Additions to

Zome Geometry

George W. Hart   and  Henri Picciotto

The Anti-Zomeball

There are two polyhedra you can make that each consist of 12 regular pentagons, 20 equilateral triangles, and 30 golden rectangles.  A large-scale model of the zomeball can be built as a distorted version of the rhombicosidodecahedron, with the length of the triangle edges being tau times the length of the pentagon edges.  (See p. 197 of Zome Geometry, where it is called a "megaball.")  The "anti-zomeball" pictured above (ignore the thread until the bottom paragraph of this page,) has the proportion the other way: the pentagon edge length is tau times the triangle edge length.  In both cases there are 30 golden rectangles, but they are oriented differently.

The uniform compound of five icosahedra

The anti-zomeball has a wonderful non-obvious property: its 60 vertices are at the positions of the vertices of five regular icosahedra. The image above shows the uniform compound of five icosahedra.  Study it to see that there are 12 concave five-pointed-stars.  Their 60 vertices are arranged exactly as in the 12 pentagons of the anti-Zomeball. One way to think of the compound is to relate it to the compound of five cubes (or the compound of five octahedra,) and inscribe one icosahedron (properly oriented) in each cube or each octahedron).

You can put a small yellow strut as a marker on 12 balls of the anti-Zomeball, to indicate the vertices of one icosahedron.  But sadly, most of the edge directions of these five icosahedra are not Zome constructible, so we can locate their vertices (as shown at the top of the page), but we can not build the edges with Zome struts.  However, if one is patient, one can form the edges of the five icosahedra by running five colors of thread between the balls.  Just use a needle, as when sewing.  (I haven't yet made this with all five icosahedra.  If you do, send me a photo to insert here.)

This image shows how to thread one face of the icosahedron.  The interesting fact which makes this an exact construction is that the two types of vertex-to-vertex distances indicated with black lines in the image above are exactly equal.  (In the model shown, with b1 x b2 rectangles, the icosahedron edge length is 2b2.)  Exercise:  Using the coordinate methods of Chapter 18, show that the edge which crosses a triangle, a rectangle, and a triangle has equal length to the edge which crosses the rectangle and the pentagon.

The compound of five 
small stellated dodecahedra
The compound of five 
great dodecahedra

The vertices of the icosahedron are also the vertices of the small stellated dodecahedron and the great dodecahedron.  (See Chapter 20 of Zome Geometry.)  So by replacing the five icosahedra with either of these polyhedra, we obtain two additional uniform compounds.  The two figures above show how their faces intersect.  Notice again how their 60 vertices are identically arranged.  Again, their edges are not all in Zome directions, so these compounds are not Zome constructible, but either can be made with thread if you are patient and careful.  The gread dodecahedron has the same edges and vertces as the icosahedron, so if you make a thread model of the five icosahedra, it is also a model of the five great dodecahedra.  The figure at the top of this page shows one of the five small stellated dodecahedra made from white thread; look for its 12 pentagrams.  Adding the other four small stellated dodecahedra, in four other colors of thread, should create a nice model of the compound at left.  (Again, I haven't made a complete model; if you do send me a photo to put here.)