# A Zome Rhombicosidodecahedron, and Some of Its Variations

Here’s a Zome rhombicosidodecahedron, made up of white Zomeballs and short blue struts.

If you replace each of the rhombicosidodecahedron’s thirty squares with golden rectangles in a certain way, by replacing certain short struts with medium struts along the triangle/rectangle edges, you get a Zomeball made of Zome — what some have called a “metazomeball.” It has enlarged triangles, compared to the pentagons.

It’s almost possible to augment each of the Zomeball’s 62 faces with all-blue pyramids. Here’s the attempt.

The Zomeball’s twelve pentagons are augmented with pyramids, using medium struts, so that each of these pyramids’ lateral faces is a golden isosceles triangle. The twenty triangles are also augmented by pyramids, with golden isosceles triangles as lateral faces. This requires the use of long blue struts. As for the augmentation performed on the thirty golden rectangles, that’s a bit more complicated.

Here’s what the golden rectangles are augmented with, with the new pieces all being short blue struts. It isn’t quite a pyramid, nor is it quite a prism. Its lateral faces are two equilateral triangles and two golden isosceles trapezoids. It’s a five-faced polyhedron in need of a good name. Until someone comes up with a better name, then, I’m going to call it a golden pentahedron.

Here’s a close-up of a golden isosceles trapezoid, by itself. It’s made of three short struts and one medium strut.

It’s also possible to do the first change in another way. When the rhombicosidodecahedron has its squares replaced with golden rectangles, rotate the golden rectangles 90 degrees, such that it increases the size of the pentagons, rather than the triangles. This has been called an “antizomeball.”

With the antizomeball, augmentation with pyramids is possible with all 62 faces.

Here’s what the three types of pyramid used in the augmentation of the antizomeball look like, separated from the main structure.

With the antizomeball, the pentagonal pyramids’ bases are made of medium struts, and long struts make the five lateral faces into golden isosceles triangles. The triangular faces of the antizomeball are made of short struts, so medium lateral edges for these pyramids make their lateral faces golden isosceles triangles. The golden rectangles in the antizomeball are different from the ones in the Zomeball, and the changes make it possible to augment the antizomeball’s golden rectangles using four medium lateral edges, forming two golden isosceles triangles and two equilateral triangles as the lateral faces of a true pyramid.

If you’d like to try Zome for yourself, which I strongly recommend, the site to visit is http://www.zometool.com.

# The Pyritohedral Golden Icosahedron

Both the Platonic icosahedron and the golden icosahedron have twenty triangular faces. In the Platonic version, these faces are all equilateral triangles. The golden icosahedron has eight such triangles, but the other twelve are golden triangles, which have a leg-to-base ratio which is the golden ratio. These golden triangles appear in pairs, and the six pairs are arranged in such a way as to make this a solid with pyritohedral symmetry: the symmetry of a standard volleyball.

A net for the golden icosahedron appears below. Both images were made using a program called Stella 4d, which you can try for free right here.

# Eight Golden Rectangles

This Euclidean construction was made using Geometer’s Sketchpad, and colored using that program, as well as MS-Paint.

# Golden Mandala

This decagonal mandala is split into fifty golden triangles (shown in yellow), and forty golden gnomons (shown in orange).

# A Golden Tessellation of Quadrilaterals

In this tessellation, golden rectangles are shown in yellow. The orange darts are each made of two golden gnomons, joined at a leg — while the blue rhombi are each made of two golden triangles, sharing a base.

# The Golden Rhombus, the Rhombic Triacontahedron, and the Rhombic Hexecontahedron

There’s a special rhombus which is called the “golden rhombus,” because its diagonals are in the golden ratio. To construct it with compass and straight edge, you first construct a golden rectangle (shown with blue edges and a yellow interior), and then connect the midpoints of its sides to form a rhombus (with edges shown in red).

Several polyhedra can be made which use golden rhombi as their faces. The most well-known of these polyhedra is the rhombic triacontahedron, which has 30 such faces. It is the dual of the icosidodecahedron.

If the rhombic triacontahedron is stellated 26 times, the result is the (non-convex) rhombic hexecontahedron. It has 60 golden rhombi as faces.

Both of these polyhedra can be constructed with Zometools (available at http://www.zometool.com). With white Zomeballs and red Zomestruts, these polyhedra look a lot like this:

The flat image at the top of this post was created using Geometer’s Sketchpad and MS-Paint. The four rotating polyhedral images were created using Stella 4d: Polyhedron Navigator, which you can purchase, or try for free, at http://www.software3d.com/Stella.php.