What Aristotle started over 2,000 years ago, a team of 30 undergraduates at the Massachusetts Institute of Technology is continuing. They’ve been capitalizing on a recent mathematical advance that has injected new life into a millennia-long quest to identify shapes that can perfectly fill, or tile, three-dimensional space.
“It’s quite exciting but also at the same time a little intimidating to know some of the greatest minds have been working on this topic,” said Yuyuan Luo, a first-year MIT student participating in the work organized by MIT professor Bjorn Poonen. (Poonen receives funding from the Simons Foundation, which also funds this editorially independent publication.)
Aristotle’s interest in the question arose as a rebuke to Plato, his teacher.
In his 360 BCE dialogue Timaeus, Plato discussed the ancient theory that the world was made of four elements: earth, water, air and fire. He conjectured that each of these elements was made of particles with a unique shape corresponding to one of the five regular solids: a particle of earth was shaped like a cube, a particle of water like the 20-sided icosahedron, a particle of air like the octahedron, and a particle of fire like the pointy, four-sided, pyramidal tetrahedron (because fire is prickly).
Aristotle objected based on his presumption (which we now know to be false) that the particles of these elements would have to be able to fill space completely. That is, he thought where there’s water, you’d need to be able to arrange copies of the icosahedral water particle such that the icosahedra perfectly occupied the entirety of the watery space without overlapping.
And there, thought Aristotle, was the catch. He explained in his 350 BCE treatise On the Heavens that copies of an icosahedron “will not succeed in filling the whole.” Therefore, he argued, particles of water can’t possibly have that shape. He doubted, for the same reason, that particles of air could be shaped like an octahedron. But he allowed that copies of a cube (earth) and of a tetrahedron (fire) do fill space, so he let Plato’s theory stand for those two elements.
Thousands of years later, it turned out Aristotle was partly wrong there as well.
As early as the 1400s scientists began to suspect that the regular tetrahedron — in which all four faces of the pyramid are equilateral triangles — can’t be used to fill space either. By the 1600s they’d established it for sure. This is something Aristotle might have recognized, too, if only he’d tried to figure it out for himself.
“If Aristotle had made models of regular tetrahedra, he would have put several around an edge by taking one tetrahedron and fitting another right to it. Within five he’d have seen that there’s a little gap that can’t be filled by another tetrahedron,” said Marjorie Senechal of Smith College.
If the regular tetrahedron doesn’t tile space, the question becomes: Do any tetrahedra?
In 1923 Duncan Sommerville confirmed the first examples that do. All told, mathematicians have now found two individual tetrahedra and three infinite families of tetrahedra that fill space. The families feature a parameter that you can adjust infinitely many ways to make some interior angles smaller and others proportionally larger while maintaining the ability to tile space. Mathematicians have not found any others. They have no idea how many might exist.
“I don’t know that this is a problem that’s going to have a theoretical solution beyond just finding these things,” said Senechal.
The fact is, most three-dimensional shapes don’t tile space. “We don’t appreciate how hard it is to tile three-dimensional space,” said Inna Zakharevich of Cornell University. “I think anything that does is pretty cool.”
That means looking for such shapes is a bit of a blind hunt. Fortunately, the search for tetrahedra that can tile three-dimensional space is aided by an elegant correspondence between the problem and two other related questions.
The first related question is: Can two flat-sided shapes of the same volume always be partitioned with straight cuts and reassembled as each other? David Hilbert asked this in 1900, and that same year his former student, Max Dehn, provided an important part of the answer.
Dehn showed that it’s possible to use the angles of any polyhedral shape — like a tetrahedron or a cube — to calculate a single quantity, now called the Dehn invariant. He proved that for two shapes to be “scissors congruent” — meaning they can be cut up and reassembled as each other — they must have the same Dehn invariant. Dehn used his new measurement to prove that the regular tetrahedron is not scissors congruent to a cube since their Dehn invariants differ.
Later in the century mathematicians proved two other key facts that tied together scissors congruence and tiling. In 1965 Jean-Pierre Sydler proved that any two shapes with the same volume and the same Dehn invariant are scissors congruent. Furthermore, in 1980 Hans Debrunner showed that any tetrahedron that tiles space must have a Dehn invariant of 0 — the same as a cube. The upshot of these discoveries is that a tetrahedron must be scissors congruent to a cube to have a chance of tiling space.
If you’re handed a tetrahedron, it’s relatively easy to calculate whether it has a Dehn invariant of 0 and therefore has the potential to tile space. However, finding all tetrahedra that have a Dehn invariant of 0 is not an easy task.
That’s where the second related question comes in.
A tetrahedron contains six “dihedral” angles formed along the edges where pairs of faces meet. In 1976 John H. Conway and Antonia J. Jones asked: Is it possible to identify all tetrahedra in which the degree measures of all six of those dihedral angles are rational numbers, meaning they can be written neatly as fractions? It’s a modern question with hints of Aristotle.
“I like to say this problem could have been asked in ancient times, but it wasn’t as far as I know,” said Kiran Kedlaya of the University of California San Diego. Kedlaya, Poonen and two other co-authors proved that exactly 59 isolated examples plus two infinite families of tetrahedra have rational dihedral angles. Quanta recently covered this result in our story “Tetrahedron Solutions Finally Proved Decades After Computer Search.”
And crucially, any tetrahedron with rational dihedral angles has a Dehn invariant of 0, meaning it’s scissors congruent to a cube and stands a chance of tiling space.
That leads to what the MIT undergraduates have been working on, together with Poonen —investigating which of these candidates fulfill their potential as three-dimensional tiles.
In mid-January, the group proved that one of the isolated rational tetrahedra does not fill space. Their result marks the first time anyone has found an example of a tetrahedron that is scissors congruent to a cube but doesn’t tile space. It’s also the latest twist in an intellectual braid that started long ago with an ancient curiosity.