There are only 5 platonic solids, but take a look at these images:
How are these things designed? How are they shaped? It looks to me like those hexagons are all the same size and shape, and evenly distributed to approximate a sphere. Same thing with the triangles in the second picture.
So how is it possible? Are some of the hexagons actually slightly smaller or irregular? Can someone show me in the blueprint exactly where the properties of a platonic solid are NOT met?
Some Hint: In order to design a polyhedron with faces evenly distributed to approximate a sphere, either a solid sphere or a spherical shell with a certain radius is taken such that sum of the spherical areas of all faces on the sphere is equal to the total surface area of sphere.
then the faces of required dimensions can be generated by facing operations with the help of (automated) machines/robots etc. Thus all the vertices of the faces will lie approximately on a sphere.
Picture 1: The hexagonal faces are evenly distributed to approximate a sphere but they all need not be regular in shape i.e. some may regular & some irregular in shape as well. In this case, three hexagonal faces meet at each vertex. There is only one convex regular polyhedron called truncated icosahedron having 20 congruent regular hexagons & 12 regular pentagons all of equal edge length.
Picture 2: The polyhedron shown is a non convex polyhedron in which has 6 stellar triangular faces are generated on each of hexagonal faces as in picture 1. Thus pyramid like stellar faces have regular triangular base & lateral faces as isosceles triangles. faces of one hexagonal groove need not be identical to that of other hexagonal groove. As far as the regular triangular shapes are concerned there is only one regular convex polyhedron called Icosahedron (platonic solid) having 20 congruent equilateral triangular faces.