Polyhedra not only occupy a prominent place in geometry, but also occur in the daily life of every person. Not to mention artificially created household items in the form of various polygons, starting with a matchbox and ending with architectural elements, crystals in the form of a cube (s alt), prism (crystal), pyramid (scheelite), octahedron (diamond), etc. e.
The concept of a polyhedron, types of polyhedra in geometry
Geometry as a science contains a section of stereometry that studies the characteristics and properties of three-dimensional figures. Geometric bodies, the sides of which in three-dimensional space are formed by limited planes (faces), are called "polyhedra". Types of polyhedra include more than a dozen representatives, differing in the number and shape of faces.
However, all polyhedra have common properties:
- They all have 3 essential components: face(surface of a polygon), vertex (corners formed at the junction of faces), edge (side of a figure or a segment formed at the junction of two faces).
- Each polygon edge connects two and only two faces that are adjacent to each other.
- Convexity means that the body is completely located only on one side of the plane on which one of the faces lies. The rule applies to all faces of the polyhedron. Such geometric figures in stereometry are called convex polyhedra. The exception is star-shaped polyhedra, which are derivatives of regular polyhedral geometric solids.
Polyhedra can be conditionally divided into:
- Types of convex polyhedra, consisting of the following classes: ordinary or classical (prism, pyramid, parallelepiped), regular (also called Platonic solids), semi-regular (second name - Archimedean solids).
- Non-convex polyhedra (star-shaped).
Prism and its properties
Stereometry as a section of geometry studies the properties of three-dimensional figures, types of polyhedra (a prism is one of them). A prism is a geometric body that necessarily has two completely identical faces (they are also called bases) lying in parallel planes, and the n-th number of side faces in the form of parallelograms. In turn, the prism also has several varieties, including such types of polyhedra as:
- Parallelepiped - formed if the base is a parallelogram -polygon with 2 pairs of equal opposite angles and 2 pairs of congruent opposite sides.
- A straight prism has edges perpendicular to the base.
- Tilted prism is characterized by the presence of non-right angles (other than 90) between the faces and the base.
- A regular prism is characterized by bases in the form of a regular polygon with equal side faces.
Basic properties of a prism:
- Congruent bases.
- All edges of the prism are equal and parallel to each other.
- All side faces are parallelogram-shaped.
Pyramid
Pyramid is a geometric body, which consists of one base and n-th number of triangular faces, connected at one point - the top. It should be noted that if the side faces of the pyramid are necessarily represented by triangles, then the base can be either a triangular polygon, or a quadrangle, or a pentagon, and so on ad infinitum. In this case, the name of the pyramid will correspond to the polygon at the base. For example, if a triangle lies at the base of a pyramid, it is a triangular pyramid, a quadrilateral is a quadrangular one, etc.
Pyramids are cone-like polyhedra. The types of polyhedra of this group, in addition to those listed above, also include the following representatives:
- A regular pyramid has a regular polygon at its base, and its height is projected to the centera circle inscribed in the base or circumscribed around it.
- A rectangular pyramid is formed when one of the side edges intersects with the base at a right angle. In this case, it is also fair to call this edge the height of the pyramid.
Pyramid properties:
- If all side edges of the pyramid are congruent (of the same height), then they all intersect with the base at the same angle, and around the base you can draw a circle with a center coinciding with the projection of the top of the pyramid.
- If the base of the pyramid is a regular polygon, then all side edges are congruent, and the faces are isosceles triangles.
Regular polyhedron: types and properties of polyhedra
In stereometry, a special place is occupied by geometric bodies with absolutely equal faces, at the vertices of which the same number of edges are connected. These solids are called Platonic solids, or regular polyhedra. Types of polyhedra with such properties have only five shapes:
- Tetrahedron.
- Hexahedron.
- Octahedron.
- Dodecahedron.
- Icosahedron.
