Three Pearls of Number Theory

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This relationship can be expressed mathematically. In the picture above, both the square S and the house H can be decomposed into the orange O , purple P , and blue B pieces. Another property that we are interested in is equicomplementability. Two shapes are equicomplementable if congruent building blocks can be added to both of them to create two equidecomposable supershapes.

In addition to being equidecomposable, our house and square are also equicomplementable. We can express this relationship mathematically as well. We can add four pieces—here a yellow piece Y , a pink piece P , an orange piece O , and a maroon piece A —to the square S in order to create a supershape M.

We can add the same pieces to the house H to create a second supershape N. These supershapes can then be decomposed into the same pieces, pictured here as a red piece R , a turquoise piece T , a brown piece B , a blue piece L , and a green piece G. Equidecomposability, equicomplementability, and equality in area are intertwined for polygons in the second dimension. Throughout the s there were many developments in these properties as they apply to two-dimensional polygons. Following preliminary work by William Wallace in , independent proofs from Farkas Bolyai in and P.

Gerwein in demonstrated that any polygon can be decomposed in such a way that its pieces can be reassembled into a square, as we illustrated earlier. This means that any pair of polygons of equal area is equidecomposable, since they can be decomposed and reassembled into the same square.

Three Pearls of Number Theory

In , Gerling furthermore showed that it does not matter if reflections are allowed in the reassembly of the decomposed shapes. The following interesting theorems and lemmas were proved in the nineteenth century. At the end of the nineteenth century, the question was settled in the second dimension and was expanded into the third. This is the topic that we will focus on: can the Bolyai-Gerwien theorem—that any two equal-area polygons are equidecomposable—be extended into the third dimension?

Instead of looking at polygons, we will look at their counterparts in the third dimension, polyhedra. Polyhedra have been defined in many ways, and not all of the definitions are compatible. A polyhedron is the union of a finite number of polygons that has the following properties:. Within the constraints of this definition, polyhedra are diverse and varied. Below are some examples. If the squished part of the second-to-last polyhedron were further squished into a single point, then by the third bullet point above it could no longer be a single polyhedron.

Just as any pair of polygons of equal area can be decomposed and reassembled into the same square, can any pair of polyhedra of equal volume be decomposed and reassembled into the same cube? Are polyhedra of equal volume equidecomposable? Are they equicomplementable? By the end of the nineteenth century there were several examples of equalvolume polyhedra that were both equidecomposable and equicomplementable, but there was no general solution.

A number theory proof

One simple example is prisms with the same height and equal area bases, stemming from the two-dimensional polygon result. In Gerling showed and then Bricard proved again in that two mirror-image polyhedra are equidecomposable by cutting them up into congruent mirror-image pieces that can then be rotated into each other. There were also some specific tetrahedra equidecomposable with a cube, shown in by M. A tetrahedron is a polyhedron with four triangular faces, six edges, and four vertices. In many current math textbooks the faces are required to be congruent.

We are not going to require that any of the faces be congruent; our definition is closer to what many current math textbooks call a triangular pyramid. Just as any polygon can be cut up into triangles, any polyhedron can be cut up into tetrahedra. First, we can cut any polyhedron into a finite number of convex polyhedra. These can each then be cut up into a finite number of pyramids with polygonal bases. Because any polygon can be cut up into a finite number of triangles, each of these polygonal pyramids can be cut up into a finite number of triangular pyramids.

This means that if we can prove or disprove the Bolyai-Gerwien theorem in the third dimension for tetrahedra, then we have also proved or disproved it more generally for polyhedra. Below is an example of a division of a polyhedron into triangular pyramids, based off of figures Unlike the area of a triangle, however, the volume of a tetrahedron and therefore the volume of a polyhedron is found through calculus, by dividing the three-dimensional polyhedron into infinitesimally thin two-dimensional cross sections and adding up their areas.

If the Bolyai-Gerwien theorem can be expanded into the third dimension, we can define the volume of any three dimensional polyhedron the same way we define the area of a two dimensional polygon, by breaking it up into discrete building blocks—tetrahedra in three dimensions and triangles in two—and reassembling the pieces into a cube or square.

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This would be an elementary solution, with no infinities or calculus required. This problem was posed by C. In a letter in , Gauss expressed that he wanted to see a proof that used finitely rather than infinitely many pieces.


