Great hexadecagram

Great hexadecagram
(OFF file)
Rank	2
Type	Regular
Notation
Bowers style acronym	Gahd
Coxeter diagram	x16/7o
Schläfli symbol	{16/7}
Elements
Edges	16
Vertices	16
Vertex figure	Dyad, length √2-√2+√2
Measures (edge length 1)
Circumradius	${\sqrt {2+{\sqrt {2}}-{\sqrt {\frac {10+7{\sqrt {2}}}{2}}}}}\approx 0.50980$
Inradius	${\frac {-1-{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}{2}}\approx 0.099456$
Area	$4(-1-{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}})\approx 0.79565$
Angle	22.5°
Central density	7
Number of external pieces	32
Level of complexity	2
Related polytopes
Army	Hed, edge length $-1-{\sqrt {2}}+{\sqrt {4-2{\sqrt {2}}}}$
Dual	Grand hexadecagram
Conjugates	Hexadecagon, Small hexadecagram, Hexadecagram
Convex core	Hexadecagon
Abstract & topological properties
Flag count	32
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(16), order 32
Convex	No
Nature	Tame

The great hexadecagram, or gahd, is a non-convex polygon with 16 sides. It's created by taking the sixth stellation of a hexadecagon. A regular great hexadecagram has equal sides and equal angles.

It is one of three regular 16-sided star polygons, the other two being the small hexadecagram and the hexadecagram.

It is the uniform quasitruncation of the octagon.

Vertex coordinates

The vertices of a regular great hexadecagram of edge length 1 are given by all permutations of:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {-1-{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {2+{\sqrt {2}}}}-1}{2}},\,\pm {\frac {1+{\sqrt {2}}-{\sqrt {2+{\sqrt {2}}}}}{2}}\right).$

External links

• Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".

• Wikipedia contributors. "Hexadecagram".