92 Dodecahedra, Arranged as a Single Rhombic Triacontahedron

With 92 dodecahedra, if you arrange them just right, you can make a model of a rhombic triacontahedron:

RTC Augmented DodecaFor purposes of comparison, here is what the rhombic triacontahedron normally looks like:

Rhombic Triaconta

Also, referring back to the first model shown, here is a picture of just one of the red rhombi-made-of-dodecahedra.

RHOMBUS Dodeca

The first polyhedron shown in this post has an interesting dual, as well. Here it is, colored by face-type (position within the overall shape):

Dual of RTC Augmented Dodeca

Here is another view of the dual, colored by number of edges per face.

Dual of RTC Augmented Dodeca

Here’s one more view of the dual, in “rainbow color mode.”

dual RTC Augmented Dodeca rainbow

Returning to the original model, at the top of this post, here’s what it looks like, if colored by face type:

RTC Augmented Dodeca face type

Here’s one more view, in “rainbow color mode.”

RTC Augmented Dodeca rainbow

All of these images were created using Stella 4d, a program you can buy, or try for free, right here.

Building a Rhombic Enneacontahedron, Using Icosahedra and Elongated Octahedra

With four icosahedra, and four octahedra, it is possible to attach them to form this figure:

Augmented Icosa

This figure is actually a rhombus, but the gap between the two central icosahedra is so small that this is hard to see. To remedy this problem, I elongated the octahedra, thereby creating this narrow rhombus:

narrow rhombus

It is also possible to use the same collection of polyhedra to make a wider rhombus, as seen below.

wide rhombus

These aren’t just any rhombi, either, but the exact rhombi found in the polyhedron below, the rhombic enneacontahedron. It has ninety rhombi as faces: sixty wide ones, and thirty narrow ones.

REC

As a result, it is possible to use the icosahedra-and-elongated-octahedra rhombi, above, to construct a rhombic enneacontahedron made of these other two polyhedra. The next several images show it under construction (I “built” it using Stella 4d, available at this website), culminating with the complete figure.

panelnof five rhombi

panel of ten rhombi

bowl towards rec

giant rec about half complete

giant rec almost finished

giant rec complete

Lastly, I made one more image — the same completed shape, but in “rainbow color mode.”

giant rec complete rainbow