The Twelfth Stellation of the Triakis Tetrahedron

Posted on 20 January 2015 by RobertLovesPi

Created with Stella 4d, available here.

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 Partially-Invisible Rhombicosidodecahedron, and One of Its Stellations

Posted on 21 December 2014 by RobertLovesPi

The polyhedron above originally had thirty yellow square faces, but I rendered them invisible so that the interior structure of this polyhedron could be seen.

When stellating such a partially-invisible figure, the new faces “inherited” from the “parent polyhedron” are either visible or invisible, depending on which type of face they are derived from. This makes for a very unusual look for some stellations, such as this, the rhombicosidodecahedron’s 50th:

I created these images using a program called Stella 4d: Polyhedron Navigator. You may try it for yourself at http://www.software3d.com/Stella.php.

A Polyhedral Journey, Beginning with the Snub Cube / Pentagonal Isositetrahedron Base/Dual Compound

Posted on 13 December 2014 by RobertLovesPi

The snub cube and its dual make an attractive compound. Since the snub cube is chiral, its chirality is preserved in this compound.

If you examine the convex hull of this compound, you will find it to be chiral as well.

Here is the mirror image of that convex hull:

These two convex hulls, of course, have twin, chiral, duals:

The two chiral convex hulls above (the red, blue, and yellow ones), made an interesting compound, as well.

This is also true of their chiral duals:

I next stellated this last figure numerous times (I stopped counting at ~200), to obtain this polyhedron:

After seeing this, I wanted to know what its dual would look like — and it was a nice polyhedron on which to end this particular polyhedral journey.

I make these transformations of polyhedra, and create these virtual models, using a program called Stella 4d. It may be purchased, or tried for free, at http://www.software3d.com/Stella.php.

A Collection of Unusual Polyhedra

Posted on 23 November 2014 by RobertLovesPi

In the post directly before this one, the third image was an icosahedral cluster of icosahedra. Curious about what its convex hull would look like, I made it, and thereby saw the first polyhedron I have encountered which has 68 triangular faces.

Still curious, I next examined this polyhedron’s dual. The result was an unusual 36-faced polyhedron, with a dozen irregular heptagons, and two different sets of a dozen irregular pentagons.

Stella 4d (the program I used to make all these images), which is available at http://www.software3d.com/Stella.php, has a “try to make faces regular” function, and I tried to use it on this 36-faced polyhedron. When making the faces regular is not possible, as was the case this time, it sometimes produce surprising results — and this turned out to be one of these times.

The next thing I did was to examine the dual of this latest polyhedron. The result, a cluster of tetrahedra and triangles, was completely unexpected.

The next alteration I performed was to create the convex hull of this cluster of triangles and tetrahedra.

Having seen that, I wanted to see its dual, so I made it. It turned out to have a dozen faces which are kites, plus another dozen which are irregular pentagons.

Next, I tried the “try to make faces regular” function again — and, once more, was surprised by the result.

Out of curiosity, I then created this latest polyhedron’s convex hull. It turned out to have four faces which are equilateral triangles, a dozen other faces which are isosceles triangles, and a dozen faces which are irregular pentagons.

Next, I created the dual of this polyhedron, and it turns out to have faces which, while not identical, can be described the same way: four equilateral triangles, a dozen other isosceles triangles, and a dozen irregular pentagons — again. To find such similarity between a polyhedron and its dual is quite uncommon.

I next attempted the “try to make faces regular” function, once more. Stella 4d, this time, was able to make the pentagons regular, and the triangles which were already regular stayed that way, as well. However, to accomplish this, the twelve other isosceles triangles not only changed shape a bit, but also shifted their orientation inward, making the overall result a non-convex polyhedron.

Having a non-convex polyhedron on my hands, the next step was obvious: create its convex hull. One more, I saw a polyhedron with faces which were four equilateral triangles, a dozen other isosceles triangles, and a dozen regular pentagons.

I then created the dual of this polyhedron, and, again, found myself looking at a polyhedron with, as faces, a dozen irregular pentagons, a dozen identical isosceles triangles, and four regular triangles. However, the arrangement of these faces was noticeably different than before.

Given this difference in face-arrangement, I decided, once more, to use the “try to make faces regular” function of Stella 4d. The results were, as before, unexpected.

Next, I created this latest polyhedron’s dual.

At no point in this particular “polyhedral journey,” as I call them, had I used stellation — so I decided to make that my next step. After stellating this last polyhedron 109 times, I found this:

I then created the dual of this polyhedron. The result, unexpectedly, had a cuboctahedral appearance.

A single stellation of this latest polyhedron radically altered its appearance.

My next step was to create the dual of this polyhedron.

This seemed like a good place to stop, and so I did.

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.

A Polyhedral Journey, Beginning with a Near-Miss Johnson Solid Featuring Enneagons

Posted on 9 October 2014 by RobertLovesPi

When Norman Johnson first found, and named, all the Johnson solids in the latter 1960s, he came across a number of “near-misses” — polyhedra which are almost Johnson solids. If you aren’t familiar with the Johnson solids, you can find a definition of them here. The “near-miss” which is most well-known features regular enneagons (nine-sided polygons):

This is the dual of the above polyhedron:

As with all polyhedra and their duals, a compound can be made of these two polyhedra, and here it is:

Finding this polyhedron interesting, I proceeded to use Stella 4d (polyhedron-manipulation software, available at http://www.software3d.com/Stella.php) to make its convex hull.

