The 16-Cell
The 24-Cell
Diagonals of Polyhedra
Ghost Symmetry

Introduction.

This page has some information on a technique I used in order to make a few mathematical models. Essentially, I adapted a construction technique I learned in a welding class to more economical (but less durable) materials. The essential materials are wooden balls, colored straws, and glue from a glue gun.

(Please note: Many of the images on these pages are free-view stereograms, with the cross-eyed pair appearing on the left and the parallel-view appearing on the right.)

An Industrial-Strength Model.

The first model in this story is that of a particular projection of the regular polytope with 5 vertices. This projection is peculiar in that even though the 3-dimensional model has no 5-fold symmetries, it has two different shadows with 5-fold symmetry. Hence, we say that this model has "ghost symmetry".

This is made from 10 steel rods of various lengths and 5 forged steel balls, each an inch in diameter. These 15 pieces were bound together with the use of a metal inert gas (MIG) welder. I took the class and completed this and a few other models in November 2007.

The First Model.

Welding steel together results in highly satisfactory models, but it takes a lot of time and money to produce very simple objects. After I took this class, I wanted to make a certain model of the 24-cell. However, with 24 vertices and 96 edges, I knew it would be a long time before I could make a similar model out of steel.

Shortly after taking the class, I found a package containing 250 drinking straws in 6 bright assorted colors sold at an extremely economical price at a locally-owned big box store. I bought a package and then thought about what I would use to bind them together. I had already made a ghost-symmetry model of the 600-cell using 720 stirring straws, but the method for binding them them together did not satisfy me.

After about a day, I remembered an inspirational thing my welding instructor said: "A MIG welder is like your basic suped-up glue gun." Subseqeuently, I got a glue gun, a whole bunch of glue sticks, and a couple dozen wooden balls, each an inch in diameter. Hence I set to work on my first balls-and-straws model, a projection of the regular 24-cell having 12-fold ghost symmetry.

A Description of the Technique.

The basic technique is so simple as to be nearly self-explanatory. To start, it is important to observe that each model represents, fundamentally, a configuration of points in 3-dimensional space. Once this observation is made, it just a matter of computing lengths, cutting straws, and gluing them together. You measure and cut a straw only if you want two points (i.e. balls) connected.

The simplest model I made is that of the projection of the 5-cell which has the same symmetry as the regular tetrahedron. One can transfer the principles used to make this to many other models. There is such a projection in 3-dimensional space where the vertices are projected to the points (1,1,1), (1,-1,-1), (-1,1,-1), (-1,-1,1), and (0,0,0). In this model, one should connect every pair of balls. Hence one needs 10 straws cut to various lengths.

The distance between (0,0,0) and (1,1,1) is sqrt(3) and the distance between (1,1,1) and (1,-1,-1) is sqrt(8). By symmetry, the model should use 4 straws corresponding to sqrt(3) and 6 straws corresponding to sqrt(8). Each ball has a positive diameter, say d, so this introduces a correction term. Thus, the model requires two different lengths of straws, say l and s, with the property that (s+d)/(l+d)=sqrt(3)/sqrt(8).

After cutting all the straws, glue everything together. Here is the model I got:

The 16-Cell
The 24-Cell
Diagonals of Polyhedra
Ghost Symmetry