Six tetrahedra whose vertices meet at the center. Blue edges are doubled with pairs of faces hidden.
Faces
24 isosceles triangles
Edges
36 (6 as degenerate pairs)
Vertices
12
Symmetry group
C3v, [3], (*33), order 6
Properties
torus
Net
A kaleidocycle before it is wrapped into a ring makes a chain of 6 disphenoids connected edge to edge.
A kaleidocycle or flextangle is a flexible polyhedron connecting six tetrahedra (or disphenoids) on opposite edges into a cycle. If the faces of the disphenoids are equilateral triangles, it can be constructed from a stretched triangular tiling net with four triangles in one direction and an even number in the other direction.
The kaleidocycle has degenerate pairs of coinciding edges in transition, which function as hinges. The kaleidocycle has an additional property that it can be continuously twisted around a ring axis, showing 4 sets of 6 triangular faces. The kaleidocycle is invariant under twists about its ring axis by , where is an integer, and can therefore be continuously twisted.
Kaleidocycles can be constructed from a single piece of paper (with dimensions ) without tearing or using adhesive. Because of this and their continuous twisting property, they are often given as examples of simple origami toys. The kaleidocycle is sometimes called a flexahedron in analogy to the planar flexagon, which has similar symmetry under flexing transformations.
A kaleidocycle or flextangle is a flexible polyhedron connecting six tetrahedra (or disphenoids) on opposite edges into a cycle. If the faces of the disphenoids...
An Amazing, Space Filling, Non-regular Tetrahedron that also includes a description of a "rotating ring of tetrahedra", also known as a kaleidocycle....
attention for her video on hexaflexagons. Cayley tree Geometric group theory Kaleidocycle Oakley, C. O.; Wisner, R. J. (March 1957). "Flexagons". The American...
Tables. Mark Leach. Retrieved June 5, 2023. Pablo Cassinello (2022). "Kaleidocycle of the Periodic Table". The INTERNET Database of Periodic Tables. Mark...
ISBN 0-521-53162-4 http://kociemba.org/themen/kaleidocycles/workingkaleidocycles.html Interactive version of kaleidocycle https://oeis.org/A338336 Mathematical...
Paul Schatz models (in German) Hans Dirnböck, Hellmuth Stachel: The Development of the Oloid About Kaleidocycles, including mention of Schatz cube v t e...
(1978) for soprano and mixed ensemble. In 1987, Baitz's composition Kaleidocycles, commissioned by iEAR Studios at Rensselaer Polytechnic Institute, was...
by making an appropriate choice of the constant c {\displaystyle c} . Kaleidocycle, a chain of tetrahedra linked edge-to-edge like a degenerate Schwarz...
Martin Gardner Lecture at MathFest in August 2021. Books M. C. Escher Kaleidocycles (with Wallace Walker), Ballantine Books, 1977, Pomegranate Artbooks...
popularised flexagons which are a form of rigid origami and the flexatube. Kaleidocycles are toys, usually made of paper, which give an effect similar to a kaleidoscope...
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