The faces of the strombic hexacontahedron, the dual of the rhombicosidodecahedron, are kites. I have no explanation for why the word “strombic” applies to it — is a kite a “stromb?”
I’ve already googled it, followed many links, etc., and it’s still as puzzling to me as it was the first time I read it. If you have a solution to this puzzle, please post it in a comment.
Software used to create this image: Stella 4d: Polyhedron Navigator, available at www.software3d.com/Stella.php.
Truncating a cube once yields an Archimedean solid with six octagonal faces, and eight triangular faces, all regular. A second truncation can be made to produce the solid shown above. It also has, as faces, six regular octagons and eight equilateral triangles — and, in addition, twenty-four isosceles triangles.
I made this using Stella 4d, software you can try for yourself at www.software3d.com/Stella.php.
I made this using Stella 4d, available at http://www.software3d.com/Stella.php.
I made this with Stella 4d, a program you can find at http://www.software3d.com/Stella.php.
Since I stumbled across this by stellating other polyhedra, I’ve never seen what a face of one component of this compound looks like, without having part of that face covered. My best guess is that the faces (of which there are eight in each part of the compound) are kites.
I used Stella 4d, available at http://www.software3d.com/Stella.php, to make this, and a free trial download is available at that site.
I used Stella 4d, available at http://www.software3d.com/Stella.php, to make this.
I used Stella 4d, available at http://www.software3d.com/Stella.php, to make this.
I’d like to find a polyhedron with the same number of faces as there are days of the year. This is the closest I’ve come, so far.
The software I used, Stella 4d, may be purchased at http://www.software3d.com/Stella.php. There is also a free trial download available.
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:
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:
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.]
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.
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