"Semi-regular polyhedra" - Pyramid. Regular polyhedra are also called Platonic solids. Snub dodecahedron. Tetrahedron. Icosahedron. Cube Correct. Rhombicosidodecahedron. Continue to next question. Let's remember. Tutorial. Control buttons. You gave the wrong answer. Snub cube. Which type of polyhedra does the following formula V=a*b*c belong to:

“Regular polyhedra in life” - History. Kusudama is a paper flower ball. Euclid. A building without corners. Examples. Goals. Johannes Kepler. Landmark of Belarus. Regular polyhedra. Unusual constructions. New wonder of the world. Polyhedra in art. Polyhedra and crystals. Application of regular polyhedra in architecture.

“Types of regular polyhedra” - Mechanical puzzles. Egyptian pyramids. Regular polyhedra and nature. Scientists who contributed to the study of regular polyhedra. Alexandrian lighthouse. Area of ​​the icosahedron. Basic formulas. Pythagoras. Halicarnassus Mausoleum. Polyhedra in nature. Hexahedron. Octahedron. Surface area of ​​a dodecahedron.

“Application of regular polyhedra” - Polyhedra in art. Use in life. Polyhedra in nature. Kepler. The world of regular polyhedra. Group "Historians". Euclid. Polyhedra in mathematics. Archimedes. Euler's theorem. The history of the emergence of regular polyhedra. Conclusion. Polyhedra in architecture. The relationship between the “golden section” and the origin of polyhedra.

“Regular polyhedra in geometry” - In crystallography there is a section called “geometric crystallography”. The rays of the crystal determine the icosahedron-dodecaeric structure of the Earth, Hypothesis of V. Makarov and V. Morozov: Tetrahedron-fire. At the intersections of the ribs there are centers of ancient cultures and civilizations. Polyhedra are all around us.

“Symmetry of regular polyhedra” - Regular dodecahedron. Each vertex of the dodecahedron is the vertex of three regular pentagons. Symmetry in art. The tetrahedron has no center of symmetry, but has 3 axes of symmetry and 6 planes of symmetry. Church of the Intercession of the Virgin Mary on the Nerl. made up of six squares. Therefore, the sum of the plane angles at each vertex is 240°.

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Definition: A convex polyhedron is called
correct if all its faces are
equal regular polygons and in
at each of its vertices the same thing converges
same number of ribs. Correct
There are only five polyhedra: tetrahedron,
hexahedron, octahedron, dodecahedron, icosahedron.

Examples of regular polyhedra:

Examples of regular polyhedra:

Characteristics and formulas:

Elements of symmetry of a regular octahedron:

Elements of symmetry of a regular icosahedron:

Cube symmetry elements:

Elements of symmetry of a regular dodecahedron:

All information taken from:

Completed by a student of group G 2-9 N.Yu. Koblyuk

Head E.V. Morozova

Tula 2010

“Mathematics possesses not only truth, but also the highest beauty - beauty that is sharpened and strict, sublimely pure and striving for true perfection, which is characteristic only of the greatest examples of art.”

Bertrand Russell

The polyhedron is called correct, If:

• It's convex.
• All its faces are equal regular polygons.
• The same number of faces converge at each of its vertices.
• All its dihedral angles are equal.

There are only five regular polyhedra :

• Tetrahedron (tetrahedron)
• Cube (hexagon)
• Octahedron (octahedron)
• Dodecahedron (dodecahedron)
• Icosahedron (twenty-hedron)

Regular polyhedron is a convex polyhedron with the greatest possible symmetry.

Since ancient times, our ideas of beauty have been associated with symmetry. This probably explains people's interest in polyhedra - amazing symbols of symmetry that have attracted the attention of outstanding thinkers.

The history of regular polyhedra goes back to ancient times. Pythagoras and his students studied regular polyhedra. They were amazed by the beauty, perfection, and harmony of these figures. The Pythagoreans considered regular polyhedrons to be divine figures and used them in their philosophical writings.

One of the oldest mentions of regular polyhedra is in Plato’s treatise (427-347 BC) “Timaus”.

Therefore, regular polyhedra are also called Platonic solids. Each of the regular polyhedra, and there are five in total, Plato associated with four “earthly” elements: earth (cube), water (icosahedron), fire (tetrahedron), air (octahedron), as well as with the “unearthly” element - sky (dodecahedron ).

By the time of Plato, the concept of four elements (elements) - the fundamental principles of the material world - had matured in ancient philosophy: fire , air , water And land .

