Further Additions to

Zome Geometry

George W. Hart   and  Henri Picciotto

Compound of 15 Golden Rectangles

This is a quick 3D "doodle" of sorts, in the form of 15 rectangles.  One way to make a compound of 15 rectangles is just to connect all fifteen opposite pairs of edges of an icosahedron.  This image shows three pairs connected:

But in the Zome construction at the top of the page, the ratio "goes the other way."  You can think of it as re-proportioning those icosahedron rectangles from 1-by-tau to tau-by-1.  Or equivalently, you can rotate each rectangle 90 degrees about the icosahedron's 2-fold axis that goes through its pair of opposite edges. In the model illustrated, each is a golden rectancle of size 2b3 by 2b4. In order to have balls at the crossing points, the long sides are built as b3+b2+b2+b3 and the short sides are b1+b2+b2+b1. To build it, you can start with a 2b2 icosahedron. Opposite pairs of icosahedron edges are the middle portions of the long sides of the rectangles, so extend them with a b3 in each direction to complete the long sides. The image below shows the view down a 5-fold axis of the compound.