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.


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

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. The majority of these things are geometrical. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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4 Responses to Building a Rhombic Enneacontahedron, Using Icosahedra and Elongated Octahedra

  1. swo8 says:

    That’s quite different, Robert. Very nice!

    Liked by 1 person

  2. Kathryn McAuley says:

    Hi Robert, Those ones i love the best. Just infinitesimus! (Really Cool)

    Liked by 1 person

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