In this video, the great dodecahedron is stellated, by twentieths, into the great stellated dodecahedron, while a selection from Ludwig van Beethoven’s Ninth Symphony plays. The images for this video were created using Stella 4d, a program you can try for yourself (free trial download available), right here: http://www.software3d.com/Stella.php.
This compound was created using Stella 4d, software you can try for yourself here.
Shown above: the compound of the icosahedron and the rhombic dodecahedron. Below is its dual, the compound of the dodecahedron and the cuboctahedron.
Both these compounds were created using the “add/blend polyhedron from memory” function in Stella 4d: Polyhedron Navigator. To check out this program for yourself, just follow this link.
The dual to this cluster-polyhedron appears below. Both virtual models were created using Stella 4d: Polyhedron Navigator, software available here.
Aren’t you glad to know that? As soon as I found out icosahedra can form a rhombic dodecahedron (see last post), I knew this would be true as well. Why? Zome explains why, actually. It’s at http://www.zometool.com. Anything buildable with yellow Zome can be built out of icosahedra. Dodecahedra con form anything buildable with red Zome. Finally, if you can make it with blue Zome, it can be built out of rhombic triacontahedra. It follows that rhombicosidodecahedra can build anything Zome-constructible — but one look at a Zomeball makes that easy to believe, since Zomeballs are modified rhombicosidodecahedra.
Anyway, here’s the rhombic triacontahedron, made of dodecahedra:
[Image created with Stella 4d; see http://www.software3d.com/Stella.php for more info re: this program.]
If a zonish dodecahedron is created with zones based on the dodecahedron’s vertices, here is the result.
If the same thing is done with edges, this is the result — an edge-distorted version of the great rhombicosidodecahedron.
Another option is faces-only. Although I haven’t checked the bond-lengths, this one does have the general shape of the most-symmetrical 80-carbon-atom fullerene molecule. Also, this shape is sometimes called the “pseudo-truncated-icosahedron.”
The next zonish dodecahedron has had zones added based on the dodecahedron’s faces and edges, both.
Here’s the one for vertices and edges.
Here’s the one for faces and vertices.
Finally, the last of this set of seven has had zones added based on all three: faces, vertices, and edges.
All seven of these were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.
For years, I have used Zometools (sold here: http://www.zometool.com) to teach geometry. The constructions for the icosahedron and dodecahedron are easy to teach and learn, due to the use of short reds (R1s) and medium yellows (Y2s) for radii for the two of them, as shown below, with short blue (B1) struts as edges for both polyhedra.
Unexpectedly, a student (name withheld for ethical and legal reasons) combined the two models, making this:
I saw it, and wondered if the two combined Platonic solids could be expanded along the edges, to stellate both polyhedra, with medium blues (B2s), to form the great and small stellated dodecahedron. By trying it, I found out that this would require intersecting blue struts — so a Zomeball needed to be there, at the intersection. Trying, however, only told me that no available combination would fit. After several more attempts, I doubled each edge length, and added some stabilizing tiny reds (R0s), and found a combination that would work, to form a compound of the great and small stellated dodecahedron in which both edge lengths would be equal. In the standard (non-stellated) compound of the icosahedron and dodecahedron, in which the edges are perpendicular, they are unequal in length, and in the golden ratio, which is how that compound differs from the figure shown directly above.
Here’s the stabilized icosahedral core, after the doubling of the edge length:
This enabled stellation of each shape by edge-extension. Each edge had a length twice as long as a B2 added to each side — and it turns out, I discovered, that 2B2 in Zome equals B3 + B0, giving the golden ratio as one of three solutions solution to x² + 1/x = 2x (the others are one, and the golden ratio’s reciprocal). After edge-stellation to each component of the icosahedron/dodecahedron quasi-compound, this is what the end product looked like. This required assembling the model below at home, where all these pictures were taken, for one simple reason: this thing is too wide to fit through the door of my classroom, or into my car.
Here’s a close-up of the central region, as well.
This uses enlarged spheres centered on the dodecahedron’s vertices, overlapping so that they obscure the edges. Also, the faces are rendered invisible. I created it using Stella 4d, available at http://www.software3d.com/Stella.php.
In this model, the usual presentation of the icosahedron/dodecahedron dual compound has been altered somewhat. The “arms” of star pentagons have been removed from the dodecahedron’s faces, and the icosahedron is rendered “Leonardo-style,” with smaller triangles removed from each of the faces of the icosahedron, with both these alterations made to enable you to see the model’s interior structure. Also, the dodecahedron is slightly larger than usual, so that its edges no longer intersect those of the icosahedron.
This model was made using Stella 4d, software you can obtain for yourself, with a free trial download available, at http://www.software3d.com/Stella.php.
The dual of the icosahedron is the dodecahedron, and a compound can be made of those two solids. If one then takes the convex hull of this solid, the result is a rhombic triacontahedron. One can then made a compound of the rhombic triacontahedron and its dual, the icosidodecahedron — and then take the convex hull of that compound. If one then makes another compound of that convex hull and its dual, and then makes a convex hull of that compound, the dual of this latest convex hull is the polyhedron you see above.
I did try to make the faces of this solid regular, but that attempt did not succeed.
All of these polyhedral manipulations were were performed with Stella 4d: Polyhedron Navigator, available at http://www.software3d.com/Stella.php.
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