other kaleidocycles 1/2

For the regular kaleidocycles the order minimum is 8; nevertheless we can build a ring with 6 tetrahedra, but which cannot turn completely.

kaleidocycles of minima orders

regular kaleidocycle of order 8

closed kaleidocycle (non regular) of order 6 may be cut using its symmetry plane (when closed) into two mirror image right-angled kaleidocycle

right-angled kaleidocycle of order 6 ("Schatz cube" below)

The "Paul Schatz cube" contains a kaleidocycle of order 6 one position of which sketches a cube which may be completed with two "bolts" (order 3 symmetry).
During the rotation we notice two interesting "triangular positions".

The net of this kaleidocycle is easy to draw: the twelve triangles which build the rectangle have one side of their right angle with a length double of that of the other side, and the twelve others are half equilateral triangles.

LiveGraphics3D has difficulties to display well all the faces

The "Konrad Schneider cube" contains a kaleidocycle of order 8 one position of which also sketches a cube which may be completed with two "bolts" (order 4 symmetry).
During the rotation we notice interesting "square positions".

The three pieces build the "invertible" cube (Umstülpwürfel in German) : we move from the cube (positive form) to the rhombic dodecahedron (negative form) with a cavity equal to the initial cube.

The net of the kaleidocycle is easy to draw starting with a strip of eight format A (a√2×2a) rectangles. net (Schneider ring)

The eight link edges (four of length a and four of length 2a) are drawn in magenta (the pairs of the small ones superimpose themselves). The dashed segments point out cuts.

references:

• the site by Jürgen Köller (special kaleidocycles, also in German)
• Umstülpkörper by Ellen Pawlowski (2005, in German)
• see also invertible polyhedra

more kaleidocycles: IsoAxis - kaleido 2

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February 2000
updated 29-09-2013