Geometria

Platonic solids
From Wikipedia, the free encyclopedia
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and angles.
There are precisely five Platonic solids(shown below):
Tetrahedron | Cube
(or hexahedron) | Octahedron | Dodecahedron | Icosahedron |
| | | | |
Polyhedron | Vertices | Edges | Faces | Schläfli symbol | Vertex
configuration |
tetrahedron | | 4 | 6 | 4 | {3, 3} | 3.3.3 |
cube | | 8 | 12 | 6 | {4, 3} | 4.4.4 |
octahedron | | 6 | 12 | 8 | {3, 4} | 3.3.3.3 |
dodecahedron | | 20 | 30 | 12 | {5, 3} | 5.5.5 |icosahedron | | 12 | 30 | 20 | {3, 5} | 3.3.3.3.3 |

The name of each figure is derived from its number of faces: respectively 4, 6, 8, 12, and 20.[1]
The aesthetic beauty and symmetry of the Platonic solids have made them a favorite subject of geometers for thousands of years. They are named for the ancient Greek philosopher Plato who theorized that the classical elements were constructed from theregular solids.

Archimedean solids
From Wikipedia, the free encyclopedia
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices. They are distinct from the Platonic solids, which are composed of only one type of polygon meeting in identical vertices, and from the Johnson solids,whose regular polygonal faces do not meet in identical vertices. The symmetry of the Archimedean solids excludes the members of the dihedral group, the prisms and antiprisms. The Archimedean solids can all be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry. See Convex uniform polyhedron.
Origin of name
The Archimedean solids take theirname from Archimedes, who discussed them in a now-lost work. During the Renaissance, artists and mathematicians valued pure forms and rediscovered all of these forms. This search was completed around 1620 by Johannes Kepler, who defined prisms, antiprisms, and the non-convex solids known as the Kepler-Poinsot polyhedra.
Classification
There are 13 Archimedean solids (15 if the mirror images of twoenantiomorphs, see below, are counted separately). Here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a vertex configuration of (4,6,8) means that a square, hexagon, and octagon meet at a vertex (with the order taken to be clockwise around the vertex).

The number of vertices is 720° divided by the vertex angle defect.
Name(Vertex configuration)   | Transparent   | Solid   | Net   | Faces   | Faces
(By type)   | Edges   | Vertices   | Symmetry group   |
truncated tetrahedron
(3.6.6) | | | | 8 | 4 triangles
4 hexagons | 18 | 12 | Td |
cuboctahedron
(3.4.3.4) | | | | 14 | 8 triangles
6 squares | 24 | 12 | Oh |
truncated cube
or truncated hexahedron
(3.8.8) | | | | 14 | 8 triangles
6 octagons | 36 | 24| Oh |
truncated octahedron
(4.6.6) | | | | 14 | 6 squares
8 hexagons | 36 | 24 | Oh |
rhombicuboctahedron
or small rhombicuboctahedron
(3.4.4.4 ) | | | | 26 | 8 triangles
18 squares | 48 | 24 | Oh |
truncated cuboctahedron
or great rhombicuboctahedron
(4.6.8) | | | | 26 | 12 squares
8 hexagons
6 octagons | 72 | 48 | Oh |
snub cube
or snub hexahedron
or snubcuboctahedron
(2 chiral forms)
(3.3.3.3.4) | | | | 38 | 32 triangles
6 squares | 60 | 24 | O |
icosidodecahedron
(3.5.3.5) | | | | 32 | 20 triangles
12 pentagons | 60 | 30 | Ih |
truncated dodecahedron
(3.10.10) | | | | 32 | 20 triangles
12 decagons | 90 | 60 | Ih |
truncated icosahedron
or buckyball
or football/soccer ball
(5.6.6 ) | | | | 32 | 12 pentagons
20 hexagons | 90 |...