A regular tridecagon
|Edges and vertices||13|
|Symmetry group||Dihedral (D13), order 2×13|
|Internal angle (degrees)||≈152.308°|
|Properties||Convex, cyclic, equilateral, isogonal, isotoxal|
The following is an animation from a neusis construction of a regular tridecagon with radius of circumcircle according to Andrew M. Gleason, based on the angle trisection by means of the Tomahawk (light blue).
Another possible animation of an approximate construction, also possible with using straightedge and compass.
GeoGebra: BME1 = 27.692307692307764°
GeoGebra: 360° ÷ 13 = 27.69230769230769°
Absolute error of BME1 ≈ 7.4E-14°
Example to illustrate the error:
At a circumscribed circle radius r = 1 billion km (the light needed for this distance about 55 minutes), the absolute error of the 1st side would be approximately 1 mm.
For details, see: Wikibooks: Tridecagon, construction description (German)
These 4 symmetries can be seen in 4 distinct symmetries on the tridecagon. John Conway labels these by a letter and group order. Full symmetry of the regular form is r26 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 g13 subgroup has no degrees of freedom but can seen as directed edges.
- Gleason, Andrew Mattei (March 1988). "Angle trisection, the heptagon, and the triskaidecagon p. 192–194 (p. 193 Fig.4)" (PDF). The American Mathematical Monthly. 95 (3): 186–194. Archived from the original (PDF) on 2015-12-19. Retrieved 24 December 2015.
- 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)
- Colin R. Bruce, II, George Cuhaj, and Thomas Michael, 2007 Standard Catalog of World Coins, Krause Publications, 2006, ISBN 0896894290, p. 81.