Created using Stella 4d, available here, by multiple stellations of a black icosidodecahedron, rendered as a rotating figure, against a black background.
Glimpses of the Invisible
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Tag Archives: polyhedra
New “Near-Miss” Candidate?
As a proposed new “near-miss” to the Johnson solids, I created this polyhedron using Stella 4d, which can be found for purchase, or trial download, here. To make it, I started with a tetrahedron, augmented each face with icosidodecahedra, created the convex hull of the resulting cluster of polyhedra, and then used Stella‘s “try to make faces regular” function, which worked well. What you see is the result.
This polyhedron has no name as of yet (suggestions are welcome), but does have tetrahedral symmetry, and fifty faces. Of those faces, the eight blue triangles are regular, although the four dark blue triangles are ~2.3% larger by edge length, and ~4.6% larger by area, when compared to the four light blue triangles. The twelve yellow triangles are isosceles, with their bases (adjacent to the pink quadrilaterals) ~1.5% longer than their legs, which are each adjacent to one of the twelve red, regular pentagons. These yellow isosceles trapezoids have vertex angles measuring 61.0154º. The six pink quadrilaterals themselves are rectangles, but just barely, with their longer sides only ~0.3% longer than their shorter sides — the shorter sides being those adjacent to the green quadrilaterals.
The twelve green quadrilaterals are trapezoids, and are the most irregular of the faces in this near-miss candidate. These trapezoids have ~90.992º base angles next to the light blue triangles, and ~89.008º angles next to the pink triangles. Their shortest side is the base shared with light blue triangles. The legs of these trapezoids are ~2.3% longer than this short base, and the long base is ~3.5% longer than the short base.
If this has been found before, I don’t know about it — but, if you do, please let me know in a comment.
UPDATE: It turns out that this polyhedron has, in fact, been found before. It’s called the “tetrahedrally expanded tetrated dodecahedron,” and is the second polyhedron shown on this page. I still don’t know who discovered it, but at least I did gather more information about it — the statistics which appear above, as well as a method for constructing it with Stella.
A Faceting of the Truncated Dodecahedron, Together with Its Dual
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.
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.
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