Here, then, is the dual of this convex hull:

Stella 4d has a “try to make faces regular” function, and I next used it on the polyhedron immediately above. If this function cannot work, though — because making the faces regular is mathematically impossible — one sometimes gets completely unexpected, and interesting, results. Such was the case here.

Next, I found the dual of this latest polyhedron.

The above polyhedron’s “wrinkled” appearance completely surprised me. The next thing I did to change it, once more, was to create this wrinkled polyhedron’s convex hull. A convex hull of a non-convex polyhedron is simply the smallest convex polyhedron which can contain the non-convex polyhedron, and this process often has interesting results.

Next, I created this latest polyhedron’s dual:

I then attempted “try to make faces regular” again, and, once more, had unexpected and interesting results:

The next step was to take the convex hull of this latest polyhedron. In the result, below, all of the faces are kites — two sets of twenty-four each.

I next stellated this kite-faced polyhedron 33 times, looking for an interesting result, and found this:

This looked like a compound to me, so I told Stella 4d to color it as a compound, if possible, and, sure enough, it worked.

The components of this compound looked like triakis tetrahedra to me. The triakis tetrahedron, shown below, is the dual of the truncated tetrahedron. However, I checked the angle measurement of a face, and the components of the above compound-dual are only close, but not quite, to being the same as the true triakis tetrahedron, which is shown below.

This seemed like a logical place to end my latest journey through the world of polyhedra, so I did.

A Polyhedral Journey, Beginning with Face-Based Zonohedrification of an Icosahedron

Posted on 28 September 2014 by RobertLovesPi

To begin this, I took an icosahedron, and made a zonish polyhedron with it, with the new faces based on the zones of the existing faces. Here’s the result.

Next, I started stellating the polyhedron above. At the sixth stellation, I found this. It’s a true zonohedron, and the first polyhedron shown here is merely “zonish,” because one has triangles, and the other does not. (One of the requirements for a polyhedron to be a zonohedron is that all its faces must have an even number of sides.)

After that, I kept stellating, finding this as the 18th stellation of the first polyhedron shown here.

With this polyhedron, I then made its convex hull.

At this point, the irregular hexagons were bothering me, so I used Stella 4d‘s “try to make faces regular” option. (Stella 4d is polyhedron-manipulation software you can try for free, or purchase, right here.)

The next step I chose was to augment all the yellow trapezoids with prisms, each with a height 1.6 times the trapezoids average edge length.

The next step was, again, to make the convex hull.

At this point, I tried “try to make faces regular” again, and was pleased with the result. The green rectangles became so thin, however, that I had to stop displaying the edges and vertices, in order for then to be seen.

Next, I augmented both the blue faces (decagons) and the yellow faces (dodecagons) with antiprisms, again using a height 1.6 times that of the augmented faces’ average edge-lengths.

Next, I made the convex hull again — a step I often take immediately after augmenting a polyhedron.

This one surprised me, as it is more complicated than I expected. To clean things up a bit, I augmented only the trapezoids (dark pink) with prisms, and dodecagons (green) with antiprisms, again using the factor 1.6 for the augmentation-height.

The next step I chose was to take the convex hull, once more. I had not yet noticed that the greater height of the trapezoidal prisms would cause the dodecagonal antiprisms to be “lost” by this step, though.

Next, “try to make faces regular” was used again.

This last result had me feeling my polyhedral journey was going in circles, so I tried augmentation again, but in a different way. I augmented this polyhedron, using prisms, on only the red trapezoids (height factor, 1.6 again) and the blue rectangles (new height factor, 2.3 times average edge length).

After that, it was time to make another convex hull — and that showed me that I had, indeed, taken a new path.

I found the most interesting faces of this polyhedron to be the long, isosceles trapezoids, so I augmented them with prisms, ignoring the other faces, using the new height-factor of 2.3 times average edge length this time.

Of course, I wanted to see the convex hull of this. Who wouldn’t?

I then started to stellate this figure, choosing the 14th stellation as a good place to stop, and making the edges and vertices visible once more.

A Zonish Icosahedron, and Some of Its “Relatives”

Posted on 27 September 2014 by RobertLovesPi

To begin this, I used Stella 4d (available here) to create a zonish polyhedron from the icosahedron, by adding zones along the x-, y-, and z-axes. The result has less symmetry than the original, but it is symmetry of a type I find particularly interesting.

After making that figure, I began stellating it, and found a number of interesting polyhedra in this polyhedron’s stellation-series. This is the second such stellation:

This is the 18th stellation:

The next one, the 20th stellation, is simply a distorted version of the Platonic dodecahedron.

This one is the 22nd stellation:

This is the 30th stellation:

The next really interesting stellation I found was the 69th:

At this point, I returned to the original polyhedron at the top of this post, and examined its dual. It has 24 faces, all of which are quadrilaterals.

This is the third stellation of this dual — and another distorted Platonic dodecahedron.

This is the dual’s 7th stellation:

And this one is the dual’s 18th stellation:

At this point, I took the convex hull of this 18th stellation of the original polyhedron’s dual, and here’s what appeared:

Here is this convex hull’s dual:

Stella 4d, the program I use to make these (available here), has a built-in “try to make faces regular” function. When possible, it works quite well, but making the faces of a polyhedron regular, or even close to regular, is not always possible. I tried it on the polyhedron immediately above, and obtained this interesting result:

While interesting, this also struck me as a dead end, so I returned to the red-and-yellow convex hull which is the third image above, from right here, and started stellating it. At the 19th stellation of this convex hull, I found this: