Enneadecagon

Regular 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

Last updated December 19, 2021

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 Approximated Enneadecagon Inscribed in a Circle.gif — Approximated enneadecagon, inscribed in a circle

Symmetry

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.

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)

Related Research Articles

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In geometry, an octagon is an eight-sided polygon or 8-gon.

In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.

In geometry, a nonagon or enneagon is a nine-sided polygon or 9-gon.

In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.

In geometry, a chiliagon or 1000-gon is a polygon with 1,000 sides. Philosophers commonly refer to chiliagons to illustrate ideas about the nature and workings of thought, meaning, and mental representation.

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In geometry, an octadecagon or 18-gon is an eighteen-sided polygon.

A megagon or 1 000 000-gon is a polygon with one million sides.

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.

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.

References

↑ 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)

enneadecagon

External links

Weisstein, Eric W. "Enneadecagon". MathWorld .

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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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