{"id":1966,"date":"2015-12-15T01:00:28","date_gmt":"2015-12-15T01:00:28","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=1966"},"modified":"2016-01-24T02:12:31","modified_gmt":"2016-01-24T02:12:31","slug":"kaleidocycle","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2015\/12\/15\/kaleidocycle\/","title":{"rendered":"Kaleidocycle"},"content":{"rendered":"
\"Kaleidocycle<\/a>

Kaleidocycle – intothecontinuum<\/p><\/div>\n

This image by intocontinuum<\/a> show how you can take 8 perfectly rigid regular tetrahedra and connect them along their edges to form a ring that you can turn inside-out. This is called a kaleidocycle<\/b>, and you can actually form a kaleidocycle with any even number of tetrahedra, as long as you have at least 8. <\/p>\n

You can see kaleidocycles with 8, 10, and 12 tetrahedra here:<\/p>\n

• Intothecontinuum, An even number of (at least 8) regular tetrahedra…<\/a>, Archery<\/i>, 19 May 20123.<\/p>\n

and this is where I got my picture. If you attempt to create a kaleidocycle with just 6 tetrahedra, the tetrahedra collide and intersect each other as they move, as shown in these animations by Greg Egan<\/a>:<\/p>\n

• Greg Egan, Collidocycle: side view<\/a> and top view<\/a>.<\/p>\n

You can also make kaleidocycles out of paper:<\/p>\n

• Jürgen Köller, Kaleidocycles<\/a>, Mathematische Basteleien<\/i>.<\/p>\n

This website shows a variety of other flexible polyhedra, as well. For example, there’s a ring of 16 pyramids, all congruent, that folds up into a perfect regular tetrahedron. And there’s another made of 16 pyramids, all congruent, that folds into an octahedron!<\/p>\n

The Rigidity Theorem<\/a> says if the faces of a convex polyhedron are made of a rigid material and the polyhedron edges act as hinges, the polyhedron can’t change shape at all: it’s rigid<\/b>. The kaleidocycles show this isn’t true for a polyhedron with a hole in it. <\/p>\n

Of course, having a hole is an extreme case of being nonconvex. And in fact there are nonconvex polyhedra without<\/i> a hole that aren’t rigid! The first of these was discovered by Robert Connelly<\/a> in 1978. Connelly’s polyhedron has 18 triangular faces. Later, Klaus Steffen found a flexible polyhedron with just 14 triangular faces. You can see it in motion here, along with a detailed description of how it works:<\/p>\n

• Greg Egan, Steffen’s polyhedron<\/a>.<\/p>\n

Later, Maksimov proved that Steffen’s polyhedron is the simplest flexible polyhedron with just triangular faces:<\/p>\n

• I. G. Maksimov, Polyhedra with bendings and Riemann surfaces, Uspekhi Matemat. Nauk<\/i> 50<\/b> (1995), 821–823.<\/p>\n

In 1997, Connelly, Sabitov and Waltz proved something even more impressive: the Bellows Conjecture<\/b>. This says that a polyhedron that’s not rigid must keep the same volume as you flex it!<\/p>\n

The famous mathematician Cauchy had claimed to prove the Rigidity Theorem in 1813. But there was a mistake in his proof. Nobody noticed it for a long time. It seems mathematician named Steinitz spotted the mistake and fixed it in a 1928 paper.<\/p>\n

Puzzle 1:<\/b> What was the mistake?<\/p>\n

Alexander Gaifullin has generalized the Rigidity Theorem and Bellows Conjecture to higher-dimensional convex polytopes. It’s also been shown that ‘generically’ polyhedra are rigid, even if they’re not convex. <\/p>\n

So, there are lots of variations on this theme: it’s very flexible.<\/p>\n

Puzzle 2:<\/b> Can you make higher-dimensional kaleidocycles out of higher-dimensional regular polytopes? For example, a regular 5-simplex has 6 corners; if you attach 3 corners of one to 3 corners of another, and so on, maybe you can make a flexible ring. Unfortunately this is in 5 dimensions—a 4-simplex has 5 corners, which doesn’t sound so good, unless you leave one corner hanging free, in which case you can just take the movie here and imagine it as the ‘bottom’ of a 4d movie where each tetrahedron is the ‘base’ of a 4-simplex: sorta boring.<\/p>\n

For more, see:<\/p>\n

• Flexible polyhedra<\/a>, Wolfram Mathworld<\/i>.<\/p>\n

• Cauchy’s theorem (geometry)<\/a>, Wikipedia<\/i>.<\/p>\n

The Bellows Conjecture was generalized to higher dimensions here:<\/p>\n

• Alexander A. Gaiufullin, Generalization of Sabitov’s theorem to polyhedra of arbitrary dimensions<\/a>, 19 October 2015.<\/p>\n

and to higher-dimensional hyperbolic spaces here:<\/p>\n

• Alexander A. Gaifullin, The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces<\/a>, 22 August 2015.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

This image by intocontinuum<\/a> show how you can take 8 perfectly rigid regular tetrahedra and connect them along their edges to form a ring that you can turn inside-out. <\/p>\n

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