In the image above, which I stumbled upon using Stella 4d (available here), the tetrahedra are elongated. If they are regular, instead, the same arrangement looks very different:
Two Compounds of Six Tetrahedra Each
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Tag Archives: tetrahedron
Two Colorings of a Hollow Stella Octangula
Both of these versions of the stella octangula, or compound of two tetrahedra, were made with Stella 4d, software available at http://www.software3d.com/Stella.php.
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:
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:
One more:
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.]
Sprawling Clusters of Truncated Tetrahedra
Truncated tetrahedra make interesting building blocks. In the images below, the truncated tetrahedron “atoms” are grouped into four-part “molecules,” each with a triangular face pointed toward the molecular center, which is found in a small tetrahedral hole between the four truncated tetrahedra. These four-part “molecules” are then attached to other, always with three coplanar triangular faces from one “molecule” meeting three from the other. If you start from a central “molecule,” and let such a cluster grow for a small number of iterations, you get this:
What does the cluster above look like if even more truncated tetrahedra are added, but without allowing overlap to occur? Like this:
Like the truncated tetrahedron itself, these sprawling clusters have tetrahedral symmetry. To keep such symmetry while building these clusters, of course, one must be careful about the exact placement of the pieces — and doing this becomes more difficult as the cluster grows ever larger. I was able to take this one more step:
All of these images were created using Stella 4d: Polyhedron Navigator. This program is available at http://www.software3d.com/Stella.php.
Multiple Octahedra, in a Rotating Cluster with Tetrahedral Symmetry
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I created this cluster using Stella 4d: Polyhedron Navigator. This program is available at http://www.software3d.com/Stella.php.
Four Octahedra, Rotating in Tetrahedral Formation
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I created this cluster by augmenting each face of a tetrahedron with an octahedron, using Stella 4d: Polyhedron Navigator. This program is available at http://www.software3d.com/Stella.php.
Seventeen Truncated Tetrahedra Rotating in Symmetrical Formation
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Software credit: you can try the free trial download of Stella 4d at http://www.software3d.com/Stella.php.
An Elongated Stella Octangula
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The Stella Octangula is another name for the compound of two tetrahedra. I made this elongated version, which uses narrow isosceles triangles in place of the usual equilateral triangles, using Stella 4d — polyhedron-manipulation software you can find at http://www.software3d.com/Stella.php.
A Variant of Kepler’s Stella Octangula
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Johannes Kepler named the compound of two tetrahedra the “stella octangula,” thus helping make it one of the best-known polyhedral compounds today. This variant uses triakis tetrahedra in place of the Platonic tetrahedra in that compound. The triakis tetrahedron is a Catalan solid, and is dual to the truncated tetrahedron.
Software credit: see http://www.software3d.com/stella.php to try or buy Stella 4d, the software I used to create this image.
600 Undulating Tetrahedra
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This is a 600-cell, one of the regular polychora (four-dimensional polytopes), with its edges and vertices rendered invisible, and its cells shrunk so that they do not touch. It’s rotating in hyperspace, and what you are seeing at any given moment is a particular three-dimensional “shadow,” or projection, of the entire figure.
It’s easy to make this sort of thing with software called Stella 4d, written by an Australian friend of mine. Here’s a link to a site where you can try it, as a free trial download, before deciding whether or not to purchase the fully-functioning version: http://www.software3d.com/Stella.php.
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