**Students are looking for special tetrahedrons that fit together**

What Aristotle began over 2,000 years ago is continued by a team of 30 MIT students. They have capitalized on recent mathematical advances that have breathed new life into a millennial search for shapes that can perfectly fill or stack **three-dimensional space**.

It’s quite interesting, but at the same time a little intimidating to know that some of the greatest minds have been working on this topic, said Yuyuan Luo, a first-year student at MIT who participates in the work organized by MIT professor Bjorn Punen. (Poonen receives funding from the Simons Foundation, which also funds this editorially independent publication.)

**Aristotle’s interest in this issue arose as a rebuke to Plato, his teacher.**

In his dialogue 360 BC. E. Timaeus Plato discussed the ancient theory according to which the world consists of four elements: earth, water, air and fire. He suggested that each of these elements consists of particles with a unique shape corresponding to one of five regular solids: a particle of earth is in the shape of a cube, a particle of water is a 20-sided icosahedron, and a particle of air is in the shape of a cube. an octahedron and a particle of fire, similar to a pointed tetrahedral pyramidal tetrahedron (because fire is thorny). Abstraction helps you navigate promising ideas in science and mathematics. Travel with us and join the conversation.

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Aristotle objected based on his assumption (which we now know is false) that the particles of these elements should be able to completely fill space. That is, he thought, where there is water, you need to be able to arrange copies of the icosahedral particle of water so that the icosahedrons completely occupy the water space without overlapping.

And this, thought Aristotle, is the snag. In his treatise 350 BC. E. About heaven, he explained that copies of the icosahedron “cannot fill everything.” Therefore, he argued, water particles cannot have this shape. For the same reason, he doubted that air particles could have the shape of an octahedron. But he admitted that the copies of the cube (earth) and the tetrahedron (fire) actually fill space, so he allowed Plato’s theory to speak for these two elements.

**Thousands of years later, it turned out that Aristotle was partly wrong about this**.

As early as the 1400s, scientists began to suspect that a regular tetrahedron, in which all four sides of the pyramid are equilateral triangles, could not be used to fill space either. By the 1600s, they had established that for sure. This is something that Aristotle could admit if he only tried to figure it out on his own.

If Aristotle had made models of regular tetrahedrons, he would have placed several around an edge, taking one tetrahedron and placing the other directly on top of it. According to Marjorie Seneschal of Smith College, in five he would have seen a small gap that cannot be filled with another tetrahedron.

If a regular tetrahedron does not take up space, the question arises: are there any tetrahedrons?

#### Regular tetrahedrons cannot completely fill or pave space.

In 1923, Duncan Sommerville confirmed the first such examples. In general, mathematicians have found two separate tetrahedra and three **infinite families** of tetrahedra that fill space. Families have an option that you can tweak in an infinite number of ways to make some interior corners smaller and others proportionally larger, while still allowing for tiling space. No other mathematicians were found. They have no idea how many there might be.

I don’t know there will be a theoretical solution to this problem other than just looking for these things, Seneschal said. The point is that most 3D shapes have no room for tiles. We don’t understand how difficult it is to lay out three-dimensional space in mosaics, said Inna Zakharevich from Cornell University. I think whatever it does is cool.

This means that the search for such forms is a kind of blind hunt. Fortunately, **finding tetrahedrons** that can tile three-dimensional space is aided by an elegant fit between the problem and two other related questions. The first related question is: can two flat forms of the same volume always be separated by straight cuts and assembled with each other? David Gilbert asked about this in 1900, and in the same year his former student Max Dehn provided an important part of the answer.