Regular polyhedra: elements, symmetry and area

Geometry is beautiful in that, unlike algebra, where it is not always clear what and why you think, it gives the object visibility. This amazing world of different bodies is decorated with regular polyhedra.

General information on regular polyhedra

According to many, regular polyhedra, or as they are called Platonic bodies, have unique properties. Several scientific hypotheses are associated with these objects. When you begin to study these geometric bodies, you understand that you know practically nothing about such a concept as regular polyhedra. The presentation of these objects in the school is not always interesting, so many do not even remember what they are called. In the memory of most people, only the cube remains. None of the bodies in geometry have such perfection as regular polyhedra. All names of these geometric bodies originated from Ancient Greece. They mean the number of faces: tetrahedron - tetrahedral, hexahedral - hexahedral, octahedron - octahedral, dodecahedron - twelve-sided, icosahedron - twenty-sided. All these geometric bodies occupied the most important place in Plato's conception of the universe. Four of them represented elements or essences: tetrahedron - fire, icosahedron - water, cube - earth, octahedron - air. The dodecahedron embodied all that exists. He was considered the main, 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 side of any of the polygons is simultaneously the side of only one other polygon along the same side;
• From each of the polygons, you can go to the others passing along adjacent polygons.

The polygons that make up the polyhedron are its faces, and their sides are the edges. The vertices of polygons are the vertices of polygons. If the concept of a polygon is understood to be flat closed polygonal lines, then one comes to the same definition of a polyhedron. In the case when this term means a part of the plane that is bounded by broken lines, it is necessary to understand a surface consisting of polygonal pieces. A convex polyhedron is a body lying on one side of the plane adjacent to its face.

Another definition of a polyhedron and its elements

A polyhedron is a surface consisting of polygons that bounds a geometric body. They are:

• Nonconvex;
• Convex (right and wrong).

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 connection between the number of edges, vertices, and faces topologically equivalent to a sphere. Adding the number of vertices and faces (B + D) for different regular polyhedra and comparing them with the number of edges, one can establish one regularity: the sum of the number of faces and vertices is equal to the number of edges (P) increased by 2. One can derive a simple formula:

• B + F = P + 2.

This formula is true for all convex polyhedra.

Basic Definitions

The concept of a regular polyhedron can not be described by one sentence. It is more polysemantic and voluminous. For a body to be recognized as such, it is necessary that it conform to a number of definitions. Thus, a geometric body will be a regular polyhedron when the following conditions are fulfilled:

• 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:

1. Cube (hexahedron) - it has a flat angle at the apex of 90 °. It has a 3-faceted angle. The sum of the planar angles at the apex is 270 °.
2. The tetrahedron is a flat angle at the apex of 60 °. It has a 3-faceted angle. The sum of the planar angles at the vertex is 180 °.
3. The octahedron is a flat angle at the apex of 60 °. It has a 4-corner angle. The sum of the flat angles at the top is 240 °.
4. The dodecahedron is a flat angle at the apex of 108 °. It has a 3-faceted angle. The sum of the planar angles at the apex is 324 °.
5. Icosahedron - it has a flat angle at the top - 60 °. It has a 5-faceted angle. The sum of the planar angles at the vertex 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 the regular pyramid, at the base of which is the regular polygon, by the number of faces, and its height is the radius of the inscribed sphere (r):

• V = 1: 3rS.

The volumes of regular polyhedra

Like any other geometric body, regular polyhedra have different volumes. Below are the formulas by which they can be calculated:

• Tetrahedron: α х 3√2: 12;
• Octahedron: α х 3√2: 3;
• Icosahedron; Α х 3;
• Hexahedron (cube): 5 х α х 3 х (3 + √5): 12;
• Dodecahedron: α х 3 (15 + 7√5): 4.

Elements of regular polyhedra

The hexahedron and the octahedron are dual geometric bodies. In other words, they can be obtained from each other in the event that the center of gravity of the face of one is taken as the vertex of the other, and vice versa. Also, the icosahedron and the dodecahedron are dual. Only a tetrahedron is dual to itself. By the method of Euclid, one can obtain a dodecahedron from a hexahedron by constructing "roofs" on the faces of the cube. The vertices of the tetrahedron are any 4 vertices of the cube not adjacent in pairs along the edge. From the hexahedron (cube) one can obtain other regular polyhedra. Despite the fact that regular polygons have an infinite number of regular polyhedra there are only 5.

Radii of regular polygons

With each of these geometric bodies, 3 concentric spheres are connected:

• Described, passing through its vertices;
• It is inscribed, touching each of its faces in the center of it;
• Middle, touching all the ribs 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 the sphere of the inscribed is calculated by the formula:

• R = a: 2 × ctg π / p × tg θ: 2,

Where θ is the two-sided angle that lies between adjacent faces.

The radius of the median sphere can be calculated by the following formula:

• Ρ = a cos π / p: 2 sin π / h,

Where h is a value of 4.6, 6.10 or 10. The ratio of the described and inscribed radii is symmetric with respect to p and q. It is calculated by the formula:

• R / r = tg π / p × tg π / q.

Symmetry of polyhedra

The symmetry of regular polyhedra causes the main interest in these geometric bodies. It is understood as the movement of a body in space, which leaves the same number of vertices, faces and edges. In other words, under the action of the symmetry transformation, the edge, vertex, or face either retains its original position, or moves to the original position of another edge, another vertex or face.

Elements of symmetry of regular polyhedra are inherent in all kinds of such geometric bodies. Here we are talking about the identity transformation, which leaves any of the points in the original position. Thus, when rotating a polygonal prism, several symmetries can be obtained. Any of them can be represented as a product of reflections. Symmetry, which is the product of an even number of reflections, is called a line. If it is the product of an odd number of reflections, then it is called the inverse. Thus, all the rotations around a straight line represent a direct symmetry. Any reflection of the polyhedron is the inverse symmetry.

To better understand the symmetry elements of regular polyhedra, we can take an example of a tetrahedron. Any straight line that passes through one of the vertices and the center of this geometric figure will pass through the center of the face opposite to it. Each of the twists at 120 and 240 ° around the line belongs to the plural number of symmetries of the tetrahedron. Since he has 4 vertices and faces, there are only eight direct symmetries. Any of the straight lines passing through the middle of the rib and the center of this body passes through the middle of its opposite edge. Any rotation of 180 °, called a half-turn, around the line is symmetry. Since a tetrahedron has three pairs of edges, then there are three more direct symmetries. Proceeding from the foregoing, it can be concluded that the total number of direct symmetries, including the identical transformation, will reach twelve. There are no other direct symmetries for the tetrahedron, but it has 12 inverse symmetries. Consequently, the tetrahedron is characterized by only 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 icosahedron are the bodies closest to the sphere. The icosahedron has the largest number of faces, the largest dihedral angle, and the closest it can be to the nested sphere. The dodecahedron has the smallest angular defect, the largest solid angle at the apex. He can fill his described sphere as much as possible.

Deploying polyhedra

The correct polyhedrons of the sweep, 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 correspond to two conditions:

• From each polygon it is possible to go through polygons having the identified side;
• Identified parties must have the same length.

It is the collection of polygons that satisfy these conditions that is called the unfolding of a polyhedron. Each of these bodies has several of them. For example, the cube has 11 pieces.