Equidistant Points on a Sphere: What is the Series and How to Draw Efficiently?

[ChemistryInclined]

Hello all,

apologies if this is obvious, but searching the internet did not provide me with a satisfactory answer.

I am trying to design a sphere with 40+ points on it, which are all equidistant.
To rephrase, what is the number series of points on spheres that are equidistant to each other?
To begin with, the series is 2 (one point at the top, one at the bottom), then 4 (equidistant pyramid), 8 (cube), then ... what follows?
What is the closest number to 40 (or whatever) where all are equidistant, and what are the angles? Illustrations welcome. No buckminster fullerenes please or whatever (since they don't fulfill this criterium), what I desire here is that all points on the sphere have an equal distance to the next.

Any budding or professional mathematicians who could help me? I imagine Riemann or the likes have worked this out... it'd be awesome to see the mathematics behind the deduction of this problem.

Regardless, I am after the real thing. How can one draw this most efficiently on a physical sphere? Preferably on the inside of a hollow sphere (otherwise I'll have to stick needles through the points drawn on the outside of the sphere)?

Any input is appreciated. Thanks.

 

[d_leet]

I can't really answer your question, but forming a cube with 8 points does not make them all equidistant from one another the ones on diagnols from one another are further apart than the ones that arent on a diagnol line from them if that makes any sense.

 

[ChemistryInclined]

No, the issue is that all *nearest* points are equidistant to each other. If it had to be equidistant to ALL points, the criterium could be only fullfilled up to the pyramid!

Wow, I stumbled upon a new word: Platonic Solids!

Sadly the descriptions I find are only up to the icosahedron... so by far not enough for my plans.

And I am still dumbfounded as to how to draw points of even a pyramid on a real physical sphere!

 

[CRGreathouse]

Wow, I stumbled upon a new word: Platonic Solids!

Sadly the descriptions I find are only up to the icosahedron... so by far not enough for my plans.

There are only five Platonic solids: the tetrahedron (triangle-base pyramid), the cube, the octohedron, the dedecahedron, and the icosahedron.