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.

Three Views of a Rotating Cluster of 33 Icosidodecahedra

33-icosidodeca

To make these three rotating cluster-polyhedra, I started with one icosidodecahedron in the center, then augmented each of its 32 faces with overlapping, additional icosidodecahedra, for a total of 33 icosidodecahedra per cluster. In the first image, only two colors are used: one for the triangular faces, and another for the pentagons. The second version, however, has the colors assigned by face-type, which is determined by each face’s placement in the overall cluster.

33-icosidodeca-ft

For the third version, I simply put Stella 4d (the program I use to make these images) into “rainbow color mode.” If you’d like to give Stella 4d a try, you can do so for free at this website.

33-icosidodeca-rc