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 separate parts must fulfill, as well as what additional global conditions must be met in order for an object to qualify as a polyhedron. The answer to this question will be presented in the article.
Problems in definition
What does this figure consist of? A polyhedron is a closed solid shape that has flat faces and straight edges. Therefore, the first problem of its definition can be called precisely the sides of the figure. Not all faces lying in planes are always a sign of a polyhedron. Let's take the "triangular cylinder" as an example. What does it consist of? Part of its surface three in pairsintersecting 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.
One more problem - planes. In the case of the "triangular cylinder" it lies in their unlimited parts. A figure is considered convex if the line segment connecting any two points in the set is also in it. Let us present 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 figures. These are non-convex 2D polyhedra that either have notches or holes.
Shapes that are not polyhedra
A flat set of points can be different (for example, non-convex) and 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 the definition excludes a figure going to infinity. An example of this would be three rays that do not meet at the same point. But at the same time, they are connected to the vertices of another figure. Traditionally, it was important for a polyhedron that it consists of flat surfaces. But over time, the concept expanded, which led to a significant improvement in understanding the original "narrower" class of polyhedra, as well as the emergence of a new, broader definition.
Correct
Let's introduce one more definition. A regular polyhedron is one in which each face is a congruent regularconvex polygons, 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 polyhedra
The Greeks knew about the polygon, which today is called the pentagram. This polygon could be called regular because all its sides are of equal length. There is also another important note. The angle between two consecutive sides is always the same. However, when drawn in a plane, it does not define 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 one of them. "Star polygons" were also allowed. Several new examples of "regular polyhedra" have been discovered. Now they are called Kepler-Poinsot polyhedra. Later, G. S. M. Coxeter and Branko Grünbaum extended the rules and discovered other "regular polyhedra".
Polyhedral formula
The systematic study of these figures began relatively early in the history of mathematics. Leonhard Euler was the first to notice that a formula relating the number of their vertices, faces and edges holds for convex 3D polyhedra.
She looks like this:
V + F - E=2, where V is the number of polyhedral vertices, F is the number of edges of the polyhedra, and E is the number of faces.
Leonhard Euler is Swissmathematician who is considered one of the greatest and most productive scientists of all time. He has been blind for most of his life, but the loss of his sight gave him a reason to become even more productive. There are several formulas named after him, and the one we just looked at is sometimes called the Euler polyhedra formula.
There is one clarification. Euler's formula, however, only works for polyhedra that follow certain rules. They lie in the fact that the form should not have any holes. And it is unacceptable for it to cross itself. A polyhedron also cannot be made up of two parts joined together, such as two cubes with the same vertex. Euler mentioned the result of his research in a letter to Christian Goldbach in 1750. Later, he published two papers in which he described how he tried to find proof 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 another aspect. Some shapes may even have an Euler characteristic that is negative
Graph Theory
Sometimes it is 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 the "father" of graph theory. He solved the problem of the Konigsberg bridge using his ideas. But the scientist did not look at the polyhedron in contextgraph theory. Euler tried to prove a formula based on the decomposition of a polyhedron into simpler parts. This attempt falls short of modern standards for proof. Although Euler did not give the first correct justification for his formula, one cannot prove conjectures that have not been made. However, the results, which were substantiated later, make it possible to use Euler's theorem at the present time as well. The first proof was obtained by the mathematician Adrian Marie Legendre.
Proof of Euler's formula
Euler first formulated the polyhedral formula as a theorem on polyhedra. Today it is often treated in the more general context of connected graphs. For example, as structures consisting of points and line segments connecting them, which are in the same part. Augustin Louis Cauchy was the first person to find this important connection. It served as a proof of Euler's theorem. He, in essence, noticed that the graph of a convex polyhedron (or what is today called such) is topologically homeomorphic to a sphere, has a planar connected graph. What it is? A planar graph is one that has been drawn in the plane in such a way that its edges meet or intersect only at a vertex. This is where the connection between Euler's theorem and graphs was found.
One indication of the importance of the result is that David Epstein was able to collect seventeen different pieces of evidence. There are many ways to justify Euler's polyhedral formula. In a sense, the most obvious proofs are methods that use mathematical induction. The result can be provendrawing it along the number of either edges, faces or vertices of the graph.
Proof of Rademacher and Toeplitz
Particularly attractive is the following proof of Rademacher and Toeplitz, based on the approach of Von Staudt. To justify Euler's theorem, suppose that G is a connected graph embedded in a plane. If it has schemas, it is possible to exclude one edge from each of them in such a way as to preserve the property that it remains connected. There is a one-to-one correspondence between the removed parts for going to the connected graph without closure and those that are not an infinite edge. This research led to the classification of "orientable surfaces" in terms of the so-called Euler characteristic.
Jordan curve. Theorem
The main thesis, which is directly or indirectly used in the proof of the polyhedra formula of the Euler theorem for graphs, depends on the Jordan curve. This idea is related to generalization. It says that any simple closed curve divides the plane into three sets: points on it, inside and outside it. As interest in Euler's polyhedral formula developed in the nineteenth century, many attempts were made to generalize it. This research laid the foundation for the development of algebraic topology and connected it with algebra and number theory.
Moebius group
It was soon discovered that some surfaces could only be "oriented" in a consistent way locally, not globally. The well-known Möbius group serves as an illustration of suchsurfaces. It was discovered somewhat earlier by Johann Listing. This concept includes the notion of the genus of a graph: the least number of descriptors g. It must be added to the surface of the sphere, and it can be embedded on the extended surface in such a way that the edges only meet 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 diagram
The scientist made another discovery, which is still used today. This so-called Euler diagram is a graphic representation of circles, usually used to illustrate relationships between sets or groups. The charts usually include colors that blend in areas where the circles overlap. Sets are represented precisely by circles or ovals, although other figures can also be used for them. An inclusion is represented by an overlap of ellipses called Euler circles.
They represent sets and subsets. The exception is non-overlapping circles. Euler diagrams are closely related to other graphic representation. They are often confused. This graphic representation 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 commonality between sets, but only a possible logical relationship if their labels are not inintersecting circle. Both options were adopted for teaching set theory as part of the new mathematical movement of the 1960s.
Fermat and Euler's theorems
Euler left a noticeable mark in mathematical science. Algebraic number theory was enriched by a theorem named after him. It is also a consequence 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 number and a is an integer not divisible by p, then:
ap-1 - 1 is divisible by p.
Sometimes the 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 only tried to prove his theory. The author hints in another letter that he was inspired by the infinite descent method. But he did not provide any evidence. Later, Eider also turned to the same method. And after him - many other famous scientists, including Lagrange, Gauss and Minkosky.
Features of identities
Fermat's Little Theorem is also called a special case of a theorem from number theory due to Euler. In this theory, the Euler identity function counts positive integers up to a given integer n. They are coprime with respect ton. 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 divisor gcd(n, k) is 1. Notation φ(n) can also be called Euler's phi function. Integers k of this form are sometimes called totative. At the heart 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 particular symbol for it. In a 1784 publication, Euler studied this function in more detail and chose the Greek letter π to represent it. James Sylvester coined the term "total" for this feature. Therefore, it is also referred to as Euler's total. The total φ(n) of a positive integer n greater than 1 is the number of positive integers less than n that are relatively prime up to n.φ(1) is defined as 1. The Euler function or phi(φ) function is a very important number-theoretic a function deeply related to prime numbers and the so-called order of integers.
