# 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
Flag orbits8
ConvexNo
NatureTame

The stephanoids or crown polyhedra are an infinite family of self-intersecting self-dual noble polyhedra, meaning that they are both vertex-transitive and face-transitive. All stephanoids have butterfly faces, which are a type of quadrilateral with a an axis of reflectional symmetry formed by crossing two edges of an isosceles trapezoid. The stephanoids have the same vertices and symmetries of the prisms and antiprisms, and they are all toroidal polyhedra with genus 1.

Each stephanoid has a degree of freedom: there are two types of edges and the edge length ratio may be continually varied without losing symmetry properties. Ignoring this variability, the stephanoids are parameterized by three integers: ${\displaystyle n}$, the number of sides on the polygonal "bases," and two additional parameters ${\displaystyle a}$ and ${\displaystyle b}$. It is constructed so 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.