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
Flag orbits	8
Convex	No
Nature	Tame

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: $n$, the number of sides on the polygonal "bases," and two additional parameters $a$ and $b$. It is constructed so 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.