DR with "pre-Van-Oss" object and shadow.
(Click for a higher-resolution image.)
Some photos tell it all, and some barely scratch the surface. This one hides a lot, so it fits into the latter category. To preface this long rant, I want to say that I have long believed the angle π/15 to be interesting, and this model, this photo, and this page represent an attempt to expound on this.
Explanation of the Photo.
First of all, the photo features me, David A. Richter, Assistant Professor of Mathematics at Western Michigan University in Kalamazoo, Michigan. In the photo, I am holding a model made from straws and pipe-cleaners, held together completely by friction, no glue. This 3-dimensional object has reflection symmetries in three pairwise perpendicular planes. Thus, the automorphism group of this object is isomorphic to a direct product of three copies of the two-element group, an abelian group of order 8. Also, the object fits snugly within a ball of a certain radius; it is nearly spherical. Ideally, the shadow of this object has 30-fold dihedral symmetry. I used this photo as a greeting to my website because, despite the inaccuracy of the construction, this shadow trick worked to my satisfaction.
Ideally, the shadow of this object is none other than Van Oss's projection of the 600-cell, the same which H. S. M. Coxeter used as the frontispiece to his widely-read book, Regular Polytopes. If you are a polytope maniac who regularly and maniacally pores through "RP", then you cannot miss it. Before the year 2005, I had known about Van Oss's projection for many years, but, shamed as I am to admit it, I learned of this "pre-Van-Oss" object only recently and with much stubborn and ignorant incredulity.
Tangent to the Zome Sphere.
Late in 2005, a colleague Scott Vorthmann alerted me that there might be a 3-dimensional object having the property that it projects to Van Oss's projection. I was skeptical, mainly because I did not understand that Van Oss's projection represents the image of an orthogonal projection from R4 to R2. Also, what I did not know for many years was that chapter 13 of Coxeter's book contained the recipe for this object. Neither I nor my hapless colleague had read (or understood) this part of Coxeter's book until about the time this model was constructed.
Scott is responsible for having written vZome, a magnificent software system which allows the user to build Zome-like models on a computer screen. He thought that it might be possible to describe this "pre-Van-Oss" object using coordinates from the Golden Field Q(√5). Consequently, Scott spent at least a a couple long exasperating nights trying to create this object in vZome, and he came incredibly close before shelving the project.
His vZome model was accurate to about 3 decimal places, with the image on the screen appearing incredibly accurate. After some analysis, I had to tell him that his model was an approximation, albeit an extremely good one. Although we don't have a proof yet, it doesn't seem possible to describe the coordinates of this image of the 600-cell using only the Golden Field. It's likely that one must use Q(cos π/15), as described in RP.
Inspired by his efforts and his beautiful vZome model, I ventured to find the magic coordinates which would project to our pre-Van-Oss object. After a few weeks of effort with Maple, I found an extremely simple way to express the coordinates using the cosine of π/15 (in particular with the use of 30th roots of unity). I had thought my formulae to be original until I read them in section 13.6 of RP. Startlingly, this clever fellow had used notation nearly identical to mine. What vanity!
Having my coordinate system confirmed by Coxeter himself, I set out to build a model of our object. I used stirring straws for edges and pipe cleaners wound around into little "spiders" as vertices. Whether or not the coordinates of its vertices lie within the Golden Field, this pre-Van-Oss object is patently non-Zomic. This model uses 23 different lengths of edges and the ratios are inconsistent with those of the Zome System. Moreover, whereas the usual Zome model of the 600-cell represents a projection which maps most vertices and edges in a 2-1 correspondence, requiring only 75 connectors and 384 struts, this object has the property that all 120 vertices and all 720 edges are represented faithfully. Thus, I had to cut 720 straws to 23 various lengths and manufacture 120 spiders, each with 12 legs. Assembling these 840 pieces correctly involved tediously labelling each of the 120 vertices using the notation that both Coxeter and I seemed to believe was appropriate. The total of manual labor involved in this construction took about 20 hours.
We say that this object serves an example of "ghost symmetry", despite the fact that we don't yet have a clear mathematical definition of the phenomenon. The symmetry group of the 600-cell has an element of order 30, and this is evident in Van Oss's projection. However, the projection from R4 to R2 which yields Van Oss's projection can be factored in such a way that the projection to 3-space does not possess this 30-fold symmetry. Another favorite example of ghost symmetry is furnished by the Zome model of the compound of fifteen 16-cells.
It should be clear what we mean by an "axis of ghost symmetry"; this shadow represents an orthogonal projection, so the image of the projection has an orthogonal complement. The pre-Van-Oss object has three different axes of ghost symmetry. Remarkably, two of these yield Van Oss's projection, with D(30) symmetry, while the other yields a projection having D(4) symmetry, the same as that of a square. The D(4) axis is perpendicular to both of the D(30) axes. By a coincidence which astonishes me, the angle between the two D(30) axes is exactly the same as the angle between a pentagonal axis and a trihedral axis of a regular icosahedron. In Zome terms, this is the same as the angle between a red axis and a yellow axis.
"RP": H. S. M. Coxeter. Regular Polytopes. 3rd ed. Dover Publications Inc., New York, 1973.
Scott Vorthmann. vZome