Geometry is beautiful because, unlike algebra, where it is not always clear what and why you think, it gives visibility to the object. This wonderful world of various bodies is decorated with regular polyhedra.
General information about regular polyhedra
According to many, regular polyhedra, or as they are also called Platonic solids, have unique properties. Several scientific hypotheses are associated with these objects. When you start to study these geometric bodies, you understand that you know practically nothing about such a concept as regular polyhedra. The presentation of these objects at school is not always interesting, so many do not even remember what they are called. Most people remember only the cube. None of the bodies in geometry are as perfect as regular polyhedra. All the names of these geometric bodies originated from ancient Greece. They mean the number of faces: tetrahedron - four-sided, hexahedron - six-sided, octahedron - octahedral, dodecahedron - twelve-sided, icosahedron - twenty-sided. All these geometric bodiesoccupied an important place in Plato's concept of the universe. Four of them personified the elements or entities: the tetrahedron - fire, the icosahedron - water, the cube - earth, the octahedron - air. The dodecahedron embodied everything that exists. It was considered the main one, because it was a symbol of the universe.
Generalization of the concept of a polyhedron
A polyhedron is a collection of a finite number of polygons such that:
- each of the sides of any of the polygons is at the same time a side of only one other polygon on the same side;
- from each of the polygons you can get to the others by passing along the polygons adjacent to it.
The polygons that make up a polyhedron are its faces, and their sides are edges. The vertices of the polyhedra are the vertices of the polygons. If the concept of a polygon is understood as flat closed broken lines, then one arrives at one definition of a polyhedron. In the case when this concept means a part of the plane that is limited by broken lines, then a surface consisting of polygonal pieces should be understood. A convex polyhedron is a body lying on one side of a plane adjacent to its face.
Another definition of a polyhedron and its elements
A polyhedron is a surface consisting of polygons that limits a geometric body. They are:
- non-convex;
- convex (correct and incorrect).
A regular polyhedron is a convex polyhedron with maximum symmetry. Elements of regular polyhedra:
- tetrahedron: 6 edges, 4 faces, 5 vertices;
- hexahedron (cube): 12, 6, 8;
- dodecahedron: 30, 12, 20;
- octahedron: 12, 8, 6;
- icosahedron: 30, 20, 12.
Euler's theorem
It establishes a relationship between the number of edges, vertices and faces that are topologically equivalent to a sphere. By adding the number of vertices and faces (B + D) of various regular polyhedra and comparing them with the number of edges, one pattern can be established: the sum of the number of faces and vertices equals the number of edges (P) increased by 2. You can derive a simple formula:
B + D=R + 2
This formula is true for all convex polyhedra.
Basic definitions
The concept of a regular polyhedron cannot be described in one sentence. It is more meaningful and voluminous. For a body to be recognized as such, it must meet a number of definitions. So, a geometric body will be a regular polyhedron if the following conditions are met:
- it is convex;
- the same number of edges converge at each of its vertices;
- all its faces are regular polygons, equal to each other;
- all its dihedral angles are equal.
Properties of regular polyhedra
There are 5 different types of regular polyhedra:
- Cube (hexahedron) - it has a flat angle at the top is 90°. It has a 3-sided angle. The sum of the flat angles at the top is 270°.
- Tetrahedron - flat angle at the top - 60°. It has a 3-sided angle. The sum of flat angles at the top is 180°.
- Octahedron - flat vertex angle - 60°. It has a 4-sided corner. The sum of flat angles at the top is 240°.
- Dodecahedron - flat angle at vertex 108°. It has a 3-sided angle. The sum of flat angles at the top is 324°.
- Icosahedron - it has a flat angle at the top - 60°. It has a 5-sided angle. The sum of flat angles at the top is 300°.
Area of regular polyhedra
The surface area of these geometric bodies (S) is calculated as the area of a regular polygon multiplied by the number of its faces (G):
S=(a: 2) x 2G ctg π/p
The volume of a regular polyhedron
This value is calculated by multiplying the volume of a regular pyramid, at the base of which there is a regular polygon, by the number of faces, and its height is the radius of the inscribed sphere (r):
V=1: 3rS
Volumes of regular polyhedra
Like any other geometric body, regular polyhedra have different volumes. Below are the formulas by which you can calculate them:
- tetrahedron: α x 3√2: 12;
- octahedron: α x 3√2: 3;
- icosahedron; α x 3;
- hexahedron (cube): 5 x α x 3 x (3 + √5): 12;
- dodecahedron: α x 3 (15 + 7√5): 4.
