# Stephanoid

(Redirected from Square stephanoid)
n-gonal stephanoid
Rank3
TypeNoble
Elements
Faces2n butterflies
Edges2n+2n
Vertices2n
Vertex figureButterfly
Measures ​
Volume${\displaystyle 0}$
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 ${\displaystyle n}$-gonal dihedral symmetry for every pair ${\displaystyle a}$ and ${\displaystyle b}$ where the faces have vertices ${\displaystyle a}$ steps apart on one base and ${\displaystyle b}$ steps apart on the other base, where ${\displaystyle a\neq b}$ and ${\displaystyle a+b\neq n}$ (those cases are degenerate). This gives ${\displaystyle \left\lfloor {\tfrac {n-2}{2}}\right\rfloor \left\lceil {\tfrac {n-2}{2}}\right\rceil }$ distinct ${\displaystyle n}$-gonal stephanoids, although if ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle n}$ share a common factor, the resulting stephanoid is a compound. If ${\displaystyle a-b}$ is even, the convex hull is a prism, else it is an antiprism.

 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.