The golden ratio, also known as φ, has a value of [1 + sqrt(5)]/2, or ~1.61803. It is associated with a great many figures in geometry, and also appears in numerous other contexts. The most well-known relationship between a geometric figure and the golden ratio is the golden rectangle, which has a length:width ratio equal to the golden ratio. An interesting property of the golden rectangle is that, if a square is removed from it, the remaining portion is simply a smaller golden rectangle — and this process can be continued without limit.
While the golden ratio is related to many polyhedra, this relationship does not always involve golden rectangles, but sometimes it does. For example, it is possible to modify a rhombicosidodecahedron, by replacing that figure’s squares with golden rectangles (with the longest side adjacent to the triangles, not the pentagons), to obtain a “Zomeball” — the node which is at the heart of the Zometool ball-and-stick modeling system for polyhedra, and other phenomena. The entire Zome system is based on the golden ratio. Zome kits are available for purchase at http://www.zometool.com, and this image of a Zomeball was found at http://www.graphics.rwth-aachen.de/media/resource_images/zomeball.png.
In some cases, the relationship between a golden rectangle, and a polyhedron, is more subtle. For example, consider three mutually-perpendicular golden rectangles, each with the same center:
While this is not, itself, a polyhedron, it is possible to create a polyhedron from it, by creating its convex hull. A convex hull is simply the smallest convex polyhedron which can contain a given figure in space. For the three golden rectangles above, the convex hull is the icosahedron, one of the Platonic solids:
In addition to the golden rectangle, there are also other quadrilaterals related to the golden ratio. For example, a figure known as a golden rhombus is formed by simply connecting the midpoints of the sides of a golden rectangle. The resulting rhombus has diagonals which are in the golden ratio.
One of the Archimedean solids, the icosidodecahedron, has a dual called the rhombic triacontahedron. The rhombic triacontahedron has thirty faces, and all of them are golden rhombi.
There are also other polyhedra which have golden rhombi for faces. One of them, called the rhombic hexacontahedron (or “hexecontahedron,” in some sources), is actually the 26th stellation of the rhombic triacontahedron, itself. It has sixty faces, all of which are golden rhombi.
Other quadrilaterals related to the golden ratio can be formed by reflecting the golden triangle and golden gmonon (described in the post right before this one) across each of their bases, to form two other types of rhombus.
In these two rhombi, the golden ratio shows up as the side-to-short-diagonal ratio (in the case of the 36-144-36-144 rhombus), and the long-diagonal-to-side ratio (in the case of the 72-108-72-108 rhombus). These two rhombi have a special property: together, they can tile a plane in a pattern which never repeats itself, but, despite this, can be continued indefinitely. This “aperiodic tiling” was discovered by Roger Penrose, a physicist and mathematician. The image below, showing part of such an aperiodic tiling, was found at https://en.wikipedia.org/wiki/Penrose_tiling.
There are also at least two other quadrilaterals related to the golden ratio, and they are also formed from the golden triangle and the golden gnomon. The procedure for making these figures, which could be called the “golden kite” and the “golden dart,” is similar to the one for making the rhombi for the Penrose tiling above, but has one difference: the two triangles are each reflected over a leg, rather than a base.
In the case of this kite and dart, it is the longer and shorter edges, in each case, which are in the golden ratio — just as is the case with the golden rectangle. Another discovery of Roger Penrose is that these two figures, also, can be used to form aperiodic tilings of the plane, as seen in this image from http://www.math.uni-bielefeld.de/~gaehler/tilings/kitedart.html.
There is yet another quadrilateral which has strong connections to the golden ratio. I call it the golden trapezoid, and this shows how it can be made from a golden rectangle, and how it can be broken down into golden triangles and golden gnomons. However, I have not yet found an interesting polyhedron, not tiling pattern, based on golden trapezoids — but I have not finished my search, either.
[Image credits: see above for the sources of the pictures of the two Penrose tilings, as well as the Zomeball, shown in this post. Other “flat,” nonmoving pictures I created myself, using Geometer’s Sketchpad and MS-Paint. The rotating images, however, were created using a program called Stella 4d, which is available at http://www.software3d.com/Stella.php.]