {\frac {\pi }{40}}={\frac {1}{4}}({\sqrt {2}}-1){\sqrt {{\frac {1}{2}}(2+{\sqrt {2}})(5+{\sqrt {5}})}}-{\frac {1}{8}}{\sqrt {2-{\sqrt {2}}}}(1+{\sqrt {2}})({\sqrt {5}}-1)$

$\cos {\frac {\pi }{40}}={\frac {1}{8}}({\sqrt {2}}-1){\sqrt {2+{\sqrt {2}}}}({\sqrt {5}}-1)+{\frac {1}{4}}(1+{\sqrt {2}}){\sqrt {{\frac {1}{2}}(2-{\sqrt {2}})(5+{\sqrt {5}})}}$

Construction of a regular tetracontagon

Regular tetracontagon with given circumcircle

Circumcircle is given

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

${\frac {\overline {JM}}{\overline {BJ}}}={\frac {\overline {BM}}{\overline {JM}}}={\frac {1+{\sqrt {5}}}{2}}=\varphi \approx 1.618$

Side length is given

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

${\frac {\overline {E_{40}E_{1}}}{\overline {E_{1}F}}}={\frac {\overline {E_{40}F}}{\overline {E_{40}E_{1}}}}={\frac {1+{\sqrt {5}}}{2}}=\varphi \approx 1.618$

Symmetry

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 Z_n representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. 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

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 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

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°	9°

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°	0°

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}.

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

External links

• Weisstein, Eric W. "Tetracontagon". MathWorld.
• Naming Polygons and Polyhedra
• tessaracontagon

Tetracontadigon

Regular tetracontadigon
A regular tetracontadigon
Type	Regular polygon
Edges and vertices	42
Schläfli symbol	{42}, t{21}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D42), order 2×42
Internal angle (degrees)	≈171.429°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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

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

One interior angle in a regular tetracontadigon is 1713⁄7°, meaning that one exterior angle would be 84⁄7°.

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

$A=10.5t^{2}\cot {\frac {\pi }{42}}$

and its inradius is

$r={\frac {1}{2}}t\cot {\frac {\pi }{42}}$

The circumradius of a regular tetracontadigon is

$R={\frac {1}{2}}t\csc {\frac {\pi }{42}}$

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

Symmetry

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 Z_n 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. 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

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. 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

Related polygons

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. However, the entire plane cannot be tiled with regular polygons while including this vertex figure, though it can be used in a tiling with equilateral polygons and rhombi.

Tetracontadigram

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

• Naming Polygons and Polyhedra

Tetracontaoctagon

Regular tetracontaoctagon
A regular tetracontaoctagon
Type	Regular polygon
Edges and vertices	48
Schläfli symbol	{48}, t{24}, tt{12}, ttt{6}, tttt{3}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D48), order 2×48
Internal angle (degrees)	172.5°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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

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}.

Further information: Trigonometric_constants_expressed_in_real_radicals § 3.75°: regular tetracontaoctagon (48-sided polygon)

One interior angle in a regular tetracontaoctagon is 1721⁄2°, meaning that one exterior angle would be 71⁄2°.

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

${\begin{aligned}A&=12t^{2}\cot {\frac {\pi }{48}}\\&=12t^{2}\left(2+{\sqrt {3}}+{\sqrt {8+4{\sqrt {3}}}}+{\sqrt {16+8{\sqrt {3}}+2{\sqrt {104+60{\sqrt {3}}}}}}\right)\\&=12t^{2}\left(2+{\sqrt {3}}+({\sqrt {6}}+{\sqrt {2}})+{\sqrt {16+8{\sqrt {3}}+10{\sqrt {2}}+6{\sqrt {6}}}}\right)\\&=12t^{2}\left(2+{\sqrt {3}}+({\sqrt {6}}+{\sqrt {2}})+2{\sqrt {4+2{\sqrt {3}}+{\sqrt {26+15{\sqrt {3}}}}}}\right).\end{aligned}}$

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

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

Symmetry

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. 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

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. 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

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

• Naming Polygons and Polyhedra

Pentacontagon

Regular pentacontagon
A regular pentacontagon
Type	Regular polygon
Edges and vertices	50
Schläfli symbol	{50}, t{25}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D50), order 2×50
Internal angle (degrees)	172.8°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

In geometry, a pentacontagon or pentecontagon or 50-gon is a fifty-sided polygon. 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

One interior angle in a regular pentacontagon is 1724⁄5°, meaning that one exterior angle would be 71⁄5°.

