Message #3660

From: mananself@gmail.com
Subject: Stellating intersection of cylinders
Date: Mon, 20 Feb 2017 20:10:47 +0000

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

http://twistypuzzles.com/cgi-bin/pdb-search.cgi?&act=inv&key=784&off=0
http://twistypuzzles.com/cgi-bin/pdb-search.cgi?&act=inv&key=784&off=15
https://www.youtube.com/channel/UCUyPm2UVtP_GQloD4U7BVeQ

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

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

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

The intersection of three orthogonal cylinders is well known and is a Steinmetz Solid. He made this puzzle based on it:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=5346

Then he extended the surfaces of the cylinders, until two meet, to get this shape:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=5605

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

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.

I searched online, and only found pictures of the union or intersection of cylinders, such as this page:
http://paulbourke.net/geometry/cylinders/

I haven’t found anything about their stellations. Have mathematicians studied them? Are the 11 shapes in this page a complete enumeration of the stellations of 10 cylinders arranged in this manner?
http://twistypuzzles.com/forum/viewtopic.php?f=15&p=361084#p361084
How many such shapes are out there?

Have you guys seen anything like this?