The shape of the cube is the atoms of the earth, because both the earth and the cube are distinguished by immobility and stability.

The shape of the icosahedron is water atoms, because. Water is distinguished by its fluidity, and of all the regular bodies, the icosahedron is the most “rolling”.

The shape of the octahedron is made up of air atoms, because the air moves back and forth, and the octahedron seems to be directed in different directions at the same time.

The tetrahedron shape is fire atoms, because. the tetrahedron is the sharpest, it seems that it is rushing in different directions.

Plato introduces the fifth element - the “fifth essence” - the world ether, the atoms of which are given the shape of a dodecahedron as the closest to a ball.

Platonic solids are called regular homogeneous convex polyhedra, that is, convex polyhedra, all faces and angles of which are equal, and the faces are regular polygons.

Platonic solids are a three-dimensional analogue of flat regular polygons. However, there is an important difference between the two-dimensional and three-dimensional cases: there are infinitely many different regular polygons, but only five different regular polyhedra.

around 429 – 347 BC

a convex polyhedron whose faces are regular

polygons with the same number of sides and each

at the vertex of which the same number of edges converge.

Icosahedron

Tetrahedron

Octahedron

Hexahedron

Dodecahedron

Plato's body

Face geometry

Number

Tetrahedron

Icosahedron

Hexahedron

Dodecahedron

Euler's formula G + B – P = 2

The surface of the tetrahedron consists of four equilateral triangles, meeting three at each vertex.

U regular tetrahedron all faces are equilateral triangles, all dihedral angles at edges and all trihedral angles at vertices are equal.

Properties of the tetrahedron :

• An octahedron can be inscribed in a tetrahedron, moreover, four (out of eight) faces of the octahedron will be combined with four faces of the tetrahedron, all six vertices of the octahedron will be combined with the centers of six edges of the tetrahedron.
• A tetrahedron with edge x consists of one inscribed octahedron (in the center) with edge x/2 and four tetrahedra (at the vertices) with edge x/2.
• A tetrahedron can be inscribed into a cube in two ways, with the four vertices of the tetrahedron aligned with the four vertices of the cube.

All six edges of the tetrahedron will lie on all six faces of the cube and are equal to the diagonal of the square face.

• A tetrahedron can be inscribed in an icosahedron, moreover, the four vertices of the tetrahedron will be combined with the four vertices of the icosahedron.

Regular polyhedron

Regular triangle

Vertex faces

Rib length

Surface area

Elements of symmetry:

The tetrahedron has no center of symmetry,

but has 3 axes of symmetry and 6 planes of symmetry

Radius of the described sphere:

Radius of inscribed sphere:

Surface area:

Tetrahedron volume:

Cube or hexahedron- a regular polyhedron, each face of which is a square. A special case of a parallelepiped and a prism. The cube has six square faces, meeting three at each vertex.

Cube properties :

• You can fit a tetrahedron into a cube in two ways, and the four vertices of the tetrahedron will be aligned with the four vertices of the cube. All six edges of the tetrahedron will lie on all six faces of the cube and are equal to the diagonal of the square face.
• The four sections of the cube are regular hexagons - these sections pass through the center of the cube perpendicular to its four diagonals.
• You can fit an octahedron into a cube, and all six vertices of the octahedron will be aligned with the centers of the six faces of the cube.
• A cube can be inscribed in an octahedron, and all eight vertices of the cube will be located at the centers of the eight faces of the octahedron.
• An icosahedron can be inscribed in a cube, and six mutually parallel edges of the icosahedron will be located respectively on six faces of the cube, the remaining 24 edges inside the cube, all twelve vertices of the icosahedron will lie on six faces of the cube.

Regular polyhedron

Vertex faces

Rib length

Surface area

Elements of symmetry:

The cube has a center of symmetry - the center of the cube, 9 axes

symmetry and 9 planes of symmetry .

Radius of the described sphere:

Radius of inscribed sphere:

Cube surface area:

Cube volume:

S= 6 a 2

V =a 3

Octahedron- one of the five regular polyhedra.

The octahedron has 8 faces (triangular),

12 edges, 6 vertices (4 edges converge at each vertex).

The octahedron has eight triangular faces, meeting four at each vertex. .

Properties of the octahedron :

• An octahedron can be inscribed in a tetrahedron, moreover, four (out of eight) faces of the octahedron will be combined with four faces of the tetrahedron, all six vertices of the octahedron will be combined with the centers of six edges of the tetrahedron.
• An octahedron with edge y consists of 6 octahedra (along the vertices) with edge y:2 and 8 tetrahedra (along faces) with edge y:2
• An octahedron can be inscribed in a cube, and all six vertices of the octahedron will be aligned with the centers of the six faces of the cube.
• A cube can be inscribed in an octahedron, and all eight vertices of the cube will be located at the centers of the eight faces of the octahedron.

