Show and Tell

Rio Grande College is proud to host the only museum of mathematics in the middle Rio Grande border region!

It features toys, models, constructions, and other interesting artifacts. Let's look at a few notable items, shall we?

We begin with the sections and net of a 24-cell, which is a regular four-dimensional polytope built from twenty-four octahedra. I printed the cells and sections on a 3D printer, using print files that I made myself. You can find them (and an account of how I made them) at my Thingiverse page.

The sections proceed as follows, with colors given as the craft paints I bought at Wal-Mart: (I) the octahedral cell at the "south pole" (Parchment); (II) the truncated octahedral section cut by a hyperplane through the midpoints of the edges "above" the south pole (Parchment and Real Brown); (III) the cuboctahedral equatorial section cut by a hyperplane through the set of vertices to which these edges connect (Look At Me Blue and Real Brown); (IV) the truncated octahedral section analogous to Section II but in the "northern hemisphere" (Look At Me Blue and Real Brown); and (V) the octahedral cell at the "north pole" (Coffee Latte).

The net has the "south pole" at the center and the "north pole" at the base. For reasons fully known only to my subconscious, but partly inspired by Dalí's Crucifixion (Corpus Hypercubus), I decided to model it after traditional depictions of the Hindu god Shiva as Nataraja or Lord of Dance, with three-fold rotational symmetry.

Here is a string-art version of the 24-cell:

It is inspired by the string art used by Mary Everest Boole in the education of her children. She was the wife of logician George Boole, and the mother of Alicia Boole Stott, to whom I owe the inspiration for my polytope sections and nets.

Actually, I relied fairly closely on Stott's 1900 paper, "On Certain Series of Sections of the Regular Four-Dimensional Hypersolids," as I built the sections and net of the 120-cell, which is a regular polytope built from 120 dodecahedral cells.

The seven sections are shown out of order. The small yellow dodecahedron is the base cell, the "south pole." Each of the succeeding sections is cut by a three-dimensional space parallel to the base cell and passing through a set of vertices. These are only half of the sections; to complete the collection, we would need to go through the first six sections again, but in reverse order, with another copy of the small dodecahedron to represent the "north pole."

This is only a partial net built from 75 cells. The red cells around the preriphery of the net represent the "equator" of the 120-cell. A four-dimensional creature could fold up this net to form the "southern hemisphere" of the polytope, plus the "equator"; a "northern hemisphere" could then be glued to the top to complete the structure.

You can read the epic tale of how I printed these things here.

Coming back down to three dimensions, we have the dodecahedron family:

Going from left to right, top to bottom, we have: the icosahedron, the dual icosahedron / dodecahedron, the triacontahedron (the convex hull of the dual pair, colored as the compound of five cubes), the dodecahedron, the icosidodecahedron (the quasiregular solid associated with the dual pair), the right-handed compound of five tetrahedra (colored as the triacontahedron), the compound of five cubes, and the skeletal compound of five tetrahedra. All designed and printed by me.

Next we have a compound of five tetrahedra, assembled, without glue, from 10 sheets of square origami paper cut into thirty 1:3 rectangles, using the instructions found here.

Each strut is one 1:3 rectangle; three struts fit into one another at each point. The five tetrahedra are not connected to one another in any way. The tricky part was getting them to intersect correctly while connecting the struts at the vertices.

The best books on modular origami I've found are by Ekaterina Lukasheva, who has a math background. The following is made according to her instructions:

It's assembled from 15 square sheets of origami paper cut into two 1:2 rectangles apiece. The 30 rectangles are all folded in exactly the same way and attached to one another without glue. The patterns are chosen according to the compound of five cubes, with six rectangles of each pattern, and one pattern for each of the five cubes. In the end, the piece is roughly the same as a triacontahedron.

Lukasheva calls it a Chameleon Eye. Isn't it beautiful?

If you liked this, please take a look at my

Fractal Art

and my

Paintings and Drawings

Michael Ortiz, Ph.D.
Associate Professor of Mathematics
Department of Natural and Behavioral Sciences

Sul Ross State University Rio Grande College

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