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.

A Radial Tessellation Using Three Types of Rhombi

radial tessellation using three types of rhombi

The yellow rhombi have angles of 40 and 140 degrees, while the blue rhombi have angles of 80 and 100 degrees, just like in the last post here. However, that post did not include the red rhombi, which have angles of 60 and 120 degrees.

Eight Kite-Rhombus Solids, Plus Five All-Kite Polyhedra — the Convex Hulls of the Thirteen Archimedean-Catalan Compounds

In a kite-rhombus solid, or KRS, all faces are either kites or rhombi, and there are at least some of both of these quadrilateral-types as faces. I have found eight such polyhedra, all of which are formed by creating the convex hull of different Archimedean-Catalan base-dual compounds. Not all Archimedean-Catalan compounds produce kite-rhombus solids, but one of the eight that does is derived from the truncated dodecahedron, as explained below.

Trunc Dodeca

The next step is to create the compound of this solid and its dual, the triakis icosahedron. In the image below, this dual is the blue polyhedron.

Trunc Dodeca dual the triakis icosahedron

The convex hull of this compound, below, I’m simply calling “the KRS derived from the truncated dodecahedron,” until and unless someone invents a better name for it.

KR solid based on the truncated dodecahedron

The next KRS shown is derived, in the same manner, from the truncated tetrahedron.

KR solid based on the truncated tetrahedron

Here is the KRS derived from the truncated cube.

KR solid based on the truncated cube

The truncated icosahedron is the “seed” from which the next KRS shown is derived. This KRS is a “stretched” form of a zonohedron called the rhombic enneacontahedron.

KR solid based on the truncated icosahedron

Another of these kite-rhombus solids, shown below, is based on the truncated octahedron.

KR solid based on the truncated octahedron

The next KRS shown is based on the rhombcuboctahedron.

KR solid derived from the rhombcuboctahedron

Two of the Archimedeans are chiral, and they both produce chiral kite-rhombus solids. This one is derived from the snub cube.

KR solid based on the snube cube

Finally, to complete this set of eight, here is the KRS based on the snub dodecahedron.

KR solid based on the snub dodecahedron

You may be wondering what happens when this same process is applied to the other five Archimedean solids. The answer is that all-kite polyhedra are produced; they have no rhombic faces. Two are “stretched” forms of Catalan solids, and are derived from the cuboctahedron and the icosidodecahedron:

If this procedure is applied to the rhombicosidodecahedron, the result is an all-kite polyhedron with two different face-types, as seen below.

all kited based on the RID

The two remaining Archimedean solids are the great rhombcuboctahedron and the great rhombicosidodecahedron, each of which produces a polyhedron with three different types of kites as faces.

The polyhedron-manipulation and image-production for this post was performed using Stella 4d: Polyhedron Navigator, which may be purchased or tried for free at http://www.software3d.com/Stella.php.