Convex regular-faced polyhedra with conditional edges

I created this page to publicize the classification of convex regular-faced polyhedra with conditional edges, a very interesting result and including some very interesting polyhedra which deserve to be better-known among polyhedron enthusiasts.

These polyhedra are a natural generalization of the Johnson solids, relaxing the requirement of strict convexity so that adjacent regular faces can be co-planar, forming a single parquet face, but requiring that all vertices be proper (informally that all vertices of the regular faces are actually vertices of the parquet faces rather than interior to them). A conditional edge is an edge, of dihedral angle π, between co-planar faces.

Allowing conditional edges but requiring proper vertices, there are five varieties of parquet faces possible: the rhombus "3+3" composed of two equilateral triangles; the square and triangle "3+4"; the pentagon and triangle "3+5"; the square and two opposite triangles "3+4+3"; and the pentagon and two opposite triangles "3+5+3".

There are 78 convex regular-faced polyhedra with conditional edges and all proper vertices. The bulk of these polyhedra (the 71 containing a rhombus 3+3) were listed by Alex Doskey, Roger Kaufman, and Steve Waterman in 2006. The listing was independently produced and proven complete in 2010 by A. V. Timofeenko.

The further generalization, allowing conditional vertices, is known to admit of a finitary classification, and is ongoing work for Timofeenko and his students. Some examples of such polyhedra have been produced by Roger Kaufman and others.

Another page gives more exposition and references.

The List

The table below shows the 78 polyhedra.

The numbering P_n,k is from Timofeenko (2011; his 2009 paper does not consider the tripartite faces 3+4+3 and 3+5+3 and so the numbering is slightly different). The numbering S_n is from Gurin-Zalgaller 2008.

The names are generally taken from Timofeenko. Some names labeled (GZ) are more fanciful names given in Gurin-Zalgaller; others labeled (S) are my own invention. I have taken some liberties with translating Timofeenko's names; in particular I use "para-augmented" and "meta-augmented" for the two ways of augmenting a pentagonal rotunda; "gyrate augmented" for augmentations which are also found in a different orientation among the Johnson solids; and various modifiers to describe the 5 different ways of biaugmenting an orthobirotunda and the 3 different ways of triaugmenting one.

The viewers require Java. The first opening of a live viewer is likely to be very slow as your browser starts Java. After the first others will start more quickly. Note (written 2020): your browser almost certainly no longer supports Java applets. I apologize for the inconvenience. You can get a copy of Java 8 which has a program "appletviewer" and run "appletviewer http://tupelo-schneck.org/polyhedra/appletviewer.html" to see, awkwardly, all 78 live views.

You can download a zip file of the data used to create these images and viewers, in ".off" and ".m" formats.

Software credits

• LiveGraphics3D, the applet used to display the polyhedra. Honorable mention to JavaView.
• Antiprism is an awesome open-source software suite for manipulating polyhedra.
• Jim McNeill's Hedron can generate a polyhedron from a text description of its net.

Inspiration

I became interested in the classification of regular-faced polyhedra due to a toy called Magformers, which is much loved by all from ages 0 to adulthood. A couple of places to see or buy them are here and here. As can be seen some parquet faces are already provided!

To the extent possible under law, Robert R Tupelo-Schneck has waived all copyright and related or neighboring rights to http://tupelo-schneck.org/polyhedra. This work is published from: United States.