Editing Deltahedron (section)

== Non-convex deltahedron ==
[[File:Stella octangula.png|thumb|upright=0.8|[[Stella octangula]] is a non-convex deltahedron]]
A ''non-convex deltahedron'' is a deltahedron that does not possess convexity, thus it has either coplanar faces or collinear edges. There are infinitely many non-convex deltahedra.<ref>{{multiref
|{{harvp|Trigg|1978}}
|{{harvp|Eppstein|2021}}
}}</ref> Some examples are [[stella octangula]], the third stellation of a regular icosahedron, and [[Boerdijk–Coxeter helix]].<ref>{{multiref
|{{harvp|Pedersen|Hyde|2018}}
|{{harvp|Weils|1991|p=[https://archive.org/details/ThePenguinDictionaryOfCuriousAndInterestingGeometry/page/n93/mode/1up?view=theater 78]}}
}}</ref>

There are subclasses of non-convex deltahedra. {{harvtxt|Cundy|1952}} shows that they may be discovered by finding the number of varying vertex's ''types''. A set of vertices is considered the same type as long as there are subgroups of the polyhedron's same group [[Transitive group|transitive]] on the set. Cundy shows that the [[great icosahedron]] is the only non-convex deltahedron with a single type of vertex. There are seventeen non-convex deltahedra with two types of vertex, and soon the other eleven deltahedra were later added by {{harvtxt|Olshevsky}},<ref>{{multiref
|{{harvp|Cundy|1952}}
|{{harvp|Olshevsky}}
|{{harvp|Tsuruta|Mitani|Kanamori|Fukui|2015}}
}}</ref> Other subclasses are the [[Isohedral figure|isohedral]] deltahedron that was later discovered by both {{harvtxt|McNeill}} and {{harvtxt|Shephard|2000}},<ref>{{multiref
|{{harvp|McNeill}}
|{{harvp|Shephard|2000}}
|{{harvp|Tsuruta|Mitani|Kanamori|Fukui|2015}}
}}</ref> and the ''spiral deltahedron'' constructed by the strips of equilateral triangles was discovered by {{harvtxt|Trigg|1978}}.<ref>{{multiref
|{{harvp|Trigg|1978}}
|{{harvp|Tsuruta|Mitani|Kanamori|Fukui|2015}}
}}</ref>