Information. Platonic Solids Platonic Solids are the tetrahedron, the cube octahedron, the dodecahedron and the icosahedron. Also known as Platonic bodies, bodies cosmic Pythagorean solids, solid perfect, Plato's polyhedra or, more accurately, convex regular polyhedra. Polyhedra are characterized by convex whose faces are equal regular polygons and whose vertices are attached to the same number of faces. Given these names after the Greek philosopher Plato who is credited if any studied in the first instance. Properties
regularity: As
expressed to define these polyhedra:
• All sides of a Platonic solid are equal regular polygons.
• In all the vertices of a Platonic solid attend the same number of faces and edges.
• All edges of a Platonic solid are the same length.
• All line angles that form the faces of a Platonic solid with each other are equal.
• All its vertices are convex to the icosahedron. Symmetrical properties
The Platonic solids are highly symmetrical:
• All of them are central symmetry about a point in space (center of symmetry) which is equidistant from their faces, vertices and edges.
• They also have axial symmetry on a number of axes of symmetry passing through the center of symmetry above.
• They also have mirror symmetry about a series of planes of symmetry (or principal planes), which divided into two equal parts. Following geometric
above, can be traced all Platonic solid three particular areas, all focused in the center of symmetry of the polyhedron:
• A sphere inscribed tangent to all sides in the middle. • A second area
tangent to all edges in the center.
• A limited area, passing through all vertices of the polyhedron.
projecting the centers of the edges of a Platonic polyhedron circumscribed about the sphere from the center of symmetry of the polyhedron is obtained a regular spherical network, composed of equal-circle arcs, which are regular spherical polygons.
polyhedra whose faces are all congruent regular polygons are called "regular polyhedra" or "Platonic solids." There are only five:
regular tetrahedron (4 vertices, 6 edges, 4 equilateral triangles as faces)
regular hexahedron or cube (8 vertices, 12 edges, 6 squares as faces)
regular octahedron (6 vertices, 12 edges, 8 equilateral triangles as faces)
regular dodecahedron (20 vertices, 30 edges, 12 pentagons as faces)
regular icosahedron (12 vertices, 30 edges, 20 equilateral triangles as faces)
solid in nature and art
Cube , the tetrahedron and octahedron appear naturally in crystal structures, there are also living this way, such a type of protozoa called radiolarians have a cube, octahedron, dodecahedron, icosahedron ... and actually receiving the scientific name incorporates the respective polyhedron of receiving the form . Too many viruses such as herpes or AIDS are shaped icosahedron. And no doubt have appeared in many paintings of different artists. When there was a greater linkage between the solid and the art was probably in the Renaissance. And no doubt have appeared in many paintings of different artists. When there was a greater linkage between the solid and the art was probably in the Renaissance. The artists began to use polyhedra as a tool to develop specific aspects of perspective. This is the case of some artists as Paolo Uccello and Piero della Francesca
Slowly the tables with solid figures were less important, until he was almost forgotten in the world of art. Escher was who, with his incredible imagination and originality rescued the Platonic solids to be incorporated into innovative paintings.
Maurits Cornelis Escher (1898-1972) was a German graphic artist known that mathematics was inspired by many of his works. Also among his works with sequences that become infinitely small, or they had enforced views that show vanishing points, we find pictures related to the Platonic solids. But his most focused on polyhedra, it is certainly between 1948 and 1954, when he draws several pictures totally focused on this issue, where the polyhedra appear as a principal and not as something merely decorative. In double planetoid planetoid tetrahedral, using tetrahedra which create strange planets, in the first two that intersect the centers of the edges and the second only one that gives rise to distorted worlds.
two lithographs in 1950 and 1952, includes the figure of the dodecahedron crashed. But perhaps the most complete work of all of Escher, as far as polyhedra are concerned, is the Stars, 1948 woodcut. The main figure is a big star, composed of 3 octahedral holes together, inside which they hold two chameleons. Apart from this figure, which already is very interesting, we see that all the stars in the background are other polyhedra. There are several tetrahedra, octahedra, icosahedra, bucket, but among them there is a recital of the more significant figures:
1. Octahedron cube intersected with
2. Two holes intersected tetrahedra
3. Dodecahedron
4. Rombicuboctedro
5. Deltoidal Icositetraedro
6. Cuboctahedron
7. Two intersecting cubes
8. Three octahedra together as the main star
9. Two tetrahedra
10. Two hollow cubes together by a vertex
