Workshop for 大阪府千里高等學校 (Dec. 27, 2013)

I gave a workshop for 大阪府千里高等學校 last Friday.

Bead model of Kaleidocycle (萬花環)

Kaleidocycle or a ring of rotating tetrahedra was invented by originally by R. M. Stalker 1933. The simplest kaleidocycle is a ring of an even number of tetrahedra. The interesting thing about the Kaleidocycle is that you can twist it inwards or outwards continually. The geometry of kaleidocycle has been studied by many people from different fields in the last 80 years:

1. Stalker, R. M. 1933 Advertising medium or toy. US Patent 1,997,022, filed 27 April 1933 and issued 9 April 1935.
2. Ball, W. W. Rouse 1939 Mathematical recreations and essays, 11th edn. London: Macmillan. Revised and extended by Coxeter, H. S. M.
3. Cundy, H. M.; Rollett, A. R. 1981 Mathematical models, 3rd edn. Diss: Tarquin Publications.
4. Fowler, P. W.; Guest, S. Proc. R. Soc. A 461(2058), 1829-1846, 2005.
5. 全仁重, Motivation Behind the Construction of Maximal Twistable Tetrahedral Torus.
6. HORFIBE Kazunori, Kaleidocycle animation.

Typically, people use paper or other solid materials to make this kind of toy. A few months ago, I discovered that you can easily make this toy by tubular beads through the standard figure eight stitch (right angle weave).

This particular model consists of 8 regular tetrahedra. You can easily extend rings that contain 10, 12, ... tetrahedra.

The procedure I used to make this 8-tetrahedra Kaleidocycle is by the standard figure-eight stitch (right angle weave) in which one just keep making triangles. Of course, some care should be paid on the sequence of these triangle.

Carbon cubes by HORFIBE Kazunori

ε Keggin structure

δ Keggin structure

γ Keggin structure

β Keggin structure

α Keggin structure

From pentaexcavated Icosidodecahedron to Mackay polyhedra

Perovskite, 4 stella octangula

Quasicrystal (準晶)