Additions to

Zome Geometry

George W. Hart   and  Henri Picciotto

Tau Towers

At left is a tower of three sizes of icosahedra and at right is a tower of three sizes of dodecahedra.  They are each built around a vertical axis that goes through opposite vertices in each component.  Standing like this, they look like a snow-man and snow-woman.  (Related constructions are made in Activity 2.2 of the book, but with the largest one surrounding the other two.)  These are made with the three sizes b1, b2, and b3, but you can imagine extending the sequence infinitely both in the larger and smaller direction, always scaling the next polyhedron by tau.  In the smaller direction, although there would be an infinite number of ever-shrinking components, it only would extend upward a finite distance, since each tower nestles nicely in a cone.  This is an example of summing a geometric series, like summing 1 + 1/2 + 1/4 + 1/8 + ..., in which there is an infinite number of things but the total is finite.  With the stacked polyhedra, each is one tau-th (instead of one half) of the previous, but still the infinite number of terms results in a finite sum.

You can visualize the surrounding cone more clearly if you notice that certain faces in the above images are coplanar.  This interesting phenomenon can be seen more clearly when the snow-man and snow-woman go to bed:

They are lying here on the floor together (just talking).  Notice that one pentagon from each dodecahedron lies in the floor plane.  Also, one triangle from each icosahedron lies in the floor plane.  While one could design towers with any scaling factor, only the golden ratio results in these coplanarity relationships.  This property for the tau-scaled icosahedron tower was pointed out to me by Richard Benish (who showed me a nice six-layer paper model) and I observed that it follows as well for a tau-scaled dodecahedron tower.  One way to prove it is to invoke the observation by Pappus that the vertices of a dodecahedron lie by fives in four parallel planes and the vertices of an icosahedron lie by threes in the same four planes.  Pappus was looking at them inscribed in the same sphere, but all we really need is to notice that the distance between planes 1 and 2 (or 3 and 4) is tau times the distance between planes 2 and 3.  So scaling down by tau brings the 3rd plane of the smaller neighbor even with the second plane of the larger neighbor, and everything else follows easily.