An Artistic Approach to Mathematics

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

I am a lover of patterns and color. I make "math manipulatives" from simple materials at hand, paper, tape, crayons, wood and of late 3D printing.  I volunteer at several after school programs in Minneapolis making and helping others make everything from counting blocks to working clocks.


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)

Fig. 3

I will detail the examples in my next post but for the time being I invite you to join in the effort to decode all of the Somap solutions so they can be compared to and cataloged with the existing table of solutions:

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

If you are interested in collaborating please contact me and I will build you a set of Soma blocks with the proper colors and codes on them.