Why are there only 5 types of platonic solids?

Question

Please tell me the difference between concave polyhedra and convex polyhedra.
PS-platonic solids are kinds of convex polyhedra only.

Answer 1

A convex shape is that which a line drawn between any two points on its surface never contains any points that's not inside the convex shape. So, for example, a cube would be a convex polyhedron, and we can create a non-convex polyhedron out of it by removing a pyramidal volume from each of its 6 sides.

A platonic solid is defined as regular convex polyhedron with one kind of a regular polygon for all of its faces. It is possible to have a non-convex regular polyhedron that also has just one kind of a regular polygon for all of its faces (see link, a stellation). Compare that with either the tetrahedron or the isocahedron, which are platonic solids.

As for why there are only 5 platonic solids, that's an interesting exercise, with numerous proofs of it, some of which can be found in the next wiki link on it. The easiest to read probably would be the one given by Euclid in the "Elements". What's even more interesting is that there are 6 "platonic solids" in 4 dimensions, but only 3 "platonic solids" for dimensions higher than 4. Hmmm?

Answer 2

Your question would not quite attempt to comprehend something yet a selection. ok, there are 5 plates. possibly one hundred million years in the past there have been 7, possibly in a 10 million years from now there will be 3. the considerable element to realize is you reside to tell the story an lively planet which has a semisolid center with solid plates in consistent action which help life. The consistent action shapes the faces of the outdoors - immediately - in specific cases inflicting cataclysmic events, like earthquakes, tsunamis, devastating volcanic eruptions... - that is astonishing in case you stand returned and inspect if from a distance.

Answer 3

Check http://mathworld.wolfram.com/PlatonicSolid.html