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

$A={\frac {25}{2}}t^{2}\cot {\frac {\pi }{50}}$

and its inradius is

$r={\frac {1}{2}}t\cot {\frac {\pi }{50}}$

The circumradius of a regular pentacontagon is

$R={\frac {1}{2}}t\csc {\frac {\pi }{50}}$

Since 50 = 2 × 5², a regular pentacontagon is not constructible using a compass and straightedge, and is not constructible even if the use of an angle trisector is allowed. 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

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 Z_n representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. 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

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. 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

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	{50⁄3}	{50⁄7}	{50⁄9}	{50⁄11}	50⁄13
Interior angle	158.4°	129.6°	115.2°	100.8°	86.4°
Picture	{50⁄17 }	{50⁄19 }	{50⁄21 }	{50⁄23 }
Interior angle	57.6°	43.2°	28.8°	14.4°

References

• Weisstein, Eric W. "Pentacontagon". MathWorld.
• Naming Polygons and Polyhedra
• Pentecontagon

Hexacontagon

Regular hexacontagon
A regular hexacontagon
Type	Regular polygon
Edges and vertices	60
Schläfli symbol	{60}, t{30}, tt{15}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D60), order 2×60
Internal angle (degrees)	174°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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

Regular hexacontagon properties

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)

$A=15t^{2}\cot {\frac {\pi }{60}}$

and its inradius is

$r={\frac {1}{2}}t\cot {\frac {\pi }{60}}$

The circumradius of a regular hexacontagon is

$R={\frac {1}{2}}t\csc {\frac {\pi }{60}}$

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

Constructible

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

Symmetry

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 Z_n 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. 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

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. 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

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°	6°

References

• Weisstein, Eric W. "Hexacontagon". MathWorld.
• Naming Polygons and Polyhedra
• Hexacontagon

Hexacontatetragon

Regular hexacontatetragon
A regular hexacontatetragon
Type	Regular polygon
Edges and vertices	64
Schläfli symbol	{64}, t{32}, tt{16}, ttt{8}, tttt{4}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D64), order 2×64
Internal angle (degrees)	174.375°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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

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 1743⁄8°, meaning that one exterior angle would be 55⁄8°.

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

$A=16t^{2}\cot {\frac {\pi }{64}}$

and its inradius is

$r={\frac {t}{2}}\cot {\frac {\pi }{64}}$

The circumradius of a regular hexacontatetragon is

$R={\frac {t}{2}}\csc {\frac {\pi }{64}}$

Further information: Trigonometric constants expressed in real radicals § 2.8125°: regular hexacontatetragon (64-sided polygon)

Construction

Since 64 = 2⁶ (a power of two), a regular hexacontatetragon is constructible using a compass and straightedge. As a truncated triacontadigon, it can be constructed by an edge-bisection of a regular triacontadigon.

Symmetry

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 Z_n 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. 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

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. 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

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

• Naming Polygons and Polyhedra

Heptacontagon

Regular heptacontagon
A regular heptacontagon
Type	Regular polygon
Edges and vertices	70
Schläfli symbol	{70}, t{35}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D70), order 2×70
Internal angle (degrees)	≈174.857°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

In geometry, a heptacontagon (or hebdomecontagon from Ancient Greek ἑβδομήκοντα, seventy) or 70-gon is a seventy-sided polygon. 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

One interior angle in a regular heptacontagon is 1746⁄7°, meaning that one exterior angle would be 51⁄7°.

