Enneadecagon

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Regular enneadecagon
Regular polygon 19 annotated.svg
A regular enneadecagon
Type Regular polygon
Edges and vertices 19
Schläfli symbol {19}
Coxeter–Dynkin diagrams CDel node 1.pngCDel 19.pngCDel node.png
Symmetry group Dihedral (D19), order 2×19
Internal angle (degrees)≈161.052°
Properties Convex, cyclic, equilateral, isogonal, isotoxal

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

Contents

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 (angle in degrees). The area, where t is the edge length, is

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

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. Symmetries of enneadecagon.png
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.

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 Regular star polygon 19-2.svg
{19/2}
Regular star polygon 19-3.svg
{19/3}
Regular star polygon 19-4.svg
{19/4}
Regular star polygon 19-5.svg
{19/5}
Interior angle ≈142.105°≈123.158°≈104.211°≈85.2632°
Picture Regular star polygon 19-6.svg
{19/6}
Regular star polygon 19-7.svg
{19/7}
Regular star polygon 19-8.svg
{19/8}
Regular star polygon 19-9.svg
{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 t0.svg
18-simplex (18D)

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References

  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)