# Thread: Equal Distances Within a Sphere

1. ## Equal Distances Within a Sphere

Hi;
My objective is to discover the exact 3D points for creating Platonic solids. Since these are the only geometries that have equal length sides and equal angles between them, I thought it might be easiest to find a formula that enables me to calculate farthest points within a sphere where the lengths and angles are equal. Does such a formula exist?
TIA,
Jonathan

2. ## Re: Equal Distances Within a Sphere

any specific platonic solid in mind or do you to be able to calculate the points for all of them?
the wiki page has a lot of useful formulas and lengths on them.

the octahedron is the simplest, it's 3d points (assuming sphere radius r and origin at (0, 0, 0) and no rotation) are: (0, r, 0), (-r, 0, 0), (0, 0, -r), (r, 0, 0), (0, 0, r), (0, -r, 0).

using the numbers at wiki, calculating the 3d points for the tetrahedron:
top point obviously at (0, r, 0), bottom y-value at -(1 - 1/sqrt(2))r.
the bottom 3 points's x- and z-values: assume it's rotated so that one of them is at x = r, z = 0.
then the other 2's coordinates are: (-(1 - 1/sqrt(2))r, -(1 - 1/sqrt(2))r, r) and (-(1 - 1/sqrt(2))r, -(1 - 1/sqrt(2))r, -r).
so the 4 points are: (0, r, 0), (r, -(1 - 1/sqrt(2))r, 0), (-(1 - 1/sqrt(2))r, -(1 - 1/sqrt(2))r, r) and (-(1 - 1/sqrt(2))r, -(1 - 1/sqrt(2))r, -r).
not sure if i could make this into an easy formula... to calculate the points for the other platonic solids, try using the numbers in wiki.

sorry it's so messy but i haven't figured out how to make nice latex formulas yet

3. ## Re: Equal Distances Within a Sphere

How did you figure out those values? Is there a page where I can go to figure out the same?

My objective is to write a formula that enables me to calculate all the nested values of Platonic solids within Metranon's Cube and to grow the cube to any size (any number of repeats of the solids) ad infinitum, so I'm going to need formulas for all the solids.

TIA,
Jonathan
PS Yeah, pity there aren't any nice, pretty LaTeX formulas lol.

4. ## Re: Equal Distances Within a Sphere

well the values for the octahedron were simple, since it got 6 points and all are the extremes following each axis.
the values for the tetrahedron i got from the wiki page on platonic solids and some trig calculations. here

hmm, i'm not sure what you mean. metatron's cube is 2d? and you get all the solids by simply drawing straight lines from each circle's center?

5. ## Re: Equal Distances Within a Sphere

> well the values for the octahedron were simple, since it got 6 points and all are the
> extremes following each axis.

Clearly, that's very simple.

> the values for the tetrahedron i got from the wiki page on platonic solids and some
> trig calculations. here

Ok, well that's not so simple. The only piece of your equation I could comprehend after studying that page was the 1/sqrt(2) part which is associated with the midradius. Perhaps if you could explain how you arrived at that formula I could calculate the other formulas myself.

> hmm, i'm not sure what you mean. metatron's cube is 2d?

No, Metranon's Cube is 3D: Metatron's Cube - Wikipedia, the free encyclopedia

> and you get all the solids by simply drawing straight lines from each circle's center?

Almost. One extends the planes and where they intersect they create the edges and verticies of the next solid Like magic. That's what's so fun about mathematics, no? Lots of that stuff to be found
Jonathan