From the Rhombic Enneacontahedron to an All-Kite Polyhedron

This is the rhombic enneacontahedron, one of the few well-known zonohedra. Its ninety faces have two types: sixty wide rhombi, and thirty narrow rhombi.

In the image above, the thirty narrow rhombi of the rhombic enneacontahedron have been augmented with prisms

The next step in my polyhedral play was to create the convex hull of this augmented rhombic enneacontahedron. This produced the solid shown above. To make the one shown below, I next used a function called “try to make faces regular.” The result is a symmetrohedron with 122 faces: 12 regular pentagons, 30 rhombi, 60 almost-square isosceles trapezoids, and thirty equilateral triangles.

Finally, I examined the dual of this symmetrohedron, which turned out to have 120 faces: two sets of sixty kites each.

The program I used to create these polyhedral images is called Stella 4d, and you can try it yourself (as a free trial download) at http://www.software3d.com/Stella.php.

A Blend of the Icosahedron and the Rhombic Enneacontahedron

This is the icosahedron, one of the Platonic solids. It has twenty faces.

The polyhedron below is the rhombic enneacontahedron, a well-known zonohedron with ninety faces.

Finally, here is a polyhedron which blends these two. It has 20 + 90 = 110 faces.

I used Stella 4d: Polyhedron Navigator to make these images. You can try this program for free at http://www.software3d.com/Stella.php.

A Twice-Zonohedrified Dodecahedron

If one starts with a dodecahedron, and then creates a zonohedron based on that solid’s vertices, the result is a rhombic enneacontahedron.

If, in turn, one then creates a new zonohedron based on the vertices of this rhombic enneacontahedron, the result is this 1230-faced polyhedron — a twice-zonohedrified dodecahedron. Included in its faces are thirty dodecagons, sixty hexagons, and sixty octagons, all of them equilateral.

Stella 4d: Polyhedron Navigator was used to perform these transformations, and to create the rotating images above. You can try this program for yourself, free, at http://www.software3d.com/Stella.php.

A Rhombic Enneacontahedron, Made of Zome

Zome is a ball-and-stick modeling system which can be used to make millions of different polyhedra. If you’d like to get some Zome for yourself, just visit http://www.zometool.com.

Normal and Expanded Versions of the Rhombic Enneacontahedron

The polyhedron above is the rhombic enneacontahedron. Sixty of its faces are wide (yellow) rhombi, while the other thirty are narrow (red) rhombi. The wider rhombi are arranged in twelve panels of five rhombi each. If those panels are moved outward from the center by just the right amount, the narrower rhombi have room to expand, becoming equilateral octagons:

Both of these rotating images were created using Stella 4d, a program you can try, for free, at http://www.software3d.com/Stella.php.

A Rhombic Enneacontahedron, Adorned with Jumpy Tessellations Which Resemble Rhombic Enneacontahedra

Zonohedrified Dodeca.gif

Software credit: I made this using Stella 4d, available 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