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

$A={\frac {35}{2}}t^{2}\cot {\frac {\pi }{70}}$

and its inradius is

$r={\frac {1}{2}}t\cot {\frac {\pi }{70}}$

The circumradius of a regular heptacontagon is

$R={\frac {1}{2}}t\csc {\frac {\pi }{70}}$

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

Symmetry

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 Z_n representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. 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

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. 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

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

• Weisstein, Eric W. "Heptacontagon". MathWorld.
• Naming Polygons and Polyhedra
• hebdomacontagon

Octacontagon

Regular octacontagon
A regular octacontagon
Type	Regular polygon
Edges and vertices	80
Schläfli symbol	{80}, t{40}, tt{20}, ttt{10}, tttt{5}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D80), order 2×80
Internal angle (degrees)	175.5°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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

Regular octacontagon

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 1751⁄2°, meaning that one exterior angle would be 41⁄2°.

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

${\begin{aligned}A=20t^{2}\cot {\frac {\pi }{80}}=\cot {\frac {\pi }{40}}+{\sqrt {\cot ^{2}{\frac {\pi }{40}}+1}}=&20\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}+{\sqrt {\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)^{2}+1}}\right)t^{2}\\=&20\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}+{\sqrt {\left(\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)+\left({\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)\right)^{2}+1}}\right)t^{2}\\=&20\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}+{\sqrt {\left(\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)^{2}+{\binom {2}{1}}\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)\left({\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)+\left({\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)^{2}\right)+1}}\right)t^{2}\\=&20\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}+{\sqrt {\left(\left(11+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}\right)+{\binom {2}{1}}\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)\left({\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)+\left(12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}\right)\right)+1}}\right)t^{2}\\=&20\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}+{\sqrt {\left(23+8{\sqrt {5}}+2\cdot {\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}+{\binom {2}{1}}\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)\left({\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)\right)+1}}\right)t^{2}\\=&20\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}+{\sqrt {24+8{\sqrt {5}}+2\cdot {\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}+{\binom {2}{1}}\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)\left({\sqrt {12+4{\sqrt {5}}+{\binom {2}{1}}\left(1+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}}\right)}}\right)t^{2}\end{aligned}}$

and its inradius is

$r={\frac {1}{2}}t\cot {\frac {\pi }{80}}$

The circumradius of a regular octacontagon is

${\begin{aligned}R={\frac {1}{2}}t\csc {\frac {\pi }{80}}={\frac {\sqrt {\left(\cot {\tfrac {\pi }{80}}\right)^{2}+1}}{2}}t\end{aligned}}$

Construction

Since 80 = 2⁴ × 5, a regular octacontagon is constructible using a compass and straightedge. 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:

$\sin {\frac {\pi }{80}}=\sin 2.25^{\circ }={\frac {1}{8}}(1+{\sqrt {5}})\left(-{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}-{\sqrt {(2+{\sqrt {2}})\left(2-{\sqrt {2+{\sqrt {2}}}}\right)}}\right)$

$+{\frac {1}{4}}{\sqrt {{\frac {1}{2}}(5-{\sqrt {5}})}}\left({\sqrt {(2+{\sqrt {2}})\left(2+{\sqrt {2+{\sqrt {2}}}}\right)}}-{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}\right)$

$\cos {\frac {\pi }{80}}=\cos 2.25^{\circ }={\sqrt {{\frac {1}{2}}+{\frac {1}{4}}{\sqrt {{\frac {1}{2}}\left(4+{\sqrt {2\left(4+{\sqrt {2(5+{\sqrt {5}})}}\right)}}\right)}}}}$

Symmetry

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 Z_n representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. 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

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. 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

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

• Weisstein, Eric W. "Octacontagon". MathWorld.
• Naming Polygons and Polyhedra
• ogdoacontagon

Enneacontagon

Regular enneacontagon
A regular enneacontagon
Type	Regular polygon
Edges and vertices	90
Schläfli symbol	{90}, t{45}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D90), order 2×90
Internal angle (degrees)	176°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

In geometry, an enneacontagon or enenecontagon or 90-gon (from Ancient Greek ἑννενήκοντα, ninety) is a ninety-sided polygon. 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

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)

$A={\frac {45}{2}}t^{2}\cot {\frac {\pi }{90}}$

and its inradius is

$r={\frac {1}{2}}t\cot {\frac {\pi }{90}}$

The circumradius of a regular enneacontagon is

$R={\frac {1}{2}}t\csc {\frac {\pi }{90}}$

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

Symmetry

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 Z_n 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. 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

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. 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

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°	8°

References

• Weisstein, Eric W. "Enneacontagon". MathWorld.
• Naming Polygons and Polyhedra
• enenacontagon

Enneacontahexagon

Regular enneacontahexagon
A regular enneacontahexagon
Type	Regular polygon
Edges and vertices	96
Schläfli symbol	{96}, t{48}, tt{24}, ttt{12}, tttt{6}, ttttt{3}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D96), order 2×96
Internal angle (degrees)	176.25°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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

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 1761⁄4°, meaning that one exterior angle would be 33⁄4°.

