Geometry is beautiful because, unlike algebra, which is not always clear as what you think, gives a visual object.This wonderful world of various bodies adorn the regular polyhedra.
Understanding regular polyhedra
According to many, regular polyhedrons, or as they are called the Platonic solids have unique properties.With these objects connected several scientific hypotheses.When you begin to study the geometric data of the body, you realize that almost do not know anything about such a concept as the regular polyhedra.The presentation of these objects in the school is not always interesting, so many do not even remember what they were called.In the memory of most people it is just a cube.None of the bodies in geometry do not possess such perfection as regular polyhedra.All the names of these geometric bodies originated from ancient Greece.They represent the number of faces: the tetrahedron - four-sided, hexahedron - Allen, octahedron - octahedron, dodecahedron - dodecahedral, icosahedron - icosahedral.All these geometric body occupies an important place in Plato's conception of the universe.Four of them embody elements or entities: the tetrahedron - fire icosahedron - water cube - earth, octahedron - air.Dodecahedron embodied all things.He was considered the main, because he was a symbol of the universe.
generalization of the concept of a polyhedron
polyhedron is a set of finite number of polygons such that:
- each side of any of the polygons is also the party of only one other polygon on the same side;
- from each of the polygons can be reached by going to the other adjacent polygons with him.
polygons constituting the polyhedron are its faces and their side - ribs.Vertices of are vertices of the polygons.If you understand by the concept of a polygon flat closed polylines, then come to one definition of a polyhedron.In the case where this notion means that part of the plane that is bounded by broken lines, it is necessary to understand the surface, consisting of polygonal pieces.Convex polyhedron is called the body lying on one side of the plane, adjacent to its faces.
Another definition of a polyhedron and its elements
polyhedron is a surface consisting of polygons, which limits the geometric body.They are:
- non-convex;
- convex (right and wrong).
regular polyhedron - is convex polyhedron with maximal 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 are topologically equivalent to a sphere.Adding the number of vertices and faces (B + D) in different regular polyhedra and comparing them with the number of ribs, you can set one rule: the sum of the number of faces and vertices equals the number of edges (F), increased by 2. You can display a simple formula:
- B + F = P + 2.
This formula holds for all convex polyhedra.
Basic definitions
concept of a regular polyhedron is impossible to describe in one sentence.It is a multi-value and volume.A body to be recognized as such, it is necessary that it meets a number of definitions.For example, the geometric body will be a regular polyhedron in the performance of these conditions:
- is convex;
- the same number of ribs converge in each of its vertices;
- all facets of it - regular polygons, equal to each other;
- all the 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 vertex is 90 °.It has a 3-sided corner.The sum of the planar angles at the tip of 270 °.
- Tetrahedron - flat angle at the top - 60 °.It has a 3-sided corner.The sum of the planar angles at the apex - 180 °.
- Octahedron - flat angle at the top - 60 °.It has a 4-sided corner.The sum of the planar angles at the apex - 240 °.
- dodecahedron - a flat angle at the top of 108 °.It has a 3-sided corner.The sum of the planar angles at the apex - 324 °.
- icosahedron - his flat angle at the top - 60 °.It has 5-sided corner.The sum of the planar angles at the tip of 300 °.
Area
regular polyhedra The surface area of geometric solids (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.
volume of a regular polyhedron
This value is calculated by multiplying the volume of a regular pyramid whose base is a regular polygon, the number of faces, and its height is the radius of the inscribed sphere (r):
- V = 1: 3rS.
volume of regular polyhedra
Like any other geometric solid, regular polyhedra have different volumes.Below are the formulas by which they can be calculated:
- tetrahedron: α x 3√2: 12;
- octahedron: α x 3√2: 3;
- icosahedron;α x 3;
- hexahedron (cube): α x 5 x 3 x (3 + √5): 12;
- dodecahedron: α x 3 (15 + 7√5): 4.
Elements regular polyhedra
hexahedron and octahedron are dual geometric bodies.In other words, they can get out of each other in the event that the centroid of one is taken as the top of the other, and vice versa.Also, it is the dual icosahedron and dodecahedron.Myself Only tetrahedron is dual.By way of Euclid can be obtained from a dodecahedron hexahedron by constructing "roofs" on the faces of the cube.The vertices of the tetrahedron are any 4 vertices of the cube, not adjacent pairs of rib.From hexahedron (cube) can be obtained, and other regular polyhedrons.Despite the fact that regular polygons have countless, regular polyhedra, there are only 5.
radii of regular polygons
With each of these geometric bodies linked 3 concentric spheres:
- described passing through its apex;
- inscribed with regard to each of its faces in the middle of it;
- median concerning all the edges in the middle.
radius of the sphere is calculated as described by the following formula:
- R = a: 2 x tg π / g x tg θ: 2.
radius of the inscribed sphere is calculated as follows:
- R = a: 2 x ctgπ / p x tg θ: 2,
where θ - dihedral angle, which is located between the adjacent faces.
median radius of the sphere can be calculated by the following formula:
- ρ = a cos π / p: 2 sin π / h,
value where h = 4.6, 6.10, or 10. The ratio of the radii as described and inscribedsymmetrically with respect to p and q.It is calculated by the formula:
- R / r = tg π / p x tg π / q.
Symmetry Symmetry polyhedra
regular polyhedra is of primary interest to these geometric bodies.It is understood as a movement of a body in space, which leaves the same number of vertices and edges.In other words, under the influence of symmetry transformations edge, vertex, face or retains its original position, or moves to the starting position of another rib, the other vertices or faces.
regular polyhedra symmetry elements common to all types of geometric solids.Here it is conducted on the identity transformation, which leaves any of the points in the original position.Thus, by rotating the polygonal prism can receive multiple symmetries.Any of these may be represented as the product of reflections.The symmetry that is the product of an even number of reflections, called direct.If it is a product of an odd number of reflections, it is called back.Thus, all the turns around the line as a straight symmetry.Any reflection of the polyhedron - a reverse symmetry.
To better understand the elements of symmetry of the regular polyhedra, you can take the example of a tetrahedron.Any line that will pass through one of the vertices and the center of this geometric figure, will pass through the center and the edge opposite her.Each of the corners 120 and 240 ° around the line belongs to the plural tetrahedral symmetry.Because he has 4 vertices and faces, we get a total of eight direct symmetries.Any of the lines passing through the middle of the edges and the center of the body, passes through the middle of its opposite edges.Every turn of 180 °, called a half-turn around the line is a symmetry.Since the tetrahedron, there are three pairs of ribs, you get three lines of symmetry.Based on the foregoing, it can be concluded that the total number of direct symmetry, and including the identity transformation, will be up to twelve.Other direct symmetry tetrahedron does not exist, but it has 12 inverse symmetry.Consequently, the tetrahedron is characterized by a total of 24 symmetries.For clarity, you can build a model of a regular tetrahedron made of cardboard and make sure it is the geometric body really has only 24 symmetry.
dodecahedron and icosahedron - closest to the area of the body.The icosahedron has the largest number of faces, the largest dihedral angle and tighter all can cling to the inscribed sphere.The dodecahedron has the lowest angular defect, the largest solid angle at the top.It can be described as much as possible to fill in the scope.
Sweep polyhedra
Regular polyhedra scan, which we all bonded in childhood have a lot of concepts.If there is a set of polygons, each side of which is identified with only one side of the polyhedron, the identification of the parties must comply with two conditions:
- of each polygon, you can go to a polygon having sides identified;
- identifiable parties must have the same length.
It is a set of polygons that meet these conditions and called scan polyhedron.Each of these bodies has several of them.For example, a cube has 11 pieces of them.
