What is a polyhedron? The modern definition of a polyhedron
The concept of a ‘polyhedron’ belongs to the primal concepts of modern mathematical thought. Unfortunately, a large percentage of students do not pay attention to this topic, citing the obviously incorrect opinion that this theme is not worth a detailed study. However, the study of modern topology is absolutely unthinkable without a full ownership of the basic mathematical concepts and axioms. In fact, the main purpose of this article is to convince the audience that this topic deserves more than a standard
Thus, what is a polyhedron from the conventional standpoint? In terms of elementary geometry, a polyhedron is a three-dimensional object, which is composed of flat polygonal faces. Neighboring faces meet along a line segment, which is called an edge. A vertex of a polyhedron is formed by neighboring edges. From the point of view of modern topology, a polyhedron may be defined as a particular example (3-dimensional example to be precise) of a polytope. On the other hand, a great number of various mathematical disciplines use other definitions of the term ‘polyhedron’, some of which may be purely abstract or particularly geometric. Nevertheless, the definition, worked out in the framework of elementary geometry, gives us the necessary basic understanding of the nature of this figure. Therefore, one should not be confused by a presence of different definitions of the term ‘polyhedron’.
What is a polyhedron: the main particular characteristics of a polyhedron which are of greatest interest to students
In fact, the main characteristic of a polyhedron is the famous Euler characteristic, which elegantly associates the numbers of edges, vertices and faces (E, V, and F) of a polyhedron. Moreover, this discovery marks a new era in mathematics. It is no exaggeration at all to say that invention of the Euler characteristic gave rise to topology, in its modern sense. The Euler characteristic demonstrates that all types of polyhedrons always satisfy a simple equation: V − E + F = 2. Thereby, according to this elegant arithmetic relationship, the amount of vertices and faces is equal to the number of edges plus two. In addition, it should be noted that this equation is not correct for some complicated shapes. In some cases, the Euler characteristic relates to the number of handles, toroidal holes and/or cross-caps in the surface. Hoverer, the basic principle of this characteristic is correct for all variations of polyhedrons. Let us prove this statement by examining the less complicated shapes that does not require the use of oversophisticated mathematical apparatus. For example, the cube (which obviously belongs to the class of polyhedrons) is probably the best-known polyhedron. Simple observation shows us that the cube features six faces: four squares on the sides, a square on the top and a square on the bottom. The abutments of these squares create the edges and the corners of the cube. The four bottom corners and the four top corners are the eight vertices of the cube. In addition, in order to complete our examination we should count the number of edges. Four edges on the top along with four edges on the bottom and four edges on the sides of our cube in sum give us twelve edges. Thereby for the cube, F = 6, V = 8 and E = 12. A simple calculation shows us that the Euler characteristic is correct for this specific polyhedron. In fact, it is easy to verify this statement: 8−12+6 = 2. Alternatively, one can perform a series of analogical calculations for other polyhedrons according to the one’s choice. Naturally, all these calculations lead us to the only possible conclusion: the Euler characteristic correctly describes the inner pattern of all polyhedrons.
In fact, the Euler characteristic is rightly considered one of the most impressive and elegant solutions in the history of mathematics because it allows us to understand the specific inner patterns of abstract mathematical models. Furthermore, its importance for modern science just cannot be overestimated. Hence, it is a small wonder that virtually all literary sources, which are dedicated to the history of topology, contain references to Leonhard Euler as a forerunner of topology. Naturally, Euler’s contribution to mathematic is not limited to this discovery. However,
It is a common place to notice that nowadays there exist an extraordinary great number of classifications of various types of polyhedrons. In fact, even accomplishing a simple research about the contemporary classification of polyhedrons, such as
- All faces of the polyhedron has to be congruent (identical) to each other.
- The polyhedron has to be convex. A polyhedron is convex if its surface (inclusive all its edges, faces, and vertices) does not traverse itself and the line segment, which connects any two points of the polyhedron, is contained in the interior or surface.
- Every face of the polyhedron is a regular polygon. Thereby, one can define an irregular polyhedron from a regular one, just by detecting one specific face, which does not satisfy all conditions that apply to regular polygons (the equal length of all its sides and the equal measure of all interties angles).
- Every vertex of the polyhedron is surrounded by the equal number of faces.
Obviously, the polyhedron, which does not satisfy even one of these demands, belongs to the class of irregular polyhedrons. In order to eschew unnecessary details and/or redundantly complicated nuances about semi-regular, quasi-regular, edge-transitive (isotoxal), vertex-transitive (isogonal), face-transitive (isohedral) or noble polyhedrons, we will not provide all information about these highly specialized types on these pages. Nevertheless, a standard
What is a polyhedron: a concise historical approach to the invention of the term
Elementary geometry (and, in truth, many modern mathematical disciplines even so oversophisticated as topology) can be traced back to the works of the Ancient Greeks. In fact, Ancient Greeks have formulated the first theorem about a regular polyhedron. It postulates that there exist five regular polyhedra: the cube, the tetrahedron, the octahedron, the icosahedron and the dodecahedron. A proof of this theorem was found in the last Euclid’s work – Elements. Unfortunately, for all mathematicians and historians, Elements is the only major mathematical work created by the Greeks, which has survived to our time. Euclid demonstrated that there exist at least five Platonic solids and that the cube, the tetrahedron, the icosahedron, the octahedron and the dodecahedron are actually regular. In addition, he proved that their number is limited to five.
For Ancient Greeks this postulate was not only hypothetical. Furthermore, their ideas about principles and patterns that unite all geometrical figures have been reflected in their philosophical views. Thus, Plato considered that the regular polyhedra were the primal elements of all matter. Plato even incorporated regular polyhedra into his atomic theory; nowadays, these polyhedra are called the Platonic solids. His beliefs were adopted, evolved and significantly expanded by Aristotle (384–322 BCE).
However, it will be incorrect to say that the concept of a ‘polyhedron’ was invented by Greek philosophers. Firstly, a solid number of archeological researches claims that not only highly developed civilizations, such as Egyptian or Chinese civilizations but also ancient commonages were familiar with basic geometrical principles. Secondly, a polyhedron is not something that cannot be found in nature. For example, there exist many natural compounds, which exist in the form of various polyhedrons. Pyrite, which is widely known as fool’s gold, can constitute crystals with twelve pentagonal faces, sodium chloride form cubical crystals and chrome alum can be found in the form of an octahedron. Naturally, we cannot affirm that ancient herders and gatherers wondered: what is a polyhedron. Nevertheless, we definitely can claim that the shape of a primitive polyhedron was familiar to our ancestries.
The further history of a ‘polyhedron’ goes back to Euler and his brilliant theorems. However, the history of the ‘polyhedron’ concept is not finished. Even nowadays, there can be discovered a great amount of mathematical researches that are dedicated to this theme. Of course, modern mathematicians are interested in much more complex problems than a simple question: ‘what is a polyhedron?’ Modern topology is highly specialized and enormously complicated sphere of science, which is closely interconnected not with neighboring mathematical disciplines, such as differential, algebraic or geometric topologies, but also with chemistry, physics (especially nuclear and quantum physics) and biology. Therefore, no one will argue about doubtless relevance of this theme.
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