Stephanoid

n-gonal stephanoid
Rank	3
Type	Noble
Elements
Faces	2n butterflies
Edges	2n+2n
Vertices	2n
Vertex figure	Butterfly
Measures ​
Volume	$0$
Central density	0
Related polytopes
Dual	n-gonal stephanoid
Convex hull	n-gonal prism or antiprism
Abstract & topological properties
Flag count	16n
Euler characteristic	0
Orientable	Yes
Genus	1
Properties
Convex	No
Nature	Tame

A stephanoid or crown polyhedron is a noble polyhedron whose faces are butterflies and which has dihedral symmetry. Their convex hulls are prisms or antiprisms. They are self-dual.

There is a stephanoid with $n$-gonal dihedral symmetry for every pair $a$ and $b$ where the faces have vertices $a$ steps apart on one base and $b$ steps apart on the other base, where $a\neq b$ and $a+b\neq n$ (those cases are degenerate). This gives $\left\lfloor {\tfrac {n-2}{2}}\right\rfloor \left\lceil {\tfrac {n-2}{2}}\right\rceil$ distinct $n$-gonal stephanoids, although if $a$, $b$, and $n$ share a common factor, the resulting stephanoid is a compound. If $a-b$ is even, the convex hull is a prism, else it is an antiprism.

Stephanoids

Square (1,2)-stephanoid
Pentagonal (1,2)-stephanoid	Pentagonal (1,3)-stephanoid
Hexagonal (1,2)-stephanoid	Hexagonal (1,3)-stephanoid	Hexagonal (1,4)-stephanoid	Hexagonal (2,3)-stephanoid
Heptagonal (1,2)-stephanoid	Heptagonal (1,3)-stephanoid	Heptagonal (1,4)-stephanoid	Heptagonal (1,5)-stephanoid	Heptagonal (2,3)-stephanoid	Heptagonal (2,4)-stephanoid

In vertex figures

Square stephanoids appear as the vertex figures of sirc and girc. Pentagonal (1,3)-stephanoids appear as the vertex figures of sriphi, mriphi, griphi, and graphi. Non-noble variants of pentagonal and hexagonal stephanoids appear as the vertex figures of sidpaxhi, gidpaxhi, and toditdy.