This polyhedron resembles a cuboctahedron, more than any other familiar polyhedra — but cuboctahedra were not used, at all, in its construction. To make it, I started with the cube of eight truncated octahedra seen in the previous post, and then stellated that figure many times. (How many? Enough times that I lost count — that’s how many.)
Stella 4d was used for this, and you may try it for free at http://www.software3d.com/stella.php.
This cubic arrangement of eight truncated octahedra has a hole in the center, and indentations in the center of each face of the cube. What would fit in these gaps? More truncated octahedra of the same size, that’s what. This wouldn’t be true for most polyhedra, but the truncated octahedron is unusual in that it can fill space without leaving gaps — much like hexagons can tile a plane, but in three dimensions.
Stella 4d was used to create this image, and you may try it for free at http://www.software3d.com/stella.php.
This cubic arrangement of eight snub cubes, one of the minority of polyhedra which are chiral, includes four “right-handed” snub cubes, and four that are “left-handed.”
Stella 4d was used to create this image, and you may try it for free at http://www.software3d.com/Stella.php.
Unlike most polyhedra, the snub cube is chiral, meaning it exists in “left-handed” and “right-handed” forms. In this fused pair of snub cubes, there is one of each type.
Stella 4d was used to create this image, and you may try it for free at http://www.software3d.com/stella.php.
Unlike most polyhedra, the snub dodecahedron is chiral, meaning it exists in “left-handed” and “right-handed” forms. In this fused pair of snub dodecahedra, there is one of each type.
Stella 4d was used to create this image, and you may try it for free at www.software3d.com/stella.php.
This is a Catalan solid. Its dual among the Archimedean solid is the snub dodecahedron, which is chiral, causing this polyhedron to be chiral as well. This simply means that these polyhedra each exist in two forms, which are mirror-images of each other.
This virtual model was made using Stella 4d, which you can find at www.software3d.com/stella.php.
Created using Stella 4d, which you may try at http://www.software3d.com/Stella.php.
Created using Geometer’s Sketchpad, MS-Paint, and Stella 4d. The last of these programs may be tried for free at http://www.software3d.com/stella.php.
Created using Stella 4d, software you can find at www.software3d.com/Stella.php.
I used Stella 4d, a program you can find at http://www.software3d.com/stella.php, to make the rotating .gif file you see here. You can many such rotating pictures of other polyhedra elsewhere on this blog.
Older versions of this program would only create still images. In those days, I would also make actual physical models out of paper (usually posterboard or card stock). However, I’ve stopped doing that, now that I can make these rotating pictures.
There is one polyhedron for which I never could construct a physical model, although I tried on three separate occasions. It’s this one, the great icosahedron, discovered, to the best of my knowledge, by Johannes Kepler. Although it only has twenty faces (equilateral triangles), they interpenetrate — and each triangle has nine regions visible (called “facelets”), with the rest of each face hidden inside the polyhedron.
To create a physical model, 180 of these facelets must be individually cut out, and then glued or taped together, and there’s very little margin for error. On my three construction-attempts, I did make mistakes — but did not discover them until I had already built much more of the model. When making paper models, if errors are made, there is a certain point beyond which repair is impossible, or nearly so.
Although I never succeeded in making a physical model of the great icosahedron myself, and likely never will, I did once have a team of three students in a geometry class successfully build one. One of the students kept the model, and all three received “A” grades.
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