Polyhedra attracted the attention of mathematicians and scientists even in ancient times. The Egyptians built the pyramids. And the Greeks studied "regular polyhedra." They are sometimes called Platonic solids. "Traditional polyhedra" consist of flat faces, straight edges and vertices. But the main question has always been what rules these individual parts must comply with, and also what additional global conditions must be met for the object to qualify as a polyhedron. The answer to this question will be presented in the article.
Problems in determining
What does this figure consist of? A polyhedron is a closed continuous shape that has flat faces and straight edges. Therefore, the first problem of its definition can be called exactly the side of the figure. Not all faces lying in the planes are always a sign of a polyhedron. Take the “triangular cylinder” as an example. What does it consist of? A part of its surface of three pairwise intersecting vertical planes cannot be considered polygons. The reason is that it has no vertices. The surface of such a figure is formed on the basis of three rays that meet at one point.
Another problem is planes. In the case of a "triangular cylinder" it consists in their unlimited parts. A figure is considered convex if a segment of a line connecting any two points in the set is also in it. We give one of their important properties. For convex sets, it is that the set of points common to the set is the same. There is another kind of figure. These are non-convex two-dimensional polyhedra that have either recesses or openings.
Non-Polyhedron Shapes
A flat set of points may be different (for example, non-convex) and may not satisfy the usual definition of a polyhedron. Even through it it is limited by sections of lines. The lines of a convex polyhedron consist of convex figures. However, this approach to definition excludes a figure that goes to infinity. Her example may be three rays that do not meet at one point. But at the same time they are connected to the vertices of another figure. Traditionally important for the polyhedron was that it consists of flat surfaces. But over time, the concept expanded, which led to a significant improvement in the understanding of the original “narrower” class of polyhedra, as well as the emergence of a new, broader definition.
Correct
We introduce one more definition. A regular polyhedron is one in which each face is a congruent regular convex polygon, and all vertices are “the same”. This means that each vertex has the same number of regular polygons. Use this definition. So you can find five regular polyhedra.
First Steps to Euler's Theorem for Polytopes
The Greeks knew about the training ground, which today is called the pentagram. This polygon could be called regular, because all its sides are of equal length. There is also one more important point. The angle between two consecutive sides is always the same. However, when drawing in a plane, it does not determine a convex set, and the sides of the polyhedron intersect each other. However, this was not always the case. Mathematicians have long considered the idea of ​​"non-convex" regular polyhedra. The pentagram was among them. “Star polygons” were also allowed. Several new examples of “regular polyhedra” have been discovered. Now they are called Kepler-Poinsot polyhedra. Later, S. S. M. Cockster and Branko Grünbaum expanded the rules and discovered other “regular polyhedra”.
Polyhedral formula
A systematic study of these figures began relatively early in the history of mathematics. Leonard Euler was the first to notice that for convex three-dimensional polyhedra a formula is valid that relates the number of their vertices, faces and edges.
It looks like this:
V + F - E = 2,
where V is the number of polyhedral vertices, F is the number of edges of polyhedra, and E is the number of faces.
Leonard Euler is a Swiss mathematician who is considered one of the greatest and most productive scientists of all time. He was blind most of his life, but his loss of vision gave him a reason to become even more productive. There are several formulas named in his honor, and the one we just examined is sometimes called the Euler polyhedron formula.
There is one clarification. Euler's formula, however, only works for polyhedrons that follow certain rules. They consist in the fact that the form should not have any holes. And it is not permissible for her to cross herself. A polyhedron also cannot consist of two parts connected together, for example, two cubes with one vertex. Euler mentioned the result of his research in a letter to Christian Goldbach in 1750. He later published two works in which he described how he tried to find evidence of his new discovery. In fact, there are forms that give a different answer to V + F - E. The answer to the sum F + V - E = X is called the Euler characteristic. She has one more aspect. Some forms may even have an Euler characteristic, which is negative
Graph theory
It is sometimes claimed that Descartes derived Euler's theorem earlier. Although this scientist discovered facts about three-dimensional polyhedra that would allow him to derive the desired formula, he did not take this additional step. Today Euler is credited with “paternity” of graph theory. He solved the problem of the Konigsberg bridge using his ideas. But the scientist did not look at the polyhedron in the context of graph theory. Euler tried to give a proof of the formula based on the decomposition of the polyhedron into simpler parts. This attempt does not meet modern standards for evidence. Although Euler did not give the first correct justification for his formula, it is impossible to prove guesses that were not made. However, the results, which found their justification later, make it possible to use Euler's theorem at present. The first proof was received by the mathematician Adrian Marie Legendre.
