If someone had asked me if it were possible to form a symmetric polyhedra out of irregular triangles and octagons, using exactly sixty of one type each, I would have guessed that it were not possible. Why does it work here? Part of the reason is that each triangle borders three octagons, and each octagon borders three triangles — a necessary, but not sufficient, condition. This is a partial truncation of an isomorph of the pentagonal hexacontahedron, the dual of the snub dodecahedron. As such, no surprise — it’s chiral.

This was made while stumbling about in the wilderness of the infinite number of possible polyhedra using Stella 4d: Polyhedron Navigator. You can get it here: http://www.software3d.com/Stella.php.

This is the dual of the polyhedron seen as the second image in the last post on this blog. If colored differently, so that only parallel faces have the same color, it looks like this (click to enlarge):

I used Stella 4d to make these images, and you can find that program at http://www.software3d.com/Stella.php.

I made this using Stella 4d, which you can find at http://www.software3d.com/Stella.php.

It doesn’t affect the coloring of this tessellation, but, in terms of vertices and edges, this tessellation is the dual of the one in the previous two posts.

After making the above polyhedron using Stella 4d (a program you can try for free at www.software3d.com/Stella.php), I checked its dual, which is shown below. I was surprised at its appearance, for it resembles a stellated polyhedron, even though it was created by a completely different process.

The image above is a compound of the rhombic triacontahedron (the dual of the icosidodecahedron) and a strombic hexacontahedron (the dual of the rhombicosidodecahedron). Below, you’ll find a compound of six square-based pyramids, all with their “centers of mass” (assuming uniform density) displaced, from the compound’s center, by equal amounts. In response to a request I have received, polyhedral images which rotate more slowly are coming soon . . . after I have finished posting my backlog of already-produced polyhedral .gif files, since there is no way to slow them down after they are already created.

The program I use for these polyhedral investigations is Stella 4d, available at www.software3d.com/Stella.php.

Each face of a great dodecahedron is a regular pentagon, and each of those pentagonal faces has an icosidodecahedron attached to it, in the figure above. The dual of this figure appears below.

Both images were created with Stella 4d, software available at www.software3d.com/Stella.php.

I recently stumbled upon some wrinkled polyhedra. Such polyhedra have an unusually high amount of surface area, compared to their volumes, relative to most more-familiar polyhedra. The next one shown, below, is the dual of the first one. shown, above.

The two above were made before I received a request to slow down the spin-rate of the polyhedra I post, most of which, in the past, have had a rotational period of four seconds. This next one has the new, slower spin-rate (with T = 6 seconds) which I am now using:

I hope all of my readers prefer this change, especially since it takes 50% more memory, per file, to slow these down to 2/3 their previous rotational speed.

These were all made using Stella 4d, software which can be tried or bought here: http://www.software3d.com/Stella.php.

The original polyhedral cluster I built using Stella 4d (available here) is above. Below is its 29th stellation.

And the 30th stellation, as well:

This is the original polyhedral cluster’s dual:

The next image is a variant of the original polyhedral cluster, rendered with only its edges, but not faces or vertices, visible. I wish I could remember exactly how I made this variant, but I simply cannot recall the exact methods I used.

This is the dual of the polyhedron shown immediately above, rendered in the same manner:

This is a compound of the two dual polyhedra right before this sentence.

To form the cluster-polyhedron above, I started with one truncated octahedron in the center, and then augmented each of its fourteen faces with another truncated octahedron. Since the truncated octahedron is a space-filling polyhedron, this cluster-polyhedron has no gaps, nor overlaps. The same cluster-polyhedron is below, but colored differently:  each set of parallel faces gets a color of its own.

This is the cluster-polyhedron’s sixth stellation, using the same coloring-scheme as in the last image:

Here’s the sixth stellation again, but with the coloring scheme that Stella 4d:  Polyhedron Navigator (the program I use to make these images) calls “color by face type.” If you’d like to try Stella for yourself, you can do so here.

Also colored by face-type, here are the 12th, 19th, and 86th stellations.

Leaving stellations now, and returning to the original cluster-polyhedron, here is its dual.

This image reveals little about this dual, however, for much of its structure is internal. So that this internal structure may be seen, here is the same polyhedron, but with only its edges visible.

Finally, here is an edge-rendering of the original cluster-polyhedron, but with vertices shown as well — just not the faces.

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:

I also found an interesting polyhedron as the 19th stellation of the dual which is three images above:

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.

I’m going to start this experiment with a single octahedron, with faces in two colors, placed so that two faces which share an edge are always of different colors.