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

$A=24t^{2}\cot {\frac {\pi }{96}}$

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

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

Symmetry

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. 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

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 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

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

• Naming Polygons and Polyhedra

Hectogon

Regular hectogon
A regular hectogon
Type	Regular polygon
Edges and vertices	100
Schläfli symbol	{100}, t{50}, tt{25}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D100), order 2×100
Internal angle (degrees)	176.4°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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

Regular hectogon

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 1762⁄5°, meaning that one exterior angle would be 33⁄5°.

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

$A=25t^{2}\cot {\frac {\pi }{100}}$

and its inradius is

$r={\frac {1}{2}}t\cot {\frac {\pi }{100}}$

The circumradius of a regular hectogon is

$R={\frac {1}{2}}t\csc {\frac {\pi }{100}}$

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. 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.

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.

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

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

Symmetry

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 Z_n representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. 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

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. 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

A hectogram is a 100-sided star polygon. There are 19 regular forms 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

• Decagon (10)
• Chiliagon (1000)
• Myriagon (10000)
• Megagon (1000000)

References

• Weisstein, Eric W. "Hectogon". MathWorld.
• Naming Polygons and Polyhedra
• hecatonagon

120-gon

Regular 120-gon
A regular 120-gon
Type	Regular polygon
Edges and vertices	120
Schläfli symbol	{120}, t{60}, tt{30}, ttt{15}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D120), order 2×120
Internal angle (degrees)	177°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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.

Regular 120-gon properties

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)

$A=30t^{2}\cot {\frac {\pi }{120}}$

and its inradius is

$r={\frac {1}{2}}t\cot {\frac {\pi }{120}}$

The circumradius of a regular 120-gon is

$R={\frac {1}{2}}t\csc {\frac {\pi }{120}}$

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

Further information: Trigonometric constants expressed in real radicals § 1.5°: regular hecatonicosagon (120-sided polygon)

Constructible

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

Symmetry

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 Z_n 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. 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

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 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

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°	3°

360-gon

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

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)

$A=90t^{2}\cot {\frac {\pi }{360}}$

and its inradius is

$r={\frac {1}{2}}t\cot {\frac {\pi }{360}}$

The circumradius of a regular 360-gon is

$R={\frac {1}{2}}t\csc {\frac {\pi }{360}}$

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

Symmetry

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 Z_n 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. 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

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 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

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.

Regular star polygons
Picture	{360}	{360/7}	{360/11}	{360/13}	{360/17}	{360/19}	{360/23}	{360/29}
Interior angle	179°	173°	169°	167°	163°	161°	157°	151°
Picture	{360/31}	{360/37}	{360/41}	{360/43}	{360/47}	{360/49}	{360/53}	{360/59}
Interior angle	149°	143°	139°	137°	133°	131°	127°	121°
Picture	{360/61}	{360/67}	{360/71}	{360/73}	{360/77}	{360/79}	{360/83}	{360/89}
Interior angle	119°	113°	109°	107°	103°	101°	97°	91°
Picture	{360/91}	{360/97}	{360/101}	{360/103}	{360/107}	{360/109}	{360/113}	{360/119}
Interior angle	89°	83°	79°	77°	73°	71°	67°	61°
Picture	{360/121}	{360/127}	{360/131}	{360/133}	{360/137}	{360/139}	{360/143}	{360/149}
Interior angle	59°	53°	49°	47°	43°	41°	37°	31°
Picture	{360/151}	{360/157}	{360/161}	{360/163}	{360/167}	{360/169}	{360/173}	{360/179}
Interior angle	29°	23°	19°	17°	13°	11°	7°	1°

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}.