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One of my favorite aspects of geometry is how seamlessly it can transition from elementary school math to the cutting edge. Modern geometry is built on fluid connections between the basic principles. This foundation was built largely by David Hilbert. The fundamentals of geometry were initially outlined by Euclid in Elements.

In the nineteenth century, geometry was becoming increasingly abstract and less and less tied to the original shapes. As an example, one such unstated assumption was that if two lines cross, they must have a point in common. Hlibert extended Elements , providing an axiomization of Euclidian geometry and proofs for the unchecked assumptions that stood in the way of geometry being as fully useful as algebra.

David Hilbert was born in Wehla, Germany. His mother was interested in philosophy, and his father was a judge and wanted him to study law. He was homeschooled for two years and began school two years late, at age eight. The subject he went on to study was mathematics, because he did not like memorization. A turning point for Hilbert was at his first presentation, at the Technische Hochschule , where he impressed and befriended Klein, 13 years his senior. In his letters back to Klein, he made comical judgements of some of the most important mathematicians of the time.

In the beginning he continued to focus on invariant theory, a branch of abstract algebra examining how algebraic expressions change in response to change in their variables. It was here, starting with lectures to his students, that Hilbert became interested in geometry.

At the second meeting of the International Congress of Mathematicians to this day the largest math conference, meeting once every four years, and the conference at which the Fields Medal is awarded , that year in Paris, David Hilbert, then 38, presented a monograph of ten of the open problems that he considered the most important for the next century.

The two that have not been solved, the axiomization of physics and the foundations of geometry, are now considered less of a priority and too vague for a definitive solution, respectively. We are interested in the third of the 23 problems, which concerns the extension of the Bolyai-Gerwien theorem into the third dimension.

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1 Introduction and History

Unlike Gauss, Hilbert did not believe that there was such a bridge: he asked simply for two tetrahedra that together formed a counterexample. In two letters to Gerling, Gauss expresses his regret that certain theorems of solid geometry depend upon the method of exhaustion, i.

Gauss mentions in particular the theorem of Euclid, that triangular pyramids of equal altitudes are to each other as their bases. Now the analogous problem in the plane has been solved. Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probable that a general proof of this kind for the theorem of Euclid just mentioned is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained as soon as we succeed in exhibiting two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra.

Specify two tetrahedra of equal volume which are neither equidecomposable nor equicomplementable.

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The proof rests on a value describing a polyhedron, the Dehn invariant, which we will look at in more detail later. The Dehn invariant does not change when the polyhedron is cut apart and reassembled into a new shape: if two polyhedra are equidecomposable, then they must have same Dehn invariant and they do. However, not all polyhedra with the same volume have the same Dehn invariant.

Hilbert’s Third Problem (A Story of Threes) | MIT Admissions

Specifically, Dehn used the example of a regular tetrahedron and a cube of equal volume, and we will examine this case as well. Two years later Dehn showed in a second paper the second part of the problem, on equicomplementability. An incomplete and incorrect proof was published by R. Bricard four years previously in It was refined by V.

Kagan from Odessa in In the s, Hadwiger, a Swiss geometer, together with his students found new properties of equidecomposability. Further progress over the past century has made it even clearer and more concise. We will be presenting a very recent version of the proof, published in in Proofs from the Book. We will present the solution in three parts: three definitions, three proofs, and three examples.

We have already gone through the definitions. A tetrahedron, as defined here, is a polyhedron with four not necessarily congruent triangular faces. Equidecomposability is a relationship between two shapes in which one can assemble one from all the pieces of another. Finally, equicomplementability is a relationship in which one can add congruent shapes to the two shapes to form two equidecomposable supershapes.

[3] Khinchin a.Y.-three Pearls of Number Theory

Afterward we will apply the result to three example tetrahedra. Our first proof is by Benko. We start by defining the segments of an edge. Each edge in a decomposed shape consists of one or more segments that, placed end to end, make up the total length of that edge. In a decomposition of a polygon, the endpoints of segments are always vertices; in a decomposition of a polyhedron, the endpoints of segments can also be at the crossing of two edges.

Otherwise, all non-endpoint points within any one segment belong to the same edge or edges. In the decomposition of the square below, for example, the hypotenuse of the larger triangle is subdivided into two segments. We examine two equidecomposable figures. They are broken up into the same pieces, which are rearranged and perhaps reflected in different ways. Imagine that we must distribute whole, indivisible tokens—which we will call pearls— on all the segments in the two decompositions.