Evidence of the Euler Formula
Euler first formulated the polyhedral formula as a polyhedron theorem. Today it is often interpreted in the more general context of related graphs. For example, as structures consisting of points and line segments connecting them, which are in one part. Augustin Louis Cauchy was the first person to find this important connection. It served as a proof of Euler's theorem. In fact, he noticed that the graph of a convex polyhedron (or what is called today) is topologically homeomorphic to a sphere and has a flat connected graph. What it is? A flat graph is one that has been drawn in a plane so that its edges meet or intersect only at the top. This was the connection between the Euler theorem and graphs.
One indication of the importance of the result is that David Epstein was able to collect seventeen different pieces of evidence. There are many options for justifying the polyhedral Euler formula. In a sense, the methods that use mathematical induction are the most obvious evidence. The result can be proved by drawing it over the number of either edges, faces or vertices of the graph.
Proof of Rademacher and Greenhouse
The following evidence of Rademacher and Toeplitz, based on the approach of Von Staudt, is especially attractive. In order to substantiate Euler’s theorem, suppose that G is a connected graph embedded in the plane. If it has schemes, it is possible to exclude one edge from each of them in such a way as to preserve the property at which it remains connected. There is a one-to-one correspondence between the remote parts for moving to a connected graph without closure and those that are not an infinite face. This study led to the classification of “orientable surfaces” in terms of the so-called Euler characteristic.
Jordan curve. Theorem
The main thesis, which is used directly or indirectly in proving the formula of the polyhedra of Euler's theorem for graphs, depends on the Jordan curve. This idea is related to generalization. It says that any simple closed curve divides a plane into three sets: points on it, inside and outside it. Since interest in Euler’s multifaceted formula developed in the nineteenth century, many attempts were made to generalize it. This study laid the foundation for the development of algebraic topology and connected it with algebra and number theory.
Mobius Group
It was soon discovered that some surfaces can be “oriented” in a consistent way only locally, but not globally. The well-known Mobius band illustrates such a surface. It was discovered a little earlier by Johann Listing. This concept includes the notion of the kind of graph: the least number of descriptors g. It must be added to the surface of the sphere, and it can be embedded on the expanded surface so that the edges meet only at the vertices. It turns out that any orientable surface in Euclidean space can be considered as a sphere with a certain number of handles.
Euler Chart
The scientist made another discovery, which is still used. This is the so-called Euler diagram - a graphical image made up of circles, commonly used to illustrate relationships between sets or groups. Charts typically include colors that blend in areas where circles overlap. The sets are depicted precisely in circles or ovals, although other figures can also be used for them. The inclusion is represented by overlapping ellipses called Euler circles.
They represent sets and subsets. The exception is non-overlapping circles. Euler diagrams are closely related to another graphic image. They are often confused. This graphic is called Venn diagrams. Depending on the sets in question, both versions may look the same. However, in Venn diagrams, overlapping circles do not necessarily indicate the commonality between the sets, but only a possible logical connection if their labels are not in the intersecting circle. Both options were adopted for teaching set theory as part of the new mathematical movement of the 1960s.
Fermat and Euler theorems
Euler left a noticeable mark on mathematical science. Algebraic number theory was enriched by a theorem named in his honor. It is also the result of another important discovery. This is the so-called general algebraic Lagrange theorem. Euler's name is also associated with Fermat's little theorem. It says that if p is a prime and a is an integer not divisible by p, then:
and p-1 - 1 is divided by p.
Sometimes this same discovery has a different name, most often found in foreign literature. It sounds like Fermat’s Christmas Theorem. The thing is that the discovery became known thanks to a letter from a scientist sent on the eve of December 25, 1640. But the statement itself has been encountered before. It was used by another scientist named Albert Girard. Fermat was only trying to prove his theory. The author hints in another letter that he was inspired by the method of endless descent. But he did not give any evidence. Later, Aider also turned to the same method. And after him - many other famous scientists, including Lagrange, Gauss and Minkosky.
Identity Features
Fermat’s small theorem is also called a special case of a theorem from number theory belonging to Euler. In this theory, the Euler identity function counts positive integers up to a given integer n. They are coprime with respect to n. Euler’s theorem in number theory is written using the Greek letter φ and looks like φ (n). It can be more formally defined as the number of integers k in the range 1 ≤ k ≤ n, for which the greatest common factor gcd (n, k) is 1. The notation φ (n) can also be called the Euler phi-function. The integers k of this form are sometimes called totative. At the core of number theory, the Euler identity function is multiplicative, meaning that if two numbers m and n are coprime, then φ (mn) = φ (m) φ (n). It also plays a key role in defining the RSA encryption system.
The Euler function was introduced in 1763. However, at that time, the mathematician did not choose any specific symbol for its designation. In a publication in 1784, Euler studied this function in more detail and chose the Greek letter π to denote it. James Sylvester coined the term “total” for this function. Therefore, it is also referred to as Euler Total. The total φ (n) of a positive integer n greater than 1 determines the number of positive integers less than n that are coprime to n.φ (1) is defined as 1. The Euler function or the phi (φ) function is a very important number-theoretic a function that is deeply related to primes and the so-called order of integers.