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.

A Space-Filling Arrangement of Polyhedra Using Truncated Cubes, Rhombcuboctahedra, Cubes, and Octagonal Prisms

Posted on 22 November 2014 by RobertLovesPi

This image above has only one polyhedron-type hidden from view, in the center: a red truncated cube. Next, more of this pattern I just found will be added.

The next step will be to add another layer of blue octagonal prisms.

And now, more yellow cubes.

This was an accidental discovery I made, just messing around with Stella 4d, a program you may try for yourself at http://www.software3d.com/Stella.php. The next cells added will be red truncated cubes.

Next up, I’ll add a set of pink rhombcuboctahedra.

The next set of polyhedra added: some yellow cubes, and blue octagonal prisms.

Now I’ll add more of the red truncated cubes.

At this point, more yellow cubes are needed.

The next polyhedra added will be pink rhombcuboctahedra.

And now, more of the blue octagonal prisms.

As long as this pattern is followed, this may be continued without limit, filling space, without leaving any gaps.

A Cluster of Nine Octahedra, and Related Polyhedra

Posted on 22 November 2014 by RobertLovesPi

If one starts with a central octahedron, then augments each of its eight triangular faces with identical octahedra, this is the result.

It is then possible to augment each visible triangle of this cluster with yet more octahedra, which produces this result, in which some octahedra overlap each other.

After making this, I wanted to see its convex hull: the smallest, tightest-fitting convex polyhedron which can contain a given non-convex polyhedron. (I use Stella 4d: Polyhedron Navigator to perform these manipulations of polyhedra, and this program makes this a fast and easy process. If you’d like to try this software, even as a free trial download, the website to visit is http://www.software3d.com/Stella.php.) Here’s what this convex hull, which bears a resemblance to the rhombcuboctahedron, looks like.

Looking for previously-unseen, and interesting, polyhedra, I then starting stellating this convex hull. I did find something interesting — to me, anyway — after only two stellations.

That concluded my latest polyhedral investigation, but I certainly don’t intend it to be my last.

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.

Two Different Clusters of Thirty-One Rhombicosidodecahedra

Posted on 16 November 2014 by RobertLovesPi

The cluster above was formed by starting with one rhombicosidodecahedron, and then augmenting each of its thirty square faces with another rhombicosidodecahedron. If you examine the single rhombicosidodecahedron below, though, you’ll see that this can be done in two different ways, each of which produces a cluster with the same degree of symmetry as the original. (If less symmetry is acceptable, there are far more than two different ways to obtain such clusters, but that does not interest me.)

When new rhombicosidodecahedra are attached to the square faces of this central rhombicosidodecahedron, they new ones can be oriented such that pentagons are placed above pentagons, and triangles above triangles — or the new ones can be oriented the other way, so that pentagons are placed above triangles, and triangles above pentagons. One of these possibilities produces the cluster at the top of this post. The other possibility produces the cluster shown below.

All of these polyhedral manipulations were performed using a program called Stella 4d: Polyhedron Navigator, which may be purchased (or tried for free, as a trial download) at http://www.software3d.com/Stella.php.

72-Faced Snub Dodecahedron Variant, and Related Polyhedra

Posted on 16 November 2014 by RobertLovesPi

Like the snub dodecahedron itself, which this resembles, this polyhedron is chiral, meaning it exists in left- and right-handed forms. One version is shown above, and its mirror-image is shown below.

With any chiral polyhedron, it is possible to make a compound out of the two mirror-images. Here is the enantiomorphic-pair compound for this polyhedron.

After making this compound, I was curious about what sort of convex hull it would have, so I used the program I employ for these polyhedral investigations, Stella 4d (available at http://www.software3d.com/Stella.php), to find out:

This polyhedron contains irregular icosagons, which are twenty-sided polygons. After playing around with this for a while, I was able to construct a related polyhedron in which the icosagons were regular — and that was one of the polyhedra seen on the post immediately before this one, which I then altered to form the others there. Had I not actually seen it happen myself, I would not have suspected there would be any connection between the snub dodecahedron, and polyhedra containing regular icosagons.

Three Polyhedra, Each Featuring One Dozen Regular Icosagons

Posted on 16 November 2014 by RobertLovesPi

Icosagons are polygons with twenty sides, and do not appear in any well-known polyhedra. The first of these three regular-icosagon-based polyhedra has 122 faces.

The second of these polyhedra, each of which bears an overall resemblance to a dodecahedron, has 132 faces.

Finally, the third of these polyhedra has a total of 152 faces.

I used Stella 4d to make each of these virtual polyhedron models — and you may try this program for free at http://www.software3d.com/Stella.php.

A Polyhedron Featuring Twelve Regular Pentadecagons, and Twenty Regular Enneagons

Posted on 16 November 2014 by RobertLovesPi

In the last post here, there were two polyhedra shown, and the second one included faces with nine sides (enneagons, also known as nonagons), as well as fifteen sides (pentadecagons), but those faces were not regular.

The program I use to manipulate polyhedra, Stella 4d (available at http://www.software3d.com/Stella.php), has a “try to make faces regular” function included. When I applied it to that last polyhedron, in the post before this one, Stella was able to make the twenty enneagons and twelve pentadecagons regular. The quadrilaterals are still irregular, but only because squares simply won’t work to close the gaps of a polyhedron containing twenty regular enneagons and twelve regular pentadecagons. These quadrilaterals are grouped into thirty panels of four each, so there are (4)(30) = 120 of them. Added to the twelve pentadecagons and twenty enneagons, this gives a total of 152 faces for this polyhedron.

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.