A Zome Model of the Compound of the Icosidodecahedron and Its Dual, the Rhombic Triacontahedron

The polyhedral compound above contains an icosidodecahedron (blue) and a rhombic triacontahedron (red). In this compound, the icosidodecahedron’s edges are bisected, while the rhombic triacontahedron’s edges are split into segments with lengths in the square of the golden ratio (~2.618 to 1).

If you want Zome of your own, the place to buy it is http://www.zometool.com.

Augmenting the Icosidodecahedron With Pyramids

Here’s an icosidodecahedron, one of the thirteen Archimedean solids.

Using a computer program called Stella 4d (available here), I augmented each face of this solid with a pyramid. Here’s the result.

Also interesting is the dual of this pyramid-augmented icosidodecahedron:

A Starry Icosidodecahedron

The stars on the pentagonal faces were drawn using Geometer’s Sketchpad and MS-Paint. The icosidodecahedron itself was created using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.

An Intermediate Form Between the Icosidodecahedron and the Rhombic Triacontahedron

This polyhedron combines the faces of an icosidodecahedron (red and blue) with the those of a rhombic triacontahedron (green). The gaps between those two sets of polygons are the yellow rectangles. I made this using the “morph duals by expansion” function of Stella 4d: Polyhedron Navigator. You can try this program for yourself, free of charge, at http://www.software3d.com/Stella.php.

Expanding the Icosidodecahedron

This is the icosidodecahedron. It’s one of the thirteen Archimedean solids. To make an expanded version of it, I first augmented each of its faces with a prism.

Next, I formed the augmented icosidodecahedron’s convex hull.

This expanded icosidodecahedron has the twelve pentagonal faces (shown in red) and twenty triangular faces (shown in blue) of the original icosidodechedron. It also has sixty rectangular faces (yellow), and sixty isosceles triangles (shown in green). That’s a total of 152 faces.

To do all of this, I used a program called Stella 4d. If you’d like to try Stella for yourself, for free, just visit this website: http://www.software3d.com/Stella.php.

Cluster of 33 Icosidodecahedra

There’s one icosidodecahedron at the center of this cluster, with more icosidodecahedra attached to each of the central figure’s 32 faces. In the first version, the coloring is simply based as the number of sides for each face.

Augmented Icosidodeca 33.gif

In the next picture, the coloring is by face-type (position in the overall cluster).

Augmented Icosidodeca 33 color by face type.gif

The last image shown here has the cluster in “rainbow color mode.”

Augmented Icosidodeca 33 rainbow.gif

I used Stella 4d to make these — a program you may try for free right here.

Using Rhombic Triacontahedra to Build an Icosidodecahedron

These two polyhedra are the icosidodecahedron (left), and its dual, the rhombic triacontahedron (right).

One nice thing about these two polyhedra is that one of them, the rhombic triacontahedron, can be used repeatedly, as a building-block, to build the other one, the icosidodecahedron. To get this started, I first constructed one edge of the icosidodecahedron, simply by lining up four rhombic triacontahedra.

ID of RTCs edge

Three of these lines of rhombic triacontahedra make one of the icosidodecahedron’s triangular faces.

ID of RTCs triangle

Next, a pentagon is attached to this triangle.

ID of RTCs pent and triangle

Next, the pentagonal ring is surrounded by triangles.

ID of RTCs star.gif

More triangles and pentagons bring this process to the half-way point. If we were building a pentagonal rotunda (one of the Johnson solids), this would be the finished product.

ID of RTCs pentagonal rotunda.gif

Adding the other half completes the icosidodecahedron.

ID of RTCs complete.gif

All of these images were created using Stella 4d: Polyhedron Navigator. You may try this program yourself, for free, at http://www.software3d.com/Stella.php. The last thing I did with Stella, for this post, was to put the finished model into rainbow color mode.

ID of RTCs complete rainbow.gif

Pluto and Charon, Adorning an Icosidodecahedron

 

icosidodeca

Images obtained by NASA’s New Horizons space probe. Geometrical rendering done using Stella 4d, available at http://www.software3d.com/Stella.php.