Kaleidocycle

Regular-based right pyramids
6 tetrahedron where vertices meet at the center. Blue edges are doubled with pairs of green faces hidden. (Click to rotate view)
Faces	24 isosceles triangles
Edges	36 (6 as degenerate pairs)
Vertices	12
Symmetry group	C_3v, [3], (*33), order 6
Properties	torus
Net

A kaleidocycle (or flextangle) is a paper-folded model connecting 6 tetrahedra (or tetragonal disphenoids with isosceles triangles) on opposite edges into cycle. Its can be constructed from a stretched triangular tiling net with 4 triangles in one direction and an even number in the other direction.

The model represents a flexible polyhedron (having degenerate pairs of coinciding edges in transition) that that can be twisted around a ring axis, showing 4 sets of 6 triangle faces that can be drawn with different colors or patterns.

Example

Variations

Beyond 6 sides, higher even number of tetrahedra, 8, 10, 12, etc, can be chained together. These models will leave a central gap, depending on the proportions of the triangle faces.^[1]

History

Wallace Walker coined the word kaleidocycle in the 1950s from the Greek kalos (beautiful), eidos (form), and kyklos (ring). In 1977 Doris Schattschneider and Wallace Walker published a book about them using M.C. Escher patterns on the faces.^[2]^[3]

Cultural uses

It was called a flextangle as a symbolic prop in the 2018 film A Wrinkle in Time.^[4]

References

^ regular tetrahedra solutions, 8, 10, 12 with Mathematica
^ Book Review:Art Meets Math in 'Kaleidocycles' May 27, 1988
^ Doris Schattschneider and Walker, M.C. Escher Kaleidocycles, 1977. ISBN 0-906212-28-6. [1]
^ Cooper, Meghan (March 8, 2018). "Disney's A Wrinkle in Time Kaleidocycle Flextangle and Activity Printables". Jamonkey. Retrieved October 15, 2018.

External links

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[1] regular tetrahedra solutions, 8, 10, 12 with Mathematica

[2] Book Review:Art Meets Math in 'Kaleidocycles' May 27, 1988

[3] Doris Schattschneider and Walker, M.C. Escher Kaleidocycles, 1977. ISBN 0-906212-28-6. [1]

[4] Cooper, Meghan (March 8, 2018). "Disney's A Wrinkle in Time Kaleidocycle Flextangle and Activity Printables". Jamonkey. Retrieved October 15, 2018.

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