Chambers inside 6 spheres around a point

[Teljkon]

dfdgdg

Edited December 19, 2021 by Teljkon

[Sheather]

I think you've left out some vital information. Could you show the image, and perhaps explain exactly what you mean by this chambers thing?

[Teljkon]

sdfs

Edited December 19, 2021 by Teljkon

[Alice]

I still don't understand what you are asking?

If the side of the cube has a lenth of 2D, then if the spheres are allowed to overlap, you can fit an infinite number. If they are not allowed to overlap, the you can only fit two. I realise this is not what you are asking though...

EDIT: OK so now I think you are asking:

"If I have 6 spheres of equal radius, and the spheres are spaced evenly such that the surface of each sphere intersects the same point in space, how many separate volumes will I create as defined by the surfaces of the 6 original spheres?"

Am I close?

Edited December 8, 2011 by Alice

[Alice]

In which case I would guess 26.

4 spheres along x and y overlap to form 8 chambers. Bring in the 5th sphere along z, which will overlap with at least some of each of the 8 chambers previously formed to form 8 new chambers. Plus the "leftover" part of sphere 5 (i.e. that which is not close to the point, the biggest bit). That's 17. Bring in sphere 6 from the other end of z. That adds another 8, plus the bit leftover. That's 26.

[Sheather]

I don't think whos surfaces intersect a point. More like whos volumes contain said point.

I'll do a sketch up and see what I can see.

EDIT: This would be much easier if I had semitransparent spheres that I could push through each other and manipulate shade and light levels with ease. As a pen-and-paper problem it's hard for me to get my head around. I'm sure there is a mathematical solution but i don't know it.

Edited December 9, 2011 by Sheather

[qualia]

do you mean this?

The number of regions into which space can be divided by mutually intersecting spheres is

giving 2, 4, 8, 16, 30, 52, 84, ... (Sloane's A046127) for , 2, ....

http://mathworld.wolfram.com/SpaceDivisionbySpheres.html

[Sheather]

That function should really have a -1 at the end to determine the number of enclosed spaces, as that equation appears to be based off number of sections including surrounding space, so the pattern would be 1,3,7,15,29 etc.

[qualia]

That function should really have a -1 at the end to determine the number of enclosed spaces, as that equation appears to be based off number of sections including surrounding space, so the pattern would be 1,3,7,15,29 etc.

 

yep. the sphere defines a closed set in R^3 so it intersects R^3 into 2 spaces. minus 1 to count out the surrounding space.

[CβL]

6 spheres that are centered around a point

1.

:D

But in all seriousness, I think qualia got it.

[Teljkon]

ssdfsfs

Edited December 19, 2021 by Teljkon

[Xenodimensional]

Does anyone know how many intersecting spheres would fit in a polytope then? :-P

[Sheather]

Is that a fractal? It looks like the compartments approach 0 size as their centre approaches the middle of the construct, but I can see no way to be sure.

[qualia]

nm. i see you were being facetious.

Edited December 13, 2011 by qualia

[Xenodimensional]

Sorry I was indeed acting in a jocular fashion, people started talking about intersecting spheres in euclidean geometries and I'll polytope at ther drop of a hat.

It's a polytope Sheather (possibly a Cayley graph), there is a limit to the self similarity/iterations so it isn't a fractal... but polychoron/polytopes are dope imo.