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:
Next, having chosen my building blocks, I need a set of rules for combining them. I choose, for this example, these three:
- 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.
- Don’t allow one tetrahedral cluster to overlap another one.
- 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:
That, in turn, leads to this:
Next, after another round of augmentation:
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.]
Given the name of this blog, and the familiarity of the number pictured above, I’m sure you recognize it as the beginning of my favorite number, pi. On various websites, you can find far more digits than are shown above. However, just the digits shown in the top row here are greater in number than that needed for any real-world, practical application. Is pi useful? Is mathematics itself useful? Of course they are . . . but those questions miss the point entirely.
Every teacher who has been in the field for long has heard the complaint, disguised as a question, “What are we ever going to use this for?” Unfortunately, most school systems, as well as teacher-training programs, have chosen to respond to this well-known complaint by repeatedly telling teachers, and teachers-in-training, that it is of extreme importance to show students how the curriculum is “relevant,” and adjust curricula to make them more so. This usually boils down, of course, to trying to convince students that learning is important because, supposedly, education = a better job in the future = more money. Sometime this “equation” works, and sometimes it does not, but it always misses a key point, one that should not be left out, but too often is.
It is a fallacy that learning has to have a practical application to be a worthwhile endeavor. There’s more to life than the fattening of bank accounts. Sadly, many of those making decisions in education do not realize this. Their attempts to reduce education to a strictly utilitarian approach are causing great harm.
This scenario has happened many times: a mathematician discovers an elegant proof to a surprising theorem, a physicist figures out something previously unknown about the nature of reality, or a researcher in another field does something comparable, and someone then asks them, “But what can this be used for?” On occasion, tired of hearing this utilitarian refrain, such researchers give unusually honest responses which surprise and confuse many people — such as, “Someone else might, at some point, find a practical application for this . . . but I sincerely hope that never happens.” Such a response is rarely understood, but it makes the person who says it feel better to vent some of their frustration with those who are obsessed with tawdry, real-world applications for everything.
Many humans — and this is a terrible shame — live almost their entire lives like rats in mazes, running down passages and around corners, chasing tangible rewards — cheese for the rats, or the ability to buy, say, a fancy new car, in the case of the people. People shouldn’t live like lab rats . . . and, unlike lab rats, we don’t have to. People are smart enough to find higher purposes in life. People can, in other words, find, understand, and appreciate beauty — in things which are useless, in the sense that they have no useful applications. We can appreciate things that transcend mere utility, if we choose to do so.
Much of life is utterly banal, for a great many people. They wake up each day, work themselves into exhaustion at horribly boring jobs, go home, numb themselves with television, massive alcohol consumption, or other hollow pursuits, fall asleep, and then get up and repeat the process the next day . . . and then they finally get old and die. Life can be so much more than that, though, and it should be.
The researchers I described earlier aren’t doing what they do for money, or even the potential for fame. Are such things as mathematics and physics useful? Yes, they are, but that isn’t why pure researchers do them. The same can be said for having sex: it’s useful because it produces replacement humans, but that isn’t why most people do it. People have sex, obviously, because they enjoy it. In simpler terms: it’s fun. Most people understand this concept as it relates to sex, but far fewer understand it when it relates to other aspects of life, particularly those of an academic nature.
Academic pursuits are of much greater value when the motivation involved is joy, and the fun involved, rather than avarice. As scrolling through this blog will show you, I enjoy searching for polyhedra which have not been seen before. I certainly don’t expect to get rich from any such discoveries I make in this esoteric branch of geometry, which is itself one subfield, among many, in mathematics. I do it because it is fun. It makes me happy.
Much of life is pure drudgery, but our lives can be enriched by finding joyous escapes from our routines. An excellent way to do that is to learn to appreciate the beauty of uselessness — uselessness of the type that elevates the human spirit, in a way that the pursuit of material goods never can.
This is the approach we should encourage students to have toward education. Learning is far too valuable an activity to be limited, in its purpose, to the pursuit of future wealth. It’s time to change our approach.
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.
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.
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:
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.
If you’d like to try Stella 4d for yourself, please visit www.software3d.com/Stella.php. A free trial download is available.