Elements of regular polyhedra
Hexahedron and octahedron are dual geometric bodies. In other words, they can be obtained from each other if the center of gravity of the face of one is taken as the vertex of the other, and vice versa. The icosahedron and dodecahedron are also dual. Only the tetrahedron is dual to itself. According to the Euclid method, you can get a dodecahedron from a hexahedron by building "roofs" on the faces of a cube. The vertices of a tetrahedron will be any 4 vertices of a cube that are not adjacent in pairs along an edge. From the hexahedron (cube) you can get other regular polyhedra. Despite the fact that there are countless regular polygons, there are only 5 regular polyhedra.
Radius of regular polygons
There are 3 concentric spheres associated with each of these geometric bodies:
- described, passing through its peaks;
- inscribed, touching each of its faces in its center;
- median, touching all edges in the middle.
The radius of the sphere described is calculated by the following formula:
R=a: 2 x tg π/g x tg θ: 2
The radius of an inscribed sphere is calculated by the formula:
R=a: 2 x ctg π/p x tg θ: 2,
where θ is the dihedral angle between adjacent faces.
The radius of the median sphere can be calculated using the following formula:
ρ=a cos π/p: 2 sin π/h,
where h value=4, 6, 6, 10 or 10. The ratio of circumscribed and inscribed radii is symmetrical with respect to p and q. Itcalculated by the formula:
R/r=tg π/p x tg π/q
Symmetry of polyhedra
The symmetry of regular polyhedra causes the main interest in these geometric bodies. It is understood as such a movement of the body in space, which leaves the same number of vertices, faces and edges. In other words, under the action of a symmetry transformation, an edge, vertex, face either retains its original position or moves to the original position of another edge, vertex, or face.
Elements of symmetry of regular polyhedra are characteristic of all types of such geometric bodies. Here we are talking about an identical transformation that leaves any of the points in its original position. So, when you rotate a polygonal prism, you can get several symmetries. Any of them can be represented as a product of reflections. A symmetry that is the product of an even number of reflections is called a straight line. If it is the product of an odd number of reflections, then it is called inverse. Thus, all rotations about a line are direct symmetry. Any reflection of a polyhedron is an inverse symmetry.
To better understand the symmetry elements of regular polyhedra, we can take the example of a tetrahedron. Any straight line that will pass through one of the vertices and the center of this geometric figure will also pass through the center of the face opposite to it. Each of the 120° and 240° turns around the line is plural.symmetry of the tetrahedron. Since it has 4 vertices and 4 faces, there are only eight direct symmetries. Any of the lines passing through the middle of the edge and the center of this body passes through the middle of its opposite edge. Any 180° rotation, called a half turn, around a straight line is a symmetry. Since the tetrahedron has three pairs of edges, there are three more direct symmetries. Based on the foregoing, we can conclude that the total number of direct symmetries, including the identical transformation, will reach twelve. The tetrahedron has no other direct symmetries, but it does have 12 inverse symmetries. Therefore, the tetrahedron is characterized by a total of 24 symmetries. For clarity, you can build a model of a regular tetrahedron from cardboard and make sure that this geometric body really has only 24 symmetries.
The dodecahedron and the icosahedron are closest to the sphere of the body. The icosahedron has the largest number of faces, the largest dihedral angle, and can be most tightly pressed against an inscribed sphere. The dodecahedron has the smallest angular defect, the largest solid angle at the vertex. He can fill his described sphere to the maximum.
Sweeps of polyhedra
Regular unwrapped polyhedra, which we all glued together in childhood, have many concepts. If there is a collection of polygons, each side of which is identified with only one side of the polyhedron, then the identification of the sides must meet two conditions:
- from each polygon, you can go over polygons that haveidentified side;
- identified sides must have the same length.
It is the set of polygons that satisfy these conditions that is called the development of the polyhedron. Each of these bodies has several of them. So, for example, a cube has 11 of them.
