Re: Spheres and images

Posted by rtharbaugh on May 18, 2003 at 15:56:55:

In Reply to: Spheres and images posted by pianowow on May 18, 2003 at 10:24:28:

Thank you Pianowow, I guess I'm getting to be like my gramma. She used to ask me "where are you living, now, Richard?" about every ten minutes. I guess it served me right because I did move around a lot back then, but not every ten minutes.

I signed onto a Sony site which hosts photo albums, but I didn't have any luck getting my little drawing on there either. I'll play with it some more later.

Meanwhile, the stacking spheres problem.

If I build a figure in free space by specifying a sphere of radius r and then place twelve spheres also of radius r around it so that each of the twelve has exactly one point in common with the origin sphere and exactly four points in common with neighboring spheres in the first layer, I have a form which will fit exactly into a sphere of radius 3r. There is a very nice picture of it in mathworld at

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

but you have to scroll down to about the last 1/3 of the text to find it.

Now I notice that the 12 contact points on the origin sphere have this same geometry, which leads me to the following questions.

1. Are there other values of r which display this geometry? ( I think there must be since the reference speaks of argonite crystals which can be grown in this form.) If so, what are they?

2. You can see that the r=1 spheres are close packed into the r=3 sphere