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The tetrahedron is the simplest polyhedron. Over the long years, this seemingly simple three-dimensional shape has extended many questions that can cause great minds to think about it. In November 2020, four mathematicians submitted a 30-page paper on the academic preprint website arXiv. They used number theory to prove an ancient problem related to tetrahedrons.

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This question can be traced back to Plato and Aristotle more than 2,000 years ago, and it aims to determine that can perfectly fill (or ” Dense paving”)A polyhedron in three-dimensional space. Plato believes that the world is composed of five types of “materials”: water, air, fire, earth, and ether. Each “material” corresponds to a specific polyhedral shape. These three-dimensional shapes with equal side lengths were later called Platonic polyhedron.

Plato uses regular polyhedrons to define ancient elements: cube (earth), icosahedron (water), regular octahedron (air), regular tetrahedron ( Fire), regular dodecahedron (ether). | Picture source: Wikipedia

But, PlatoAristotle’s student does not agree with this hypothesis. He believes that if the world is really made of these substances, then these corresponding shapes must be able to completely fill the space. He believes that although the cubes and tetrahedrons corresponding to earth and fire can fill the space, the icosahedrons and octahedrons corresponding to water and air cannot do this.

Of course, Aristotle’s judgment on this issue is not entirely correct. Since the 15th century, scientists have begun to question the possibility of a regular tetrahedron that can fill space. Scientists in the 17th century have confirmed that the regular tetrahedron cannot do this. This is actually very easy to verify. You only need to place several regular tetrahedron models side to side, and you will find that within the five regular tetrahedrons, there will inevitably be a gap that cannot be filled.

In fact, most three-dimensional shapes cannot be densely packed. Then, a new question arises: If a regular tetrahedron cannot fill the space, can other tetrahedrons do it?

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The answer is yes. In 1923, the mathematician Duncan Sommerville proved the first tetrahedron that can be densely packed. So, how many such tetrahedrons are there? However, it is very difficult to find such a tetrahedron. But fortunately, mathematicians discovered that the problem of finding a tetrahedron that can be densely paved in three-dimensional space is related to two other problems.

The first question is the third question of the 23 questions asked by David Hilbert in (David Hilbert) in 1900 : For any two polyhedrons of equal volume, is it always possible to cut one of the polyhedrons into a finite number of polyhedrons, and then recombine them into another polyhedron?

Two-dimensional example of scissors congruent: Two-dimensional polygons with the same area are scissors congruent. | Picture reference source: QuantaMagazine

In other words, the question can be formulated as: Is any pair of polyhedrons with the same volume all scissors congruent? One shape is congruent with another shape scissors, which means that one of them can be cut in a straight line to form another shape.

In the same year, Max Dehn (Max Dehn) put forward a key concept to answer this question. He proved that this question is related to polyhedral Angle is related to side length. He found that from the perspective of a polyhedron, a quantity called Dern invariant can be calculated. When two shape scissors are congruent, then their Dern invariants must be equal.

The congruent shape of three-dimensional scissors requires that the two shapes have the same volume and the same Deen invariant. The figure shows a regular tetrahedron and a cube with the same volume, but they are not scissors congruent because they have different Dern invariants. | PictureReference source: Wikipedia Commons

In 1980, Hans Debrunner proved that for any tetrahedron that may be densely paved, its Dern invariant must be the same as a cube-equal to zero. This means that the tetrahedron congruent with the cube scissors can be densely packed. The mathematicians then discovered that all the dihedral angles of the tetrahedron congruent with the cube scissors are rational numbers.

At this point, another related problem also appeared.

In 1976, John Conway(John H. Conway) and Antonia Jones(Antonia JJones) published a paper in which they asked the question: Is it possible to identify all tetrahedrons whose dihedral angles are all rational numbers? ?

They thought that they could find this kind of rational tetrahedron by solving a specific polynomial equation. There are six variables in their equation, corresponding to the six dihedral angles of a tetrahedron; it has 105 terms, reflecting the relationship between the six dihedral angles. This polynomial equation has an infinite number of solutions, corresponding to an infinite number of different tetrahedral configurations.

A tetrahedron has 6 dihedral angles. | Image source: Wikipedia Commons

Conway and Jones believed that in order to find solutions where all dihedral angles are rational degrees by solving equations, one type of equation must be foundA special solution that corresponds exactly to the rational tetrahedron. But they don’t know how to do this.

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In 1995, mathematicians Bjorn Poonen, Michael Rubinstein and other mathematicians searched and discovered these special rational tetrahedra through computers. Their results show that 59 tetrahedrons satisfy these conditions, plus two tetrahedrons from infinite families. The tetrahedrons in the infinite family all have an angle parameter that can be adjusted infinitely, so that these tetrahedrons can maintain the ability of densely-paved space no matter what adjustment they undergo.

However, Poonen and others could not prove that the tetrahedrons they have found are all tetrahedrons that can be densely packed. Up to now, the method clarified by the four mathematicians in a new paper confirmed that what was discovered 25 years ago are all rational tetrahedra, and there are no other examples that have not been discovered.

In the method provided by the new research institute, the mathematician first proved that the complex polynomial equation used to represent the tetrahedron can be expressed as many simpler polynomials. They transformed a complex 6-variable equation into hundreds of relatively simple equations and solved these equations. Then, based on the prediction of some properties of the equation solution, they made more targeted settings in the solution process, and obtained an algorithm that can search for the solution of the equation quickly and efficiently.

In the end, they found the 59 independent tetrahedrons and two infinite tetrahedrons. Moreover, these tetrahedrons with rational dihedral angles all have a Dern invariant of zero, which means that they are all congruent with the cube scissors and may be densely packed.

Now, a group of undergraduates from the Massachusetts Institute of Technology are continuing to study this problem, trying to find out which of them can do three-dimensional dense paving. In January 2021, they found a counterexample, proving that an independent rational tetrahedron cannot be densely packed. This is the first time a mathematician has discovered an example of a tetrahedron that is congruent with the cube scissors but cannot be densely packed.

#Reference source:

https://www.quantamagazine.org/mit-math-students-continue-aristotles-tetrahedra-tiling-20210209/

https://www.quantamagazine.org/mathematicians-finally-prove-rational-tetrahedron-solutions-20210202/

http://www-math.mit.edu/~poonen/papers/press_release.pdf

https://arxiv.org/pdf/2011.14232.pdf

This article is from WeChat official account:Principle (ID: principle1687), author: Zuoyou