# Message #3663

From: Melinda Green <melinda@superliminal.com>

Subject: Re: [MC4D] Stellated intersection of cylinders

Date: Mon, 20 Feb 2017 20:16:04 -0800

I’ve not examined these very closely, but to me they look more like unions of intersections than stellations of intersections. As you suggested, both views may be correct though it seems more natural for me to see this approach as starting with a symmetric set of intersecting cylinders and unioning all possible intersections of N of them. For example, the 6-cylinder Magaminx <http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=5735> is the union of all possible triplets of cylinders (6 choose 3). This 3-cylinder 2x2x2 <http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=5605> is the union of all pairs of 3 (3 choose 2)., whereas this one <http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=5346> is 3 choose 3. Twists are always around one cylinder. For 10 cylinders, that doesn’t add up to 11 choices though. Maybe it’s every possible fully-symmetric way to choose 3, 4, or 5 cylinders out of 10?

This does seem like a very clever idea and may even extend nicely into higher dimensions.

-Melinda

On 2/20/2017 12:12 PM, mananself@gmail.com [4D_Cubing] wrote:

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> Recently I saw these puzzles created by Alexandr Abalikhin (twistypuzzles ID: TER)

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> http://twistypuzzles.com/cgi-bin/pdb-search.cgi?&act=inv&key=784&off=0

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> http://twistypuzzles.com/cgi-bin/pdb-search.cgi?&act=inv&key=784&off=15

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> https://www.youtube.com/channel/UCUyPm2UVtP_GQloD4U7BVeQ

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> The puzzles with names including fourcylinder, sixcylinder, tricylinder intrigued me. Recently he constructed tencylinder puzzles:

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> http://twistypuzzles.com/forum/viewtopic.php?f=15&p=361084#p361084

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> These puzzles are shape mods. But the shapes are very interesting. I consider them as different levels of stellation of the intersection of cylinders.

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> The intersection of three orthogonal cylinders is well known and is a Steinmetz Solid. He made this puzzle based on it:

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> http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=5346

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> Then he extended the surfaces of the cyli nders, until two meet, to get this shape:

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> http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=5605

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> You can think of the new shape as the region contained by at least two of the three cylinders. If you think of the original intersection as a "curvy" rhombic dodecahedron, then the new shape is a "curvy" stellated rhombic dodecahedron:

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> https://en.wikipedia.org/wiki/First_stellation_of_rhombic_dodecahedron

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> And Alexandr created stellations of intersections of more cylinders. I can see some corresponding "flat" stellated/compound polyhedra. But the curvy shapes are neat because one can construct them just out of cylinders.

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> I searched online, and only found pictures of the union or intersection of cylinders, such as this page:

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> http://paulbourke.net/geometry/cylinders/

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> I haven’t found anything about their stellations. Have mathematicians studied them? Are the 11 shapes in thi s page a complete enumeration of the stellations of 10 cylinders arranged in this manner?

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> http://twistypuzzles.com/forum/viewtopic.php?f=15&p=361084#p361084

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> How many such shapes are out there?

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> Have you guys seen anything like this?

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