Man first discovered regular polyhedrons in natural environments. The tetrahedron, cube and octahedron can be found among crystallised minerals while the dodecahedron and the icosahedron solids are found in the animal world.
Tetrahedron a mineral of copper crystallizes in a tetrahedron shape zink-sulfide. On the other hand a retains an incomplete tetrahedron shape.
For example galenite, gold and silver cristalizes in a cube or octahedron shape.
Solt and galenite cristals can either retain the shape of cube or an incomplete cube. Criolite also cristalizes as an incomplete cube.
The cuprit cristal in octahedron shape. Often tetrahedron and calcine can be twin cristals.
Icosahedron shaped cristals can only be made syntheticaly for instance synthetic borillium cristals.
With a microscope Haeckel in 1887 discovered among sea animals a type of parasite called radiolarias and he found among them certain ones that retain an icosahedron dodecahedron and octahedron shape. This fact is visiable in their latin names: corcogonia icosahedra, circorhegma dodecahedra, circoporus octahedrus.
The story of polyhedron goes back to prehistoric times.Even in the antiquity people were using polyhedrons.
In the British Museum in the Egyption collection there are 2 icosahedron shaped dice which come from the time of Ptolemaios-dinastia. At an archeological dig in Italy next to Padua in Monta Loffa they found a dodecahedron shaped atrusk toy which was made out of talc in B.C 500.
The pitagoreuses in B.C. 500 already knew about cubes, tetrahedrons and dodekahedrons. Theaitetos in B.C. 400 discovered 5 different regular solids. It is likely that the theory and the discovery of the 5 separate regular polyhedra were written by Theaitetos in the 13th, 14th and 15th book called Euklides Parts.
In the antiquity regular solids were often connected to myths.
Platon continued this tradition when he used them in his own theories. This is why regular polyhedras are often called Platon shapes. The cube, tetrahedron, octahedron and icosahedron were often associated with the four elements earth, fire, air and water. The The 5th regular solid, the dodekahedron was supposed to be the shape of the universes lining.
The first written version of the theory of regular shapes was written in Euklides's book, Particles. The XIIIth book explains how we can work out the diametre of a paraphrasing globe which will give us the measurement of the edges of a regular polyhedra. The XVth book explains how we can draw a tetrahedron and an octahedron into a cube, an octahedron into a tetrahedron, a cube into an octahedron, a dodecahedron into an icosahedron and it also explains how we can calcolate surface angles.
The generalization of semiregular solids comes from the antiquity . Pappusz discovered the generalization of regular shapes in B.C 400, whereupon not long afterwards Archimedes (B.C 287-212) studied them. Nowadays the 13 semiregular shapes are called Archimedes's solids. These polyhedral angles are congrudes, however their lateral faces are a regular poligon.
Dürer(1471-1528) and Leonardo da Vinci (1452-1529) approached regular solids through the arts in the Middle Ages. Dürer wrote down how to manufacture the 5 regular solids from their net.
Leonardo da Vinci first of all approached regular solids from the angle of drawing . He made the skeleton of regular solids from sticks. To this skeleton he needed to imagine the faces.
If we look at this kind of model from away outside the centre point one of the faces this way we can see this chosen face to a big polygon from this perspective. This polygon is made up by all the other faces. Nowadays this type of drawing of the shape is called Schlegel-diagram and is used for describing regular polyhedrons with this Schläfli number marked with n, m symbols. N represents the facenumber of the polygons boundaring the polyhedron. M represents the numbers of edges which meet into one vertice.
Kepler (1571-1630) was the first who noticed regular-starsolids (outside of the 5 regular convex solids) He discovered one of the regular compositions, the stella-octangula. The tetrahedrids' twin cristals' shape is like this is any natural environment.
Descartes (1596-1650) already knew about the connection between the convex polyhedrons' faces, vertices and edges. Euler (1707-1783) generalized this connection to regular polyhedrons. Euler's theory was proved by more then one person. For example Legendre (1752-1833), Cauchy (1789-1852) and Staudt (1748-1867).
The theory of polyhedrons are the base of more than one science.. For example Euler's generalized rule to used in topologie. Felix Klein point out the connection of between the theory's of algebra and polyhedrons. He proved that because of the qualitys of icosahedrons and dodecahedrons the general fifthdegree equation hasn't got a root formula.
It is also possible to generalize the concept of regular solids into a higher dimension. This type of solids we call politopes. six regular solids exsist as a 4 dimensional shape. It is interesting, that only 3 regular politops exsist as a 5 dimensional shape. These 3 are a version of generalized the tetrahedron the cube and the octahedron.