# Stephanoid

(Redirected from Square stephanoid)
n-gonal stephanoid Rank3
TypeNoble
Elements
Faces2n butterflies
Edges2n+2n
Vertices2n
Vertex figureButterfly
Measures ​
Volume$0$ Central density0
Related polytopes
Dualn-gonal stephanoid
Convex hulln-gonal prism or antiprism
Abstract & topological properties
Flag count16n
Euler characteristic0
OrientableYes
Genus1
Properties
ConvexNo
NatureTame

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.

## 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.