Hexadecagram

Hexadecagram
(OFF file)
Rank	2
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Had
Coxeter diagram	x16/5o
Schläfli symbol	{16/5}
Elements
Edges	16
Vertices	16
Vertex figure	Dyad, length √2-√2-√2
Measures (edge length 1)
Circumradius	$\sqrt{2-\sqrt2-\sqrt{\frac{10-7\sqrt2}{2}}} ≈ 0.60134$
Inradius	$\frac{1-\sqrt2+\sqrt{4-2\sqrt2}}{2} ≈ 0.33409$
Area	$4(1-\sqrt2+\sqrt{4-2\sqrt2}) ≈ 2.67271$
Angle	67.5°
Central density	5
Number of external pieces	32
Level of complexity	2
Related polytopes
Army	Hed, edge length $1-\sqrt{2-\sqrt2}$
Dual	Hexadecagram
Conjugates	Hexadecagon, Small hexadecagram, Great 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 hexadecagram, or had, is a non-convex polygon with 16 sides. It's created by taking the fourth stellation of a hexadecagon. A regular 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 great hexadecagram.

It is the uniform quasitruncation of the octagram.

Vertex coordinates

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

• $\left(±\frac12,\,±\frac{1-\sqrt2+\sqrt{4-2\sqrt2}}{2}\right),$
• $\left(±\frac{1-\sqrt{2-\sqrt2}}{2},\,±\frac{1-\sqrt2+\sqrt{2-\sqrt2}}{2}\right).$

External links

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

• Wikipedia Contributors. "Hexadecagram".