INQ's Trisected Cube: A Lesson In Geometric Dissection

What you’re holding in your hand is a cube split symmetrically along a helicoid surface–or as we prefer to call it, a trisected cube. Creating figures like this is a part of geometry known as geometric dissection. It has long been known that for any two polygons of equal area, there is a set of smaller polygons that can be arranged to form either of the two (this is the basis of tangrams, among other things). This partition into smaller pieces is called a dissection.  In three dimensions this doesn’t necessarily hold true and the dissection of 3D figures is arguably even more complex, and creating them takes an exceedingly clever mind.

This cubic trisection was invented by a self-described “recreational mathematician” named Robert Reid in the 1980s. Reid came up with ten separate ways to dissect a cube using three identical smaller pieces, an incredibly difficult mathematical task. And the man himself was an interesting character; though trained in mathematics at school, he was not a professional mathematician. Born and raised in England, he relocated to Peru after WWII, where he operated an import-export business, ran a movie theatre, a high-end antique shop, and acted in a Peruvian chewing gum commercial and at least one 1970s Spanish-language horror movie so radical it was banned in Argentina. He moved back to England near the end of his life, and became a regular at London-area puzzle gatherings.    

Reid was inspired by famous author and puzzler Martin Gardner, and they kept up a longtime correspondence. One of Gardner’s 1966 columns on MC Escher inspired Reid to pick up geometric dissection and it’s companion art, tessellation. Some of Escher’s best-known creations are tessellations–patterns created when the same shape is repeated over and over again, covering a plane without any gaps or overlaps. Reid went on to create hundreds of these patterns (also called “tilings”), even publishing a book of them titled “The Gentle Art of Filling Space.” 

Interestingly, though trisecting a cube is its own incredible puzzle, it is reminiscent of two of the three classical problems of ancient Greek mathematics: 1) trisecting an angle and 2) doubling a cube. The first question involves attempting to trisect an arbitrary angle into three smaller angles, using only a straightedge and a compass (it’s amazing how much ancient Greece was like middle school geometry class). People tried to do it for literally over a thousand years, until Pierre Wantzel came up with a proof in 1837 that showed it was generally impossible with the exception of a few very special angles.

Doubling a cube is a similar problem. Again using only a straightedge and a compass, if given the length of an edge of a cube, to try and construct a second cube having double the volume of the first. Again, this one is possible using other tools, but not the ones specified. Actually, part of what is interesting for this puzzle is the origin of the challenge: the citizens of the ancient city of Delos were suffering from a plague, so they consulted the oracle of Apollo for advice on ending it. The oracle made a pronouncement: they must build for Apollo a new altar of double the size. So the Delians built one two times longer on each side, which of course was eight times the volume. It turned out Apollo was trying to teach them a lesson; by “size” he meant twice the volume, not edge size, and so the plague kept raging. According to ancient sources, “he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt for geometry.”  So you heard it straight from Apollo, kids: stay in school!