Software credit: see http://www.software3d.com/stella.php.
Polyhedral Cage Based On an Icosahedron / Dodecahedron Compound
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Variant of the Compound of Five Cubes
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In this compound (unlike the classical compound of five cubes), the cubes are not complete. I found it while repeatedly-stellating the rhombic triacontahedron, which you can see in the previous post.
(Software credit: see http://www.software3d.com/stella.php
Reflection-Planes of the Rhombic Triacontahedron
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Each circle shown is in one of the reflection-planes of this polyhedron, which is the dual of the icosidodecahedron, an Archimedean solid.
(Software credit: see http://www.software3d.com/stella.php)
Expanded Truncated Icosahedron III
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This version of an expanded truncated icosahedron was made in a similar fashion to the one two posts ago — but with the augmentation-by-prisms step altered by using prisms of somewhat greater height, with the goal being to make the rectangular faces closer to “squareness.”
As a result, this polyhedron is closer than the others to being a near-miss to the Johnson Solids — a category of polyhedra which has been discussed on this blog before. “Near-misses” are almost Johnson solids, and must therefore have faces which are regular (as the hexagon and pentagons here are) are nearly-regular (as the rectangles and triangles come close to being).
Is this a near-miss, or is it not? That’s difficult to answer, for that set of polyhedra has no precise definition, and cannot have one — it only has a “fuzzy” definition.
In my opinion, it isn’t quite a near-miss, for the triangles are too non-equilateral to qualify, to my eye — but others might disagree. In the unlikely event that mathematicians wish to start talking about “near-near-misses,” I offer this as a member of that set.
(Software credit: see http://www.software3d.com/stella.php for a free trial download of the software used to make these polyhedral images.)
Expanded Truncated Icosahedron II
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In this version of an expansion of the truncated icosahedron, the faces are simply moved far apart — but with nothing filling the space between them.
Like the last post, this was made using Stella 4d, which you may download and try for free (as a trial version) at http://www.software3d.com/stella.php.
Expanded Truncated Icosahedron
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To create this polyhedron, I started with a truncated icosahedron (soccer ball shape), then augmented each face with prisms, then took the convex hull of the resulting non-convex figure.
All of this was done in about two minutes with Stella 4d, a program which makes manipulating polyhedra both quick and easy. You can try it here: http://www.software3d.com/stella.php.
For Tumblr-Refugees
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If you’ve never had a Tumblr-blog, you probably won’t get the joke here. If you have, though, you will.
Dodecahedron with Turquoise Stars
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Programs used: Geometer’s Sketchpad and MS-Paint, to make the turquoise stars; and Stella 4d, to place that image of each face of this polyhedron, and then make this rotating .gif file.
Stella 4d may be tried for free at http://www.software3d.com/stella.php.
Polyhedron with 240 Faces, in Four Sets of Sixty
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The 240 faces are:
- 60 small isosceles triangles
- 60 large isosceles triangles
- 60 small kites
- 60 larger irregular pentagons, with one side each that is so short that they strongly resemble kites
Created with Stella 4d, available at http://www.software3d.com/stella.php.
Great Rhombicosidodecahedron with Interference Patterns
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Programs used: Geometer’s Sketchpad and MS-Paint, to make the image seen in the previous post; and Stella 4d, to place that image of each face of this polyhedron, and then make this rotating .gif file.
Stella 4d may be tried for free at http://www.software3d.com/stella.php.
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