A Dodecahedron with Four Symetrically-Truncated Vertices

dodeca with 4 verts truncated tet symm

Dodecahedra have icosahedral (also called icosidodecahedral) symmetry. In the figure above, this symmetry is changed to tetrahedral, by truncation of four vertices with positions corresponding to the vertices (or, instead, faces) of a tetrahedron. The interchangeability of vertices and faces for the tetrahedron is related to the fact that the tetrahedron is self-dual.

[Image created using Stella 4d, available here.]

A Fashionable Tetrahedron

Tetrahedron with four pyramidal hats

You can tell this is a fashionable tetrahedron because he’s wearing four pyramidal hats — one to cover each vertex.

This bit of polyhedral silliness was created with Stella 4d, software you may try for free right here.

The Compound of the Pyramid-Excavated Dodecahedron and Tetrahedrally-Excavated Icosahedron, Together with Its Interesting Dual

Compound of Augmented Dodeca and Augmented Icosa

This is the familiar dodecahedron/icosahedron compound, but with each face of both components of the compound altered by the excavation of an equal-edge-length pyramid. To make it, as well as the rotating image below, I used Stella 4d, which you can find here.

Also, here is the dual of the compound above:

Compound of Augmented Dodeca and Augmented Icosa

Three Polyhedral Clusters of Icosahedra

In the last post on this blog, there were three images, and the first of these was a rotating icosahedron, rendered in three face-colors. After making it, I decided to see what I could build, using these tri-colored icosahedra as building blocks. Augmenting the central icosahedron’s red and blue faces with identical icosahedra creates this cubic cluster of nine icosahedra:

cube of icosahedra

If, on the other hand, this augmentation is performed only on the blue faces of the central icosahedron, the result is a tetrahedral cluster of five icosahedra:

5 icosa

The next augmentation I performed started with this tetrahedral cluster of five icosahedra, and added twelve more of these icosahedra, one on each of the blue faces of the four outer icosahedra. The result is a cluster of 17 icosahedra, with an overall icosahedral shape.

icosa made of icosa

All of these images were made using Stella 4d, which is available at http://www.software3d.com/Stella.php.

Two Compounds of Six Tetrahedra Each

compound of six elongated tetrahedra

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:

Tetrahedra 6

Two Colorings of a Hollow Stella Octangula

hollow stella octangula 2

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:

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.]

 

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:

Cluster Truncateed Tetra

What does the cluster above look like if even more truncated tetrahedra are added, but without allowing overlap to occur? Like this:

Image

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:

Image

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

Image

Multiple Octahedra, in a Rotating Cluster with Tetrahedral Symmetry

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

Image

Four Octahedra, Rotating in Tetrahedral Formation

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.