Combining Octahedral and Icosahedral Symmetry to Form Pyritohedral Symmetry

Posted on 8 April 2015 by RobertLovesPi

Pyritohedral symmetry, seen by example both above and below, is most often described at the symmetry of a volleyball:

[Image of volleyball found here.]

To make the rotating polyhedral compound at the top, from an octahedron and an icosahedron, I simply combined these two polyhedra, using Stella 4d, which may be purchased (or tried for free) here.

In the process, I demonstrated that it is possible to combine a figure with octahedral (sometimes called cuboctahedral) symmetry, with a figure with icosahedral (sometimes called icosidodecahedral) symmetry, to produce a figure with pyritohedral symmetry.

Now I can continue with the rest of my day. No matter what happens, I’ll at least know I accomplished something.

The Final Stellation of the Icosahedron

Posted on 9 January 2015 by RobertLovesPi

This is what you get if you stellate an icosahedron seventeen times. The eighteenth stellation “loops” back around to the original figure, the icosahedron. For this reason, the figure above is often called “the final stellation of the icosahedron,” as well as “the complete icosahedron.” Its faces are twenty irregular star enneagons, of the type shown below. The red areas are the “facelets” which can be seen, while the other parts of the star enneagon are hidden inside the figure.

Both of these images were made using Stella 4d: Polyhedron Navigator, which you can try for yourself right here. A free trial download is available.

A Rhombic Dodecahedral Lattice, Made of Icosahedra

Posted on 22 December 2014 by RobertLovesPi

I used Stella 4d: Polyhedron Navigator to make this. You can find this program at http://www.software3d.com/Stella.php.

A Special Type of Compound, Built with Zome, of the Great and Small Stellated Dodecahedra

Posted on 17 December 2014 by RobertLovesPi

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.

An Unusual Presentation of the Icosahedron/Dodecahedron Base/Dual Compound

Posted on 5 December 2014 by RobertLovesPi

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.

Three Polyhedral Clusters of Icosahedra

Posted on 23 November 2014 by RobertLovesPi

In the last post on this blog, there were three images, and the first of these was a rotating icosahedron, rendered in three face-colors. After making it, I decided to see what I could build, using these tri-colored icosahedra as building blocks. Augmenting the central icosahedron’s red and blue faces with identical icosahedra creates this cubic cluster of nine icosahedra:

If, on the other hand, this augmentation is performed only on the blue faces of the central icosahedron, the result is a tetrahedral cluster of five icosahedra:

The next augmentation I performed started with this tetrahedral cluster of five icosahedra, and added twelve more of these icosahedra, one on each of the blue faces of the four outer icosahedra. The result is a cluster of 17 icosahedra, with an overall icosahedral shape.

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

On Icosahedra, and Pyritohedral Symmetry

Posted on 23 November 2014 by RobertLovesPi

In this icosahedron, the four blue faces are positioned in such a way as to demonstrate tetrahedral symmetry. The same is true of the four red faces. The remaining twelve faces demonstrate pyritohedral symmetry, which is much less well-known. It was these twelve faces that I once distorted to form what I named the “golden icosahedron” (right here: https://robertlovespi.wordpress.com/2013/02/08/the-golden-icosahedron/), but, at that point, I had not yet learned the term for this unusual symmetry-type.

To most people, the most familiar object with pyritohedral symmetry is a volleyball. Here is a diagram of a volleyball’s seams, found on Wikipedia.

Besides the golden icosahedron I found, back in 2013, there is another, better-known, alteration of the icosahedron which has pyritohedral symmetry, and it is called Jessen’s icosahedron. Here’s what it looks like, in this image, which I found at http://en.wikipedia.org/wiki/Jessen%27s_icosahedron.

The rotating icosahedron at the top of this post was made using Stella 4d, a program which may be purchased, or tried for free (as a trial version) at http://www.software3d.com/Stella.php.

An Alteration of the Icosahedron/Dodecahedron Compound

Posted on 18 November 2014 by RobertLovesPi

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.

A Polyhedral Demonstration of the Fact That Twenty Times Four Is Eighty

Posted on 16 November 2014 by RobertLovesPi

The Platonic solid known as the icosahedron has twenty triangular faces. This polyhedron resembles the icosahedron, but with each of the icosahedron’s triangles replaced by a panel of four faces: three isosceles trapezoids surrounding a central triangle. Since (20)(4) = 80, it is possible to know that this polyhedron has eighty faces — without actually counting them.

To let you see the interior structure of this figure, I next rendered its triangular faces invisible, to form “windows,” and then, just for fun, put the remaining figure in “rainbow color mode.”

I perform these manipulations of polyhedra using software called Stella 4d. If you’d like to try this program for yourself, the website to visit for a free trial download is http://www.software3d.com/Stella.php.

Icosidodecahedra, Icosahedra, and Dodecahedra

Posted on 13 November 2014 by RobertLovesPi

If one starts with a single icosidodecahedron, and then augments its pentagonal faces with dodecahedra, and its trianguar faces with icosahedra, this is the result.

This figure has gaps in it where two pentagons and two triangles meet around a vertex. If one puts icosidodecahedra in those gaps, this is the resulting figure.