Geometry/Polygon -> Polyhedron
I would like to know the equation/function that would link some or any of the properties of a regular polygon (nš of vertex and edges of a square or pentagon, for example) to the properties of their "associated" polyhedron (a cube or dodecahedron's vertex, edges and faces).
Thanks for your time!
ANSWER: Hi Eric,
Regular polygons don't have an "associated" polyhedron, per se. With exception perhaps of the equilateral triangle and the square, there isn't really a way to make a correspondence between a regular polygon and a polyhedron. Suppose we take a pentagon. What is its polyhedron? A pentagonal prism? A pentagonal pyramid? Some other solid with pentagonal faces? There are rules particular to prisms and pyramids, but there isn't any associated polyhedron for a regular polygon in the general case.
Or perhaps I'm misunderstanding your question. If you think I am, feel free to ask a follow-up.
---------- FOLLOW-UP ----------
QUESTION: By "associated" polyhedron I meant the regular polyhedron whose faces are said polygon, for example pentagon -> dodecahedron. By knowing the nš of edges or vertices of the pentagon, in this case, would you be able to know the number of faces and/or edges and/or vertices of the dodecahedron? Is there such a function?
There are very few regular polyhedra whose faces are the regular polygon. What you are asking is closely related to the Platonic solids, of which the dodecahedron is one. The others are the tetrahedron, the cube, and the icosahedron. For the Platonic solids, there are some relations between vertices, edges, and faces (such as Euler's formula), but they are a step removed from the functions you seek. However, you should definitely read up on the Platonic solids as they may very well be of interest to you.
I would like to point out that the tetrahedron and the icosahedron are both regular polyhedra whose faces are all equilateral triangles. A correspondence between sides of a polygon and a polyhedron whose faces are that polygon is very limited. The pentagon -> dodecahedron is the exception rather than the rule.
Thanks for asking,