Piet Hein

The Soma cube

The Danish physicist, puzzle inventor and writer Piet Hein (1905-1996) is - among (many) other things - the inventor of the 'Soma cube', which consists of 3x3x3 unit cubes, divided into 7 pieces.

There are 240 essentially different ways to reassemble the cube. Some people have difficulty finding even one solution...

And of course one can try to assemble the 7 pieces into other shapes. Over 2000 have been listed; if you like it, have fun! See the links below.

Some nice sites on the Soma cube:

- The animated solution here is from Jürgen Köller.

He has them for dozens of games! - The world's leading sites of Thorleif Bundgård (also on super-ellipse and super-egg) and his companion Courtney McFarren
- Christian Eggermont shows over 50 figures that can be made using the 7 pieces

Martin Gardner's column in *Scientific American* of September 1958 contributed strongly to the success of the Soma cube. Reprinted in his compilation *More Mathematical Puzzles and Diversions*, ch. 6. My edition is from Pelican, 1961.

Super-ellipse and super-egg

Piet Hein also promoted the 'super-ellipse', a curve that is somewhere in between an ellipse and a rectangle:

Normal ellipse | Super-ellipse | |

This shape has for instance been used in a traffic circle in Stockholm. It seems to be easier to 'round' such a circle.

When a super-ellipse is rotated around its major axis, a 'super-egg' is formed. A real Columbus' egg, as it can be balanced on one end!

The ellipse and the super-ellipse are special cases of the Lamé curves, named after the French mathematician Gabriel Lamé (1795-1870), who described them in 1818. Hein thus did not 'invent' the super-ellipse, as many seem to believe; he just advocated its usefulness in architecture, the design of furniture and household goods, etc.

The general mathematical form of the Lamé curves is:

|x/a|^{n} + |y/b|^{n} = 1

in which the vertical bars mean that the absolute value is taken. An ellipse results for n=2 (or a circle if a=b). Hein took n=2.5 for his super-ellipse. Other values, like n=2.2 or n=3, also yield pretty curves. I saw some websites stating that Hein used n=3 (for instance Bundgård, mentioned above). I guess that's a mistake.

It is very informative to play around with __n__ yourself. The website of the University of St. Andrews (Scotland) allows you to do so. Great fun!

Martin Gardner's column in *Scientific American* of September 1965 about super-ellipse and super-egg also is a classic. Reprinted in his compilation *Mathematical Carnival*, ch. 18. My copy is from Pelican, 1978.

Piet Hein is also the inventor of the helical dial. It can be found in the gardens of Egeskov Castle in Denmark.