The Cundy Deltahedra

One day on a MathWorld page about Deltahedra I read this statement: "Cundy (1952) identified 17 concave deltahedra with two kinds of polyhedron vertices". The source of the statement was from a 1952 paper that H. Martyn Cundy published in the Mathematical Gazette titled "Deltahedra" [Ref]. This led down a path which led to the discovery that there are at least 25 such examples.

Deltahedra are polyhedra composed entirely of equilateral triangles. There are only eight convex deltahedra. They were enumerated in 1915 O. Rausenberger [Ref] and later in 1947 by H. Freudenthal and B. H. van der Waerden [Ref]. The well known deltahedra, the tetrahedron, octahedron, and icsoahedron are all isogonal which means they have one form of vertex. The other five convex deltahedrons all have two forms of vertices. While polyhedra that have one form of vertex and regular faces are called uniform, those with two forms of vertices are sometimes termed biform.

Here are models of the 5 biform convex deltahedra. Of these, only the Snub Disphenoid (J84) cannot be made by augmentation of simpler polyhedra and is elementary.

Convex Biform Deltahedra
Triangular Dipyramid J12 S=D3 F=6     E=9    V=5 off     wrl     switch	Pentagonal Dipyramid J13 S=D5 F=10     E=15    V=7 off     wrl     switch	Snub Disphenoid J84 S=D2 F=12     E=18    V=8 off     wrl     switch	Triaugmented Triangular Prism J51 S=D3 F=14     E=21    V=9 off     wrl     switch	Gyroelongated Square Dipyramid J17 S=D4 F=16     E=24    V=10 off     wrl     switch

Cundy proposed a relaxation of the problem so as to enumerate nonconvex deltahedra. Cundy noted that his table included "only those solids in which the triangles are totally on the outside". Such is the case, this condition is also in force. In total, his table included 17 deltahedra. (To see what happens when self-intersecting faces are allowed see Coptic Biform Deltahedra)

Three of the original 17 are actually incorrect. Number 11 and 12 have 3 kinds of vertices. Number 10 has coincident edges and vertices. These are presented but noted as invalid.

Cundy made the following comments on various one in his list. (Thanks to Branko Grünbaum for this information and the list itself)

• No.1 is the net of the regular 4-dimensional simplex
• No. 9 is one of the 59 Icosahedra
• No. 10 can be considered to have triangular and hexagonal faces, and is then a stellated "Archimedean polyhedron" with the hexagons in diametral planes through the origin. If the pyramids are "everted" in this case the result is an octahedron with its faces divided into four equilateral triangles
• similarly if the pyramids of No. 13 are everted the result is an icosahedron with divided faces
• If, in case of No. 7, the pyramids are described inwards, they overlap and a peculiar re-entrant polyhedron results. The same will happen in a number of other cases; the table includes only those solids in which the triangles are totally on the outside