Piet Hein and the Soma Cube
This Danish polymath had a formative influence on my brain: not the least because of his invention of a famous construction puzzle
In my early teens I discovered this great polymath, inventor and poet. His writings never failed to tickle my fantasy and fuel my sense of humour. In fact you might notice my style of writing and narration still bears traces of the PH influence.
Piet Hein, born in 1905, studied philosophy (like me) and theoretical physics in Copenhagen. During the German occupation of his country he decided not to flee but to stay and work in the resistance. He used the pen in his battle against the occupiers.
One of my first encounters with Piet Hein was through his “grooks”. They were originally published in Danish newspapers. The totally invented word (“gruks” in Danish) came to mean short aphoristic poems characterized by brevity, irony, satire, unexpected use of language, rhyme and rhythm. In his lifetime Hein wrote 7000 of them, mostly in Danish or English. Here is a typical example, one of the earliest I encountered:
Losing one glove is certainly painful,
but nothing compared to the pain,
of losing one, throwing away the other,
and finding the first one again.
This was possibly his first grook. It appeared in a Danish newspaper and on graffiti all over town. The censors missed its deeper meaning, but the population did not: if you lose your freedom under Nazi occupation you should not abandon your self-respect by collaborating with the enemy. Because that will be even more painful when you find your freedom some day.
Piet Hein’s grooks have been collected in maybe 20 volumes. Here are some I have known for most of my adult life. I have replaced the line feeds with slashes to save space:
- Problems worthy of attack / prove their worth by hitting back.
- Naive you are if you believe / life favours those who aren’t naive.
- Sun that givest all things birth, / shine on everything on earth! / If that’s too much to demand, / shine at least on this our land. / If even that’s too much for thee, / shine at any rate on me.
- As Pastor X steps out of bed, / he slips a neat disguise on: / that halo round his priestly head / is really his horizon.
- It may be observed, in a general way, / that life would be better, distinctly / If more of the people with nothing to say / were able to say it succinctly.
In 1959 the city of Stockholm was looking for the optimum design for a roundabout on the town centre. A circle wasted space, an ellipse had overly narrow ends. The Swedes consulted Piet Hein, who proposed using a superellipse, also known as a Lamé curve after Gabriel Lamé. It is defined by an equation that allows for various shapes between a rectangle and an ellipse — for the mathematically inclined here is a description. That worked fine for the roundabout, and I am pleased to note that Stockholm has retained the shape to this day.
Subsequently Piet Hein used the geometry of the superelipse to make a solid object which he dubbed a “superegg.” It has interesting properties and became a novelty item.
In Denmark you can see giant supereggs in parks and historical places. In 1971 a one-ton superegg made of steel and aluminium was placed outside Kelvin Hall in Glasgow (above right).
The Soma Cube
In 1933 Piet Hein was waiting for a lecture on quantum mechanics by Werner Heisenberg to begin. He started speculating: if you construct all irregular pieces that are possible with four cubes, and add the only irregular piece consisting of three cubes, you have a total of 6x4 = 24, plus 3 = 27 cubes.
Can these pieces be arranged into a 3x3x3 cube? Instead of listening carefully to all that Heisenberg had to say, he puzzled over his cube problem, and at home he constructed the first “Soma Cube” (he took the name from Aldous Huxley’s “Brave New World”). Piet soon discovered that you can indeed arrange the seven pieces into a 3x3x3 cube — and do this in 240 distinct ways.
Now I want you to go to Amazon, or Ebay, or to some local toy shop and buy a nice Soma Cube. They cost just $6 to $20 and will give you and your family years of pleasure. I think that no child should grow up without one.
Above you see a very remarkable young boy trying to construct a 3x3x3 cube for the first time. He is Gukesh, twelve year old at the time, and already a full chess grandmaster — I predict that in around eight years he will be fighting for the World Championship title. [Addendum in June 2024: I was wrong! Gukesh is now the official challenger, three years earlier than I had anticipated. Addendum November 2024: And now he has become World Champion, at 18 the youngest in chess history.] On the left is his father, an ENT surgeon. They were visiting me in Hamburg, and I was giving the boy all kinds of non-chess puzzles. It was the start of a small research project I am engaged in.
Gukesh worked out a solution in ten or fifteen minutes. He may be one of the most talented chess players in the world, but he is not as experienced with puzzles and general logical problems as my grandson Enders (take a look at the “Magic of Logic” articles in my bio-blog). Today I pulled out the cube for the seven-year-old, and he solved it in just a few minutes.
After he had solved the basic 3x3x3 cube I had a different problem for Enders: I sketched the following mini-figure on a piece of paper and asked him to construct it — in his mind. Without touching any of the pieces. In the above picture you can see him thinking.
It was fascinating to hear Enders’ reasoning: okay, there are eight cubes in the figure, so the three-cube piece is eliminated. The T-shaped piece is also out, since we would need two of them. Then suddenly, after just four or five minutes, he said “I have it!”
I told Enders to point out the required pieces, and then construct the figure — which he did with without any hesitation. He had really worked it out in his mind. I told him that the great biologist Konrad Lorenz called this “Hantieren im Vorstellungsraum” — manipulation of objects in your imagination. It is something that only humans can do, apart from very rudimentary examples observed in crows and monkeys.
It was in any case quite a feat — you can try this yourself: do the same with the cube pieces, if you have them, or try to find the solution just looking at the seven pieces shown in the first Soma picture above.
So, playing with the Piet Hein’s cube will help train your brain in 3D thinking. I did it for many years, and corresponded in my teens with the great Martin Gardner on the subject. I started making figures for people to solve, and in fact won a prize for one of them. I have scanned some of my figures from the time and present them to you. Hope you enjoy solving them as much as I had composing them.
The following composition (click to enlarge) was published in a technical magazine that Martin Gardner recommended it to — and it won a prize.
There is a trick to reconstructing it: if you count the number of small cubes the figure must contain it comes to 28 — one too many. You figure it out.
If you become hooked and are looking for more figures to construct you will find literally hundreds with a simple google search (for soma cube figures).
Finally a word of caution: there is a slanting soma cube version called Rhōma, where each block is a rhombic hexahedron (with the same angles as organic carbon compounds). When the Rhōma cube, which is said to have only one solution, is put together, it is in itself a rhombic hexahedron. I advocate caution because it is devilishly difficult to solve. For some reason I find it very much harder than in a regular soma cube to visualise how the individual pieces can fit together.
And it’s addictive!
