A kaleidocycle or flextangle is a

flexible polyhedron In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such ...

connecting six

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

(or

disphenoid In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same sh ...

s) on opposite edges into a cycle. If the faces of the disphenoids are equilateral triangles, it can be constructed from a stretched

triangular tiling In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...

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

k\pi/2

, where

k

is an integer, and can therefore be continuously twisted. Kaleidocycles can be constructed from a single piece of paper (with dimensions

l \times 2l

) without tearing or using adhesive. Because of this and their continuous twisting property, they are often given as examples of simple

origami ) is the Japanese paper art, art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of pape ...

toys. The kaleidocycle is sometimes called a flexahedron in analogy to the planar

flexagon In geometry, flexagons are flat models, usually constructed by folding strips of paper, that can be ''flexed'' or folded in certain ways to reveal faces besides the two that were originally on the back and front. Flexagons are usually square or ...

, which has similar symmetry under flexing transformations.

Examples

This animation demonstrates the flexing of a kaleidocycle around its ring axis. The four sets of 6 triangular faces are shown in different colours (with every other face in each set of six shown in grey for contrast).

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.

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 Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made Mathematics and art, mathematically inspired woodcuts, lithography, lithographs, and mezzotints. Despite wide popular interest, Escher was for ...

patterns.

Cultural uses

The shape was called a ''flextangle'' by characters in the 2018 science fantasy adventure film ''

A Wrinkle in Time ''A Wrinkle in Time'' is a Young adult fiction, young adult science fantasy novel written by American author Madeleine L'Engle. First published in 1962, the book won the Newbery Medal, the Sequoyah Book Award, the Lewis Carroll Shelf Award, and ...

'', which depicted a paper model with its inner faces colored with hearts and patterns which become hidden when those faces are folded together. The paper toy suggested how space and time could be folded to explain the magical travels of the story. The toy is given to the daughter by her father at the start of the movie and its hearts show how love can be ''enfolded'' and still be there, even after the father mysteriously disappears.

References

External links