Software credit: I made this image using Stella 4d, available at http://www.software3d.com/Stella.php (free trial download available).
Stellar Artifact III
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Stellar Artifact II
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Software credit: I made this image using Stella 4d, available at http://www.software3d.com/Stella.php (free trial download available).
Stellar Artifact
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Software credit: I made this image using Stella 4d, available at http://www.software3d.com/Stella.php (free trial download available).
A Rhombicosidodecahedron, Augmented with Thirty Icosakaipentagonal Prisms
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When adding icosakaipentagonal prisms (those where the bases have 25 sides) to the thirty square faces of a rhombicosidodecahedron, the prisms can have one of two orientations. One is above, and here is the other one (click to enlarge):
Software credit: I made these images using Stella 4d, available at http://www.software3d.com/Stella.php (free trial download available).
A Dodecahedron, Augmented with Twelve Pentagonal Cupolae
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Software credit: I made this using Stella 4d, available at http://www.software3d.com/Stella.php (free trial download available).
The Compound of Five Rhombic Dodecahedra
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This compound is unusual in that it is most attractive as a ball-and-stick model, with the faces rendered invisible, rather than the traditional coloring for compounds. In the traditional coloring, no faces are hidden, and each component of the compound is given faces of a different color. Here’s the same compound, rendered in the traditional manner:
Of course, matters of aesthetics are not subject to mathematical proof. Some might prefer the second version to the first.
Software credit: please see www.software3d.com/Stella.php to try or buy Stella 4d, the software I use to make these polyhedral images.
Tessellation Featuring Regular Dodecagons, Regular Octagons, Squares, and Hexaconcave, Equilateral Dodecagons
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Three-Color Arrangement for the Dodecahedron
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There are well-known symmetrical 4- and 6-color arrangements for the dodecahedron, and the rhombic dodecahedron has such arrangements in 3, 4, and 6 colors. What’s different about the (Platonic) dodecahedron that the 3-color arrangement you see here doesn’t make the cut, yet there is one for the rhombic dodecahedron?
The answer: in the other arrangements mentioned, faces of the same color do not share edges. Here, they do, so this one is usually not listed with the others.
Software credit: please visit http://www.software3d.com/Stella.php and try the free download of Stella 4d available there. It’s the program I used to make this image. And, as for the color arrangements mentioned above, they’re pre-loaded into the Stella interface as easy-to-find options.
James Randi, on the Limited Effectiveness of Belief
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One of the high points of my life was the day I got to have several conversations with James Randi. I enjoyed them. Some others who were there, though, not so much.
An example of how one of the question-and-answer sessions went:
Question: What happens to us after we die?
Randi’s answer: What happens to a computer after you turn off the power?
Apparently the questioner was rather upset by this reply, but I didn’t figure that out myself, even though I was present. I learned about it later, from others. Randi’s response simply made sense to me.
The Vacuum Cleaner Enigma
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A vacuum is, by definition, a region of space devoid of matter. While a perfect vacuum is a physical impossibility, very good approximations exist. Interplanetary space is good, especially far from the sun. Interstellar space is better, and intergalactic space is even better than that.
Along come humans, then, and they invent these things:
. . . and call them “vacuum cleaners.”
Now, this makes absolutely no sense. There isn’t anything cleaner than a vacuum — and the closer to an ideal vacuum a real vacuum comes, the cleaner it gets. Since vacuums are the cleanest regions of space around already, why would anyone pay good money for a machine that supposedly cleans them? They’re already clean!
Even cleaning in general is a puzzle, without vacuums being involved at all. To attempt to clean something — anything — is, by definition, an attempt to fight the Second Law of Thermodynamics. Isn’t it obvious that any such effort is, in the long run, doomed from the outset?
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[Image note: I didn’t create the images for this post, but found them using Google. I assume they are in the public domain.]
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