An Experiment Involving Augmentation of Octahedra with More Octahedra, Etc.

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.

1

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

2

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.

3

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

4

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.

5

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!

6

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.

Convex hull

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.)

dual of Convex hull

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

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

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.

1 face-based zonish icosahedron

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.)

2 6th stellation face-based zonish icosahedronAfter that, I kept stellating, finding this as the 18th stellation of the first polyhedron shown here.

3 18th stellation face-based zonish icosahedron

With this polyhedron, I then made its convex hull.

4 Convex hull of 18th stellation of face-based zonish icosahedronAt 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.)

5 spring model of convex hull of 18th stellation of face-based icosahedron

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.

6 Augmented sping model of convex hull of 18th stellation of FBZI

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

7 Convex hull of augmented 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.

8 spring model of last oneNext, 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.

9 Augmented Poly 9th in series

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

10 Convex hull

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.

11 augmented Convex hull

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.

12 convex hull

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

13 spring model

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).

14 augmented spring model

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

15 Convex hullI 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.

16 augmented Convex hull

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

17 Convex hull

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.

18 the 14th stellation of the previous Convex hull

A Simulation of Crystalline Growth Using Polyhedral Augmentation

Crystals and crystalline growth have been studied for centuries because of, at least in part, their symmetry. Crystals are cut in such a way as to increase this symmetry even more, because most people find symmetry attractive. However, where does the original symmetry in a crystal come from? Without it, jewelers who cut gemstones would not exist, for the symmetry of crystalline minerals themselves is what gives such professionals the raw materials with which to work.

To understand anything about how crystals grow, one must look at a bit of chemistry. The growth of crystals:

  • Involves very small pieces:  atoms, molecules, ions, and/or polyatomic ions
  • Involves a small set of simple rules for how these small pieces attach to each other

Why small pieces? That’s easy:  we live in a universe where atoms are tiny, compared to anything we can see. Why is the number of rules for combining parts small, though? Well, in some materials, there are, instead, large numbers of ways that atoms, etc., arrange themselves — and when that happens, the result, on the scale we can see, is simply a mess. Keep the number of ways parts can combine extremely limited, though, and it is more likely that the result will possess the symmetry which is the source of the aesthetic appeal of crystals.

This can be modeled, mathematically, by using polyhedral clusters. For example, I can take a tetrahedron, and them augment each of its four faces with a rhombicosidodecahedron. The result is this tetrahedral cluster:

Image

Next, having chosen my building blocks, I need a set of rules for combining them. I choose, for this example, these three:

  1. Only attach one tetrahedral cluster of rhombicosidodechedra to another at triangular faces — and only use those four triangles, one on each rhombicosidodecahedron, which are at the greatest distance from the cluster’s center.
  2. Don’t allow one tetrahedral cluster to overlap another one.
  3. When you add a tetrahedral cluster in one location, also add others which are in identical locations in the overall, growing cluster.

Using these rules, the first augmentation produces this:

Image

That, in turn, leads to this:

Image

Next, after another round of augmentation:

Image

One more:

Image

In nature, of course, far more steps than this are needed to produce a crystal large enough to be visible. Different crystals, of course, have different shapes and symmetries. How can this simulation-method be altered to model different types of crystalline growth? Simple:  use different polyhedra, and/or change the rules you select as augmentation guidelines, and you’ll get a different result.

[Note:  all of these images were created using Stella 4d: Polyhedron Navigator. This program is available at http://www.software3d.com/Stella.php.]

 

An Octahedron, Augmented with Eight Great Icosahedra, and the Dual of this Augmented Polyhedral Cluster

Image

An Octahedron, Augmented with Eight Great Icosahedra, and the Dual of this Augmented Polyhedral Cluster

I made these using Stella 4d, which you can try at http://www.software3d.com/Stella.php. Here is its dual, also:

GrIcosadual-Augmented Octa

 

 

A 182-Faced Convex Hull, with an Explanation of that Term, As It Relates to Polyhedra

Image

A 182-Faced Convex Hull

I made this polyhedron, using Stella 4d: Polyhedron Navigator, by taking the convex hull of a different polyhedron, one not shown here. To those who don’t already know, though, that just raises a question:  what does it mean to “take the convex hull” of a polyhedron? Precisely-worded mathematical definitions of “convex hull” are easy to find, using Google and/or Wikipedia, but I have a more informal definition — one which matches the way I actually think about this operation one can perform on polyhedra.

Here’s how I picture the process: imagine a thin, spherical rubber sheet is surrounding, but not touching, the starting polyhedron. Next, start shrinking the rubber sheet. It can touch the polyhedron inside it (which might be non-convex), but it cannot penetrate any of its faces. Keep shrinking the sheet until it gets caught at points on the polyhedron inside, and then keep shrinking it further. When it starts to stretch, keep going. Stop just before the rubber sheet starts to burst from being over-stretched — and the shape of the rubber sheet, at that point, is the convex hull of the polyhedron inside it. Stretching the rubber sheet, to the limit, ensures that the convex hull will only have flat, polygonal faces — not any sort of curved surfaces.

Here’s an example — one that will end with a different convex hull than the one found at the top on this post. I’ll start with a great rhombcuboctahedron, which is also known as a great rhombicuboctahedron, as well as a truncated cuboctahedron.

Trunc Cubocta

If I take the convex hull of this great rhombcuboctahedron, the result is simply another great rhombcuboctahedron — the very thing I started with — which doesn’t explain much. Therefore, before taking the convex hull, I’m going to alter it. This can be done in many ways, of course. I’m choosing augmentation of each face with prisms, and setting the prism-height at twice the edge length of these faces. Here’s the result.

Trunc Cubocta

Taking the convex hull of this doesn’t return this same polyhedron, as it would have before the augmentation-with-prisms. Instead, after the “stretching of the imaginary rubber sheet,” this is the result:

Convex hull 1

In this image, the faces that are unmoved still have their original colors. There are also many new faces, of varying types, which were created in the “convex hulling” process. All of these new faces are shown in the same drab-green color.

The next step, changing the color scheme, has little (if any) mathematical significance, but it certainly does increase the attractiveness of the result — and admiration of beauty is, and always has been, a major motivating force in the millenia-old study of polyhedra. I’m choosing a color scheme which gives each face-type a separate color, and also lets the red, yellow, and dark blue keep their same colors.

Convex hull 2

If you’d like to try Stella 4d for yourself, please visit www.software3d.com/Stella.php. A free trial download is available.