A Cuboctahedral Cluster of Rhombic Dodecahedra

cuboctahedron of Rhombic Dodeca

It is well-known that the cuboctahedron and the rhombic dodecahedron are dual polyhedra. However, until I stumbled upon this, I was unaware that rhombic dodecahedra could actually be arranged into a cluster with the overall shape of a cuboctahedron.

[Software credit: see http://www.software3d.com/Stella.php for more information about Stella 4d, the program I use to make these rotating images. A free trial download is available at that website.]

Cuboctahedral Cluster of Rhombic Triacontahedra

Augmented Rhombic Triaconta

Due to their high number of planes of symmetry, rhombic triacontahedra make excellent building blocks to build other polyhedra. To make this, I used a program called Stella 4d, which you can try right here.

A Collection of Four Polyhedra Decorated with Mandalas

First, a cuboctahedron.

Rotating Cubocta with rotating mandalasNext, its dual, the rhombic dodecahedron.

Rotating RD with rotating mandalas

And, after that, the icosidodecahedron.

Rotating Icosidodeca with rotating mandalas

And finally, its dual, the rhombic triacontahedron.

Rotating RTC with rotating mandalas

All of these rotating images were assembled using Stella 4d, available at http://www.software3d.com/Stella.php.

Dodecahedral Cluster of Cuboctahedra and Icosidodecahedra

Augmented IcosidodDSJFGSca

I made this using Stella 4d:  Polyhedron Navigator, software you may try for yourself at http://www.software3d.com/Stella.php.

Pulsating Cuboctahedron, Featuring Enneagrammic Mandalas

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Pulsating Cuboctahedron, Featuring Enneagrammic Mandalas

The enneagramic mandalas on the square faces of this cuboctahedron are from the last post, with inverted-color, smaller versions of the same image on the triangular faces. These mandalas were created using Geometer’s Sketchpad and MS-Paint. Projecting them onto the faces of the cuboctahedron, and then creating this rotating, pulsating .gif image, however, took a third program: Stella 4d, which you can buy, or try for free, at http://www.software3d.com/Stella.php.

A Non-Convex Variant of the Cuboctahedron

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A Non-Convex Variant of the Cuboctahedron

The convex hull of this solid is the cuboctahedron. To me, it looks like a hybrid of that solid, and the Stella Octangula. I created it using Stella 4d, which is available (including a free trial download) at http://www.software3d.com/Stella.php.

The Zonish Cuboctahedron: A New Near-Miss Discovery?

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The Zonish Cuboctahedron:  A New Near-Miss Discovery?

If one starts with a cuboctahedron, and then creates a zonish polyhedron from it, adding zones (based on the faces) to the faces which already exist, here is the result, below, produced by Stella 4d: Polyhedron Navigator (software you may buy or try at http://www.software3d.com/Stella.php):

new nearmiss before making faces regular its a face based zonish cuboctahedron

The hexagons here, in this second image, are visibly irregular. The four interior hexagon-angles next to the octagons each measure more than 125 degrees, and the other two interior angles of the hexagons each measure less than 110 degrees — too irregular for this to qualify as a near-miss to the Johnson solids. However, Stella includes a “try to make faces regular” function, and applying it to the second polyhedron shown here produces the polyhedron shown in a larger image, at the top of this post.

It is this larger image, at the top, which I am proposing as a new near-miss to the 92 Johnson solids. In it, the twelve hexagons are regular, as are the eight triangles and six octagons. The only irregular faces to be found in it are the near-squares, which are actually isosceles trapezoids with two angles (the ones next to the octagons) measuring ~94.5575 degrees, and two others (next to the triangles) measuring 85.4425 degrees. Three of the edges of these trapezoids have the same length, and this length matches the lengths of the edges of both the hexagons and octagons. The one side of each trapezoid which has a different length is the one it shares with a triangle. These triangle-edges are ~15.9% longer than all the other edges in this proposed near-miss.

My next step is to share this find with others, and ask for their help with these two questions:

    1. Has this polyhedron been found before?
    2. Is it close enough to being a Johnson solid to qualify as a near-miss?

Once I learn the answers to these questions, I will update this post to reflect whatever new information is found. If this does qualify as a near-miss, it will be my third such find. The other two are the tetrated dodecahedron (co-discovered, independently, by myself and Alex Doskey) and the zonish truncated icosahedron (a discovery with which I was assisted by Robert Webb, the creator of Stella 4d).

More information about these near-misses, one of my geometrical obsessions, may be found here:  https://en.wikipedia.org/wiki/Near-miss_Johnson_solid

Cuboctahedron with Mandalas

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Cuboctahedron with Mandalas

The images on the faces of this polyhedron may be seen in still black and white in the previous post. I used Geometer’s Sketchpad and MS-Paint to make the flat image, and then Stella 4d to put it all together. You may try Stella for free at http://www.software3d.com/Stella.php.

A Space-Filling Pair of Polyhedra: The Cuboctahedron and the Octahedron

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A Space-Filling Pair of Polyhedra:  The Cuboctahedron and the Octahedron

There are only a few polyhedra which can fill space without leaving gaps, without “help” from a second polyhedron. This filling of space is the three-dimensional version of tessellating a plane. Among those that can do this are the cube, the truncated octahedron, and the rhombic dodecahedron.

If multiple polyhedra are allowed in a space-filling pattern, this opens new possibilities. Here is one: the filling of space by cuboctahedra and octahedra. There are others, and they are likely to appear as future blog-posts here.

Software credit: I made this virtual model using Stella 4d, polyhedral-manipulation software you can buy, or try as a free trial download, at http://www.software3d.com/Stella.php.