This faceting of the truncated dodecahedron, one of many, was made with Stella 4d, software you can buy, or try for free, here. Here is its dual, below.
A Faceting of the Truncated Dodecahedron, Together with Its Dual
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Tag Archives: Mathematics
On Consistent and Inconsistent Combining of Chiralities, Using Polyhedral Augmentation
For any given chiral polyhedron, a way already exists to combine it with its own mirror-image — by creating a compound. However, using augmentation, rather than compounding, opens up new possibilities.
The most well-known chiral polyhedron is the snub cube. This reflection of it will be referred to here using the letter “A.”
To avoid unnecessary confusion, the same direction of rotation is used throughout this post. Apart from that, though, the image below, “Snub Cube B,” is the reflection of the first snub cube shown.
There are many ways to modify polyhedra, and one of them is augmentation. One way to augment a snub cube is to attach additional snub cubes to each square face of a central snub cube, creating a cluster of seven snub cubes. In the next image, all seven are of the “A” variety.
If one examines the reflection of this cluster of seven “A” snub cubes, all seven, in the reflection, are of the “B” variety, as shown here:
Even though one is the reflection of the other, both clusters of seven snub cubes above have something in common: consistent chirality. As the next image shows, inconsistent chirality is also possible.
The cluster shown immediately above has a central snub cube of the “A” variety, but is augmented with six “B”-variety snub cubes. It therefore exhibits inconsistent chirality, as does its reflection, a “B” snub cube augmented with “A” snub cubes:
With simple seven-part snub-cube clusters formed by augmentation of a central snub cube’s square faces by six snub cubes of identical chirality to each other, this exhausts the four possibilities. However, multiplying the possibilities would be easy, by adding more components, using other polyhedra, mixing chiralities within the set of polyhedra added during an augmentation, and/or mixing consistent and inconsistent chirality, at different stages of the growth of a polyhedral cluster formed via repeated augmentation.
All the images in this post were created using Stella 4d, which you can try for yourself at this website.
Compound of the Great and Small Stellated Dodecahedra
In this compound, as shown above, the small stellated dodecahedron is yellow, while the red polyhedron is the great stellated dodecahedron. Below, the same compound is colored differently; each face has its own color, unless faces are in parallel planes, in which case they have the same color.
Making a physical model of this compound would have taken most of the day, if I did it using such things as posterboard or card stock, compass, ruler, tape, scissors, and pencils. For the first several years I built models of polyhedra, starting about nineteen years ago, that was how I built such models. The virtual polyhedra shown above, by contrast, took about ten minutes to make, using Stella 4d: Polyhedron Navigator, which you can try for free, or purchase, here.
There’s also a middle path: using Stella to print out nets on cardstock, cutting them out, and then taping or gluing these Stella-generated nets together to make physical models. I haven’t spent much time on this road myself, but I have several friends who have, including the creator of Stella. You can see some of his incredible models here, and some amazing photographs of other Stella users’ paper models, as well as some in other media, at this website.
Two Different Forty-Part Polyhedral Compounds
The polyhedron above is a compound of twenty cubes and twenty octahedra, colored by symmetry-based face-type. If the same compound is viewed in “rainbow color mode,” it looks like this:
With this particular compound, though, there are two versions — without taking coloring into consideration at all. The other version simply has the twenty cubes and twenty octahedron in a different, but still symmetrical, arrangement:
The compound above uses this second arrangement, colored by face type, and the next image is the same (second) compound, but in “rainbow color mode.”
These rotating polyhedral images were made with Stella 4d, software you can try for yourself, right here.
Another Faceting of the Great Rhombicosidodecahedron
This could also be called one of many possible faceted truncated icosidodecahedra. I made it using Stella 4d, which you can try and/or buy here. Faceting is the reciprocal operation of stellation, and involves connecting the vertices of a polyhedron into faces which are unlike those of the original polyhedron. At least some, and sometimes all, of the faceted faces intersect each other, inside the polyhedron’s convex hull, as is the case here.
For comparison, here is that convex hull: a (non-faceted) great rhombicosidodecahedron, also made using Stella.
For a different faceting of this polyhedron, just look here: https://robertlovespi.wordpress.com/2013/11/19/a-faceting-of-the-great-rhombicosidodecahedron/
Two Polyhedral Meta-Compounds
The polyhedral compound above is actually a compound of two compounds: the compound of three cubes (red, yellow, and blue), as well as the cube/octahedron base/dual compound (green and purple). The dual of this five-part compound is shown below, still with the cube/octahedron compound in green and purple (it is its own dual), and with the three parts of the compound of three octahedra in red, yellow, and blue.
I created these using the “add/blend from memory” function of Stella 4d: Polyhedron Navigator, one of this program’s capabilities which I have only recently begun to explore. You may try this software for yourself, for free, right here.
The Greatly Augmented Icosidodecahedron, and Its Dual
If a central polyhedron’s pentagonal and triangular faces are augmented by great dodecahedra and great icosahedra, I refer to it as a “greatly augmented” polyhedron. Here, this has been done with an icosidodecahedron. The same figure appears below, but in “rainbow color” mode.
In the next image, “color by face type,” based on symmetry, was used.
The next image shows the dual of this polyhedral cluster, with face color chosen on the basis of number of sides.
Here is another version of the dual, this one in “rainbow color” mode.
Finally, this image of the dual is colored based on face type.
These six images were made with Stella 4d, which may be found here.
The Greatly Augmented Rhombicosidodecahedron
I call this variant of the rhombicosidodecahedron “greatly augmented” because it was formed by augmenting each pentagonal face of a central rhombicosidodecahedron with a great dodecahedron, while each triangular face is augmented with a great icosahedron. It was made using Stella 4d, which may be found here.
A Fashionable Tetrahedron
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.
Two Polyhedral Compounds: the Dodecahedron / Truncated Octahedron, and Its Dual, the Icosahedron / Tetrakis Cube
That’s the compound of the dodecahedron and the truncated octahedron above. Shown next is its dual, the compound of the icosahedron and the tetrakis cube. Both compounds were made using Stella 4d: Polyhedron Navigator, which you may try here.
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