The Cundy Deltahedra

The Cundy Deltahedra

or Biform Deltahedra

or Biform Deltahedra

Roger's Polyhedra, (c) 2006-2024, Roger Kaufman

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.

Links below display various models

Clicking on pictures opens an Interactive OFF Viewer
OFF files open in an online browser. Downloaded files may be viewed with an application such as Antiprism
STEL files may be saved and opened with Stella
VRML files (wrl,switch,cyl) open in an online browser. Downloaded files may be viewed with a VRML browser such as FreeWRL

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

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

Not including the invalid ones, some statistics are:

Including the 5 convex biforms, there 30 unique biform deltahedra which have been discovered
It has not been proven that all the nonconvex biform deltahedra have been found
Of these, 8 are chiral
Symmetry distributions are: 8 - Dihedral, 11 - Tetrahedral, 4 - Octahedral and 7 - Icosahedral
Out of the 11 new ones which were added to the compilation, 4 of those have 44 faces

Each figure in the following tables lists the symmetry (S) Dn - Dihedral, T - Tetrahedral, O - Octahedral, I - Icosahedral. The total Face, Edge and Vertex counts are given. A (C) after the name denotes that solid is chiral.

Models from Cundy's Table

Models from Cundy's Table

Models from Cundy's Table

Models from Cundy's Table

George Olshevsky conducted an extensive search and presented an unpublished paper "Breaking Cundy's Deltahedra Record" [Ref] available here by permission. An additional 11 examples were found. The models below are only those not found in Cundy's table.

The names for the polyhedra were taken form the Olshevsky paper (these names were also used for models in Cundy's tabulation above). The (#) number in parentheses after the name are refer to the indexing used in the Olshevsky paper.

It should be noted that #23, #24 and #25 cannot be made by simple augmentation or excavation as all the others have been. These had to be made by a process called spring modeling which adjusted all the edges to unity. #23 was discovered by Mason Green and called it a triangular cingulated antiprism.

Additional Examples Found

Additional Examples Found

Additional Examples Found

Question or comments about the web page should be directed to PolyhedraSmith@gmail.com.

The deltahedra were created in Robert Webb's Stella application and Antiprism. The generation of OFF and VRML files were processed with Antiprism. The Hedron application by Jim McNeill was used to generate switch files. The image files were created with off2pov and POV-ray.

History:

2024-06-17 Use base.css
2024-06-13 Switch to Multi OFF Viewer
2023-09-04 Switch to Simple OFF Viewer and X_Ite VRML Viewer
2023-03-13 Open Interactive Viewer from model pictures
2019-03-12 Changed email address from defunct bigfoot.com
2008-01-21 Fixed VEF counts on 05_Icosagyraugmented_Icosahedron
2008-01-17 A few edits
2008-01-16 Initial Release

Back to the main Polyhedron Page.
Link to this page as http://www.interocitors.com/polyhedra/Deltahedra/Cundy

Triangular Dipyramid J12

F=6 E=9 V=5

off stel wrl switch