Next, I will augment the red faces — and only the red faces — with identical octahedra.

The regions with four blue, adjacent faces look as though they might hold icosahedra — but I checked, and they don’t quite fit. I will therefore continue the same process — augmenting only the red faces with more octahedra of the original type.

I’ve now decided that I definitely like this game, so I’ll keep playing it.

Immediately above, at the fourth of these images, some of the octahedra have started to overlap slightly, but I’m choosing to not be bothered by that — I’m continuing the now-established pattern, just in order to see where it takes me.

The regions of overlap are now far more obvious, but I’m continuing, anyway. Why? Because this is fun, that’s why! Right now, Stella 4d, the program I use to do these polyhedral manipulations, is chugging away on the next one. (This program is avilable at http://www.software3.com/Stella.php.) Ah, it’s ready — here it is!

Rather than repeat this process again, I now have another question: what would the convex hull of this figure look like? (A convex hull of a non-convex polyhedron is the smallest convex polyhedron which can contain a given non-convex polyhedron.) With Stella 4d, that’s easily answered.

I must admit this: that was nothing like what I expected — but such unexpected discoveries are a large part of what makes these polyhedral investigations with Stella 4d so much fun. And now, to close this particular polyhedral journey, I will have Stella 4d produce, for me, the dual of the convex hull shown above. (In case you aren’t familiar with duality regarding polyhedra, it describes the relationship between the octahedron, with which this post began, and the familar cube. Basically, with duals, faces and verticies are “flipped” over edges, although that is an extremely informal and imprecise way to describe the at the process.)

And with that, my friends, I bid you good night!

One of the thirteen rhombic dodecahedra in this cluster cannot be seen, for it is hidden in the middle. The other twelve are each attached to a face of the central rhombic dodecahedron.

If one then creates the convex hull of this cluster — the smallest convex polyhedron which can contain it — this is the result:

This polyhedron has fifty faces:  the six square faces of a cube, the eight triangular faces of an octahedron, the twelve rhombic faces of a rhombic dodecahedron, and twenty-four rectangles to fill the gaps between the other faces.

This fifty-faced polyhedron also has an interesting dual, with 48 faces, all of which are kites. Half of these 48 kites are of one type, and arranged into eight panels of three kites each, while the other half are arranged into six panels of four kites each:

Returning to the fifty-faced polyhedron, two images above, here is what happens if one tries to make each face as regular as possible:

In this polyhedron, the six squares are still squares, the eight triangles are still regular, and the twelve rhombi are still rhombi, although these rhombi are wider than before. The 24 rectangles, however, have now been transformed into isosceles trapezoids.

[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.]

It would really be a pain to count the faces of this polyhedron, in order to verify that there are 270 of them. Fortunately, it isn’t necessary to do so. The polyhedron above is made of rhombus-shaped panels which correspond to the thirty faces of the rhombic triacontahedron. Each of these panels contains nine faces: one square, surrounded by eight triangles. Since (9)(30) = 270, it is therefore possible to see that this polyehdron has 270 faces, without actually going to the trouble to count them, one at a time.

The software I used to make this polyhedron may be found at http://www.software3d.com/Stella.php, and is called Stella 4d. With Stella 4d, a single mouse-click will let you see the dual of a polyhedron. Here’s the dual of the one above.

This polyhedron is unusual, in that it has faces with nine sides (enneagons, or nonagons), as well as fifteen sides (pentadecagons). However, these enneagons and pentadecagons aren’t regular — yet — but they will be in the next post.

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.

If one starts with a central truncated octahedron, leaves its six square faces untouched, and augments its eight hexagonal faces with trianglular cupolae, this is the result.

Seeing this, I did a quick check of its dual, and found it quite interesting:

After seeing this dual, I next created its convex hull.

After seeing this convex hull, I next creating its dual:  one of several 48-faced polyhedra I have found with two different sets of twenty-four kites as faces, one set in six panels of four kites each, and the other set consisting of eight sets of three kites each. I think of these recurring 48-kite-faced polyhedra as polyhedral expressions of a simple fact of arithmetic: (6)(4) = (8)(3) = 24.

I use Stella 4d (available at http://www.software3d.com/Stella.php) to perform these polyhedral transformations. The last one I created in this particular “polyhedral journey” is shown below — but, unfortunately, I cannot recall exactly what I did, to which of the above polyhedra, to create it.

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, 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 irregaular 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 irregaular 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.

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 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. If may be purchased, or tried for free, at http://www.software3d.com/Stella.php.