Regular polyhedron

triangle

Vertex faces

Dual polyhedron

Elements of symmetry:

The octahedron has a center of symmetry - the center of the octahedron, 9 axes of symmetry and 9 planes of symmetry.

Radius of the described sphere:

Radius of inscribed sphere:

Surface area:

Octahedron volume:

Icosahedron- regular convex polyhedron, twenty-sided polyhedron, one of the Platonic solids. Each of the 20 faces is an equilateral triangle. The number of edges is 30, the number of vertices is 12. The surface of the icosahedron consists of twenty equilateral triangles, meeting five at each vertex.

Properties :

• The icosahedron can be inscribed in a cube, in this case, six mutually parallel edges of the icosahedron will be located respectively on six faces of the cube, the remaining 24 edges inside the cube, all twelve vertices of the icosahedron will lie on six faces of the cube
• A tetrahedron can be inscribed in an icosahedron, moreover, the four vertices of the tetrahedron will be combined with the four vertices of the icosahedron.
• An icosahedron can be inscribed into a dodecahedron; moreover, the vertices of the icosahedron will be aligned with the centers of the dodecahedron’s faces.
• A dodecahedron can be inscribed into an icosahedron; moreover, the vertices of the dodecahedron will be aligned with the centers of the icosahedron’s faces.

Regular polyhedron

Regular triangle

Vertex faces

Dual polyhedron

dodecahedron

Elements of symmetry:

The icosahedron has a center of symmetry - the center of the icosahedron, 15 axes of symmetry and 15 planes of symmetry.

Radius of the described sphere:

Radius of inscribed sphere:

Surface area:

Volume of icosahedron:

Dodecahedron(dodecahedron) - a regular polyhedron, a three-dimensional geometric figure made up of twelve regular pentagons. Each vertex of the dodecahedron is the vertex of three regular pentagons. It has twelve pentagonal faces, converging at the vertices in threes.

Thus, the dodecahedron has 12 faces (pentagonal), 30 edges and 20 vertices (3 edges converge in each. The sum of the plane angles at each of the 20 vertices is 324°.

The dodecahedron is used as a random number generator (along with other dice) in tabletop role-playing games.

Regular polyhedron

Regular pentagon

Vertex faces

Dual polyhedron

icosahedron

Elements of symmetry:

The dodecahedron has a center of symmetry - the center of the dodecahedron, 15 axes of symmetry and 15 planes of symmetry.

Radius of the described sphere:

Radius of inscribed sphere:

Surface area:

Dodecahedron volume:

Regular polyhedra are found in living nature. For example, the skeleton of the single-celled organism Feodaria ( Circjgjnia icosahtdra ) The shape resembles an icosahedron.

What caused this natural geometrization of feodaria? Apparently, because of all the polyhedra with the same number of faces, it is the icosahedron that has the largest volume with the smallest surface area. This property helps the marine organism overcome the pressure of the water column.

Regular polyhedra are the most “profitable” figures. And nature makes extensive use of this. This is confirmed by the shape of some crystals.

Take table salt, for example, which we cannot do without. It is known that it is soluble in water and serves as a conductor of electric current. And table salt crystals ( NaCl ) have the shape of a cube.

Aluminum-potassium quartz is used in the production of aluminum. ( K [ Al ( SO 4 ) 2 ]  12 H 2 O ), the single crystal of which has the shape of a regular octahedron.

The production of sulfuric acid, iron, and special types of cement is not complete without sulfur pyrites ( FeS ). The crystals of this chemical are dodecahedron shaped.

Antimony sodium sulfate is used in various chemical reactions ( Na 5 ( SbO 4 ( SO 4 )) - a substance synthesized by scientists. The crystal of sodium antimony sulfate has the shape of a tetrahedron.

The last regular polyhedron - the icosahedron - conveys the shape of boron crystals (IN). At one time, boron was used to create first-generation semiconductors.

Feodaria

( Circjgjnia icosahtdra )

“There are a shockingly small number of regular polyhedra, but this very modest squad managed to get into the very depths of various sciences.”

L. Carroll

Materials used:

http://www.vschool.ru

http://center.fio.ru

http://gemsnet.ru

http://alzl.narod.ru

http://ru.wikipedia.org

Used