An Artistic Approach to Mathematics

Fig. 1 Twenty-seven Soma Puzzle Cubes that form "The Crystal" used to form a much larger crystal. |

My favorite manipulative is the Soma Cube by Piet Hein:

https://en.wikipedia.org/wiki/Soma_cube

There are 240 unique solutions to the cube puzzle which I have been pondering since I got the puzzle as a Christmas gift in 1969. How would one even approach the cataloging solutions and making sure there were no repeats?

If one is lost one certainly can use a map and it turns out Soma has one. It is called the Somap and can be found here:

http://www.fam-bundgaard.dk/SOMA/NEWS/N030518.HTM

The instructions are not super easy to understand but another Soma enthusiast has sorted it out. I am forever in your debt Merve Eberhardt. Merv has also done some other interesting work normalizing the existing solutions:

http://www.fam-bundgaard.dk/SOMA/NEWS/N151008.HTM

Fig. 2 The Somap |

Above (Fig. 2) is the Somap, a summary of Soma cube solutions and the structure of the solution relationships, I also find it to be a beautiful work of art. Large and small symmetries and departures from symmetry. From a graph theory perspective the vertices are shorthand(based on key piece properties) solutions and the edges are labeled with the Soma piece swaps to get from one solution to the next.

It is explained here:

http://www.fam-bundgaard.dk/SOMA/NADDICT/Addict42x.gif

Let's look at some worked out examples. It helps to make a Soma set that has the colors and codes on them (Fig. 3)

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