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Concept
Pentadecagon
Summary
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.
A regular pentadecagon is represented by Schläfli symbol {15}.
A regular pentadecagon has interior angles of 156°, and with a side length a, has an area given by
As 15 = 3 × 5, a product of distinct Fermat primes, a regular pentadecagon is constructible using compass and straightedge:
The following constructions of regular pentadecagons with given circumcircle are similar to the illustration of the proposition XVI in Book IV of Euclid's Elements.
Compare the construction according Euclid in this image: Pentadecagon
In the construction for given circumcircle: is a side of equilateral triangle and is a side of a regular pentagon.
The point divides the radius in golden ratio:
Compared with the first animation (with green lines) are in the following two images the two circular arcs (for angles 36° and 24°) rotated 90° counterclockwise shown. They do not use the segment , but rather they use segment as radius for the second circular arc (angle 36°).
A compass and straightedge construction for a given side length. The construction is nearly equal to that of the pentagon at a given side, then also the presentation is succeed by extension one side and it generates a segment, here which is divided according to the golden ratio:
Circumradius Side length Angle
The regular pentadecagon has Dih15 dihedral symmetry, order 30, represented by 15 lines of reflection. Dih15 has 3 dihedral subgroups: Dih5, Dih3, and Dih1. And four more cyclic symmetries: Z15, Z5, Z3, and Z1, with Zn representing π/n radian rotational symmetry.
On the pentadecagon, there are 8 distinct symmetries. John Conway labels these symmetries with a letter and order of the symmetry follows the letter. He gives r30 for the full reflective symmetry, Dih15. He gives d (diagonal) with reflection lines through vertices, p with reflection lines through edges (perpendicular), and for the odd-sided pentadecagon i with mirror lines through both vertices and edges, and g for cyclic symmetry.
Official source
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