# the kaleidohedron from the IsoAxis grid

Kaleidohedron? Comes from the Greek: "kalos" (nice) + "eidos" (look) + "hedra" (faces).
IsoAxis, created and patented by the graphic designer Wallace Walker, is a rectangular grid divided into 36+24=60 isosceles right triangles allocated into 6 identical pairs of  strips. When folded front against front along the 11 sides of the strips, and back against back along the 2x(5+2)=14 lines which form a diagonal square grid, it takes almost naturally shape; assembling the two ends is enough to get a curious ring shaped form which can have the look of a non convex polyhedron with 36 faces (the remaining 24 triangles are folded inside). It is very astonishing to discover that this "ring" can be rotated around itself, transforming its appearance, like a blowing flower.
remark : studying the movement of this object, J. A. Gutierrez has shown that this is only possible with a slightly flexible material (study in Spanish, translated in French).

 Other outstanding positions and an animation of this amazing object are presented on two other pages.

If we extend it using an even number of strips, and then distort the grid by bringing the extreme strips closer, we get other astonishing forms which handling becomes tricky. The possibilities are infinite.

 This grid of 18 strips of golden triangles (36°-108°-36°) leads to a similar object ; four views of it are presented on an other page.

To discover this little wonder, you will have to reproduce the grid, fold it and assemble the extremes (as you would do to create a regular right prism). The beauty has to be deserved!  IsoAxis will greatly reward your efforts. You may color or decorate the triangles to increase the visual pleasure.

# the kaleidocycles

kaleidocycle: "kalos"=nice + "eidos"=aspect + "kuklos"=circle
If we stretch IsoAxis, so that the 36 large triangles have all their angles acute, the object we get by folding the grid and assembling its opposite sides, is a ring of 12 tetrahedra!  This kaleidocycle can still turn around itself.
Below the grid has been stretched in order to get 36 equilateral triangles. With a further modification 48 identical triangles appear, in 12 strips of four (this pattern produces the same regular kaleidocycle): each strip is then the pattern of a regular tetrahedron, and the sides (vertical lines) turn into opposite orthogonal edges which are the links with the two neighboring tetrahedra.
To modify the diameter of the ring we can add or suppress pairs of strips (at least 8 regular tetrahedra).

All the nets presented are intended to be reproduced by geometric constructions (ruler and compasses or drawing software); they come with sufficient indications to do so. I don't use tabs to be glued; by experience I prefer adhesive tape which allows more precise assemblings and leads to more robust objects.

The ring of tetrahedra closes up without problem if their number is even; actually the tetrahedra are equifacial with two opposite orthogonal edges which are used as links between them. So we can create an infinite variety of kaleidocycles.
It has been proved that the number of (non regular) tetrahedra can be reduced to 6, and that the "eye" of the ring can be reduced to a point when the link edges are coplanar.
There are also kaleidocycles where all the faces of the tetrahedra are right triangles; the "invertible cube" (Schatz cube) is an outstanding such configuration of order 6.

Be patient during the initialization! (reload the page if an animation doesn't start)

 regular kaleidocycle of order 12(regular tetrahedra) closed kaleidocycle of order 12(non regular tetrahedra) right-angled kaleidocycle of order 12(right-angled faces)

If we use an oblique grid of triangles as pattern, we may get a twisted kaleidocycle; the chain of tetrahedra looks then like a Möbius ribbon. Describing it mathematically was nice challenge taken up by Marcus Engel; his parametrized animation embraces different types of kaleidocycles, included the twisted ones (whose number of tetrahedra can be odd).

 references: •  The outstanding SITE BY MARCUS ENGLE (http://www.kaleidocycles.de/intro.shtml) is no longer accessible; here are copies of his work about the theory of kaleidocycles with the applet which allows to visualize all types of kaleidocycles, and a few nets •  M.C.Escher kaleidocycles  by Doris Schattschneider and Wallace Walker (Tarquin Publications - 1978)    with 17 models of polyhedra and kaleidocycles decorated with "repeating patterns" of M.C.Escher •  Metamorphs  by Robert Byrnes (Tarquin Publications - 2004)    with 14 models of objects which can be rotated on themselves (among them 7 kaleidocycles) •  To see: the nice kaleidocycles by Nicolas Hannachi ("Math à mâter "). An elementary study of the closed kaleidocycle of order 6 by Xavier Hubaut (in French) •  Les kaléïdocycles irréguliers fermés  by Carole Le Beller (in French) •  A Group Theoretic Approach to Kaleidocycles and Cubeocycles  by Lisa Marie Bush

more kaleidocycles: kaleido 1 - kaleido 2 - AniKA (Marcus Engel)

 home page convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects February 2000updated 13-03-2009