Regular polyhedra owe their name to the ancient Greek philosopher Plato, who described these geometric bodies in his writings and connected them with the natural elements: earth, water, fire, air. The fifth figure was awarded the similarity with the structure of the universe. In his opinion, the atoms of natural elements in shape resemble the types of regular polyhedra. Due to its most exciting property -symmetry, these geometric bodies were of great interest not only to ancient mathematicians and philosophers, but also to architects, artists and sculptors of all times. The presence of only 5 types of polyhedra with absolute symmetry was considered a fundamental discovery, they were even awarded a connection with the divine principle.
Hexahedron and its properties
In the form of a hexagon, Plato's successors assumed a similarity with the structure of the atoms of the earth. Of course, at present, this hypothesis has been completely refuted, which, however, does not prevent the figures from attracting the minds of famous figures with their aesthetics in modern times.
In geometry, a hexahedron, also known as a cube, is considered a special case of a parallelepiped, which, in turn, is a kind of prism. Accordingly, the properties of the cube are related to the properties of the prism, with the only difference being that all the faces and corners of the cube are equal to each other. The following properties follow from this:
- All edges of the cube are congruent and lie in parallel planes with respect to each other.
- All faces are congruent squares (there are 6 in total in a cube), any of which can be taken as a base.
- All interface angles are 90.
- An equal number of edges emanates from each vertex, namely 3.
- The cube has 9 axes of symmetry, which all intersect at the intersection point of the diagonals of the hexahedron, called the center of symmetry.
Tetrahedron
A tetrahedron is a tetrahedron with equal faces in the form of triangles, each of the vertices of whichis the junction point of three faces.
Properties of regular tetrahedron:
- All faces of a tetrahedron are equilateral triangles, which means that all faces of a tetrahedron are congruent.
- Since the base is represented by a regular geometric figure, that is, it has equal sides, the faces of the tetrahedron converge at the same angle, that is, all angles are equal.
- The sum of flat angles at each of the vertices is 180, since all angles are equal, then any angle of a regular tetrahedron is 60.
- Each of the vertices is projected to the intersection point of the heights of the opposite (orthocenter) face.
The octahedron and its properties
Describing the types of regular polyhedra, one cannot fail to note such an object as an octahedron, which can be visually represented as two quadrangular regular pyramids glued together by bases.
Properties of the octahedron:
- The very name of a geometric body suggests the number of its faces. The octahedron consists of 8 congruent equilateral triangles, in each of the vertices of which an equal number of faces converge, namely 4.
- Since all faces of an octahedron are equal, its interface angles are also equal, each of which equals 60, and the sum of the plane angles of any of the vertices is thus 240.
Dodecahedron
If we imagine that all the faces of a geometric body are a regular pentagon, then we get a dodecahedron -a figure of 12 polygons.
Properties of the dodecahedron:
- Three faces intersect at each vertex.
- All faces are equal and have the same edge length and equal area.
- The dodecahedron has 15 axes and planes of symmetry, and any of them passes through the vertex of the face and the middle of the opposite edge.
Icosahedron
No less interesting than the dodecahedron, the figure of the icosahedron is a three-dimensional geometric body with 20 equal faces. Among the properties of a regular twenty-hedron, the following can be noted:
- All faces of the icosahedron are isosceles triangles.
- Five faces converge at each vertex of the polyhedron, and the sum of the adjacent angles of the vertex is 300.
- The icosahedron, like the dodecahedron, has 15 axes and planes of symmetry passing through the midpoints of opposite faces.
Semi-regular polygons
Besides Platonic solids, the group of convex polyhedra also includes Archimedean solids, which are truncated regular polyhedra. The types of polyhedra of this group have the following properties:
- Geometric bodies have pairwise equal faces of several types, for example, a truncated tetrahedron has 8 faces, like a regular tetrahedron, but in the case of an Archimedean solid, 4 faces will be triangular and 4 will be hexagonal.
- All angles of one vertex are congruent.
Star polyhedra
Representatives of non-volumetric types of geometric bodies are stellated polyhedra whose faces intersect each other. They can be formed by merging two regular 3D solids or by extending their faces.
Thus, such stellated polyhedra are known as: stellated forms of the octahedron, dodecahedron, icosahedron, cuboctahedron, icosododecahedron.
