The following is an animation from a neusis construction of a regular tridecagon with radius of circumcircle according to
Andrew M. Gleason,[1] based on the
angle trisection by means of the
Tomahawk (light blue).
An approximate construction of a regular tridecagon using
straightedge and
compass is shown here.
Another possible animation of an approximate construction, also possible with using straightedge and compass.
Up to the maximum precision of 15 decimal places, the absolute error is
Constructed central angle of the tridecagon in GeoGebra (display significant 13 decimal places, rounded)
Central angle of tridecagon
Absolute angular error of the constructed central angle:
Up to 13 decimal places, the absolute error is
Example to illustrate the error
At a circumscribed circle of radius r = 1 billion km (a distance which would take light approximately 55 minutes to travel), the absolute error on the side length constructed would be less than 1 mm.
Symmetry
The regular tridecagon has
Dih13 symmetry, order 26. Since 13 is a
prime number there is one subgroup with dihedral symmetry: Dih1, and 2
cyclic 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 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 be seen as
directed edges.
A tridecagram is a 13-sided
star polygon. There are 5 regular forms given by
Schläfli symbols: {13/2}, {13/3}, {13/4}, {13/5}, and {13/6}. Since 13 is prime, none of the tridecagrams are compound figures.
^John H. Conway, Heidi Burgiel,
Chaim Goodman-Strauss, (2008) The Symmetries of Things,
ISBN978-1-56881-220-5 (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,
ISBN0896894290, p. 81.