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| Regular tridecagon | |
|---|---|
 A regular tridecagon | |
| Type | Regular polygon |
| Edges andvertices | 13 |
| Schläfli symbol | {13} |
| Coxeter–Dynkin diagrams |    |
| Symmetry group | Dihedral (D13), order 2×13 |
| Internal angle (degrees) | ≈152.308° |
| Properties | Convex,cyclic,equilateral,isogonal,isotoxal |
| Dual polygon | Self |
Ingeometry, atridecagon ortriskaidecagon or 13-gon is a thirteen-sidedpolygon.
Aregular tridecagon is represented bySchläfli symbol {13}.
The measure of each internal angle of aregular tridecagon is approximately 152.308degrees, and the area with side lengtha is given by
As 13 is aPierpont prime but not aFermat prime, the regular tridecagon cannot beconstructed using acompass and straightedge. However, it is constructible usingneusis, or angle trisection.
The following is an animation from aneusis construction of a regular tridecagon with radius of circumcircle according toAndrew M. Gleason,[1] based on theangle trisection by means of theTomahawk (light blue).
Theregular tridecagon hasDih13 symmetry, order 26. Since 13 is aprime number there is one subgroup with dihedral symmetry: Dih1, and 2cyclic group symmetries: Z13, and Z1.
These 4 symmetries can be seen in 4 distinct symmetries on the tridecagon.John Conway labels these by a letter and group order.[2] Full symmetry of the regular form isr26 and no symmetry is labeleda1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), andi when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled asg for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only theg13 subgroup has no degrees of freedom but can be seen asdirected edges.
The regular tridecagon is used as the shape of theCzech 20 korun coin.[3]
Atridecagram is a 13-sidedstar polygon. There are 5 regular forms given bySchläfli symbols: {13/2}, {13/3}, {13/4}, {13/5}, and {13/6}. Since 13 is prime, none of the tridecagrams are compound figures.
| Tridecagrams | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Picture |  {13/2} |  {13/3} |  {13/4} |  {13/5} |  {13/6} | ||||||
| Internal angle | ≈124.615° | ≈96.9231° | ≈69.2308° | ≈41.5385° | ≈13.8462° | ||||||
Although 13-sided stars appear in theTopkapı Scroll, they are not of these regular forms.[4]
The regular tridecagon is thePetrie polygon of the12-simplex:
| A12 |
|---|
 12-simplex |
