A Hollow Faceting of the Rhombicosidodecahedron, and Its Hollow Dual

The images above all show a particular faceting of the rhombicosidodecahedron which, to my surprise, is hollow. It has the vertices of a rhombicosidodecahedron, but two different face-types, as seen in the smaller pictures: yellow hexagons, and red isosceles trapezoids. (To enlarge any image in this post, simply click on it.)

The dual of this polyhedron is even more obviously hollow, as seen below. Its faces, as seen in the still picture, are crossed hexagons — with edges which cross three times per hexagon, no less.

The software I used to make these polyhedra, Stella 4d, will return an error message if the user attempts to make a polyhedron which is not mathematically valid. When I’ve made things that look (superficially) like this before, I used “hide selected faces” to produce hollow geometrical figures which were not valid polyhedra, but that isn’t what happened here (I hid nothing), so this has me confused. Stella 4d (software you can buy, or try for free, here) apparently considers these valid polyhedra, but I am at a loss to explain such familiar concepts as volume for such unusual polyhedra, or how such things could even exist — yet here they are. Clarifying comments would be most appreciated.

A Hollow Octahedron Made of Rhombic Dodecahedra, with Variations


The original polyhedral cluster I built using Stella 4d (available here) is above. Below is its 29th stellation.

hOLLOW oCTAHEDRON mADE OF Rhombic Dodeca 29th stellation

And the 30th stellation, as well:

hOLLOW oCTAHEDRON mADE OF Rhombic Dodeca 30th stellation

This is the original polyhedral cluster’s dual:

hOLLOW oCTAHEDRON mADE OF Rhombic Dodeca dual

The next image is a variant of the original polyhedral cluster, rendered with only its edges, but not faces or vertices, visible. I wish I could remember exactly how I made this variant, but I simply cannot recall the exact methods I used.

hollow octahedron variant

This is the dual of the polyhedron shown immediately above, rendered in the same manner:

hollow object made of cubes -- dual of hollow octahedron variantThis is a compound of the two dual polyhedra right before this sentence.

hollow octahedron variant compound of it and dual

Four Hollow Polyhedron-Variants Related to the Rhombic Triacontahedron and the Icosahedron


Four Hollow Polyhedron-Variants Related to the Rhombic Triacontahedron and the Icosahedron

To make any of these smaller images as large as the first one, simply click on the smaller image of your choice.


These are all members of the same stellation-sequence, just like the two octahedron-variants in the last post.


I made these in just a few minutes with software, written by a friend of mine, called Stella 4d: Polyhedron Navigator. You can check it out for yourself at http://www.software3d.com/Stella.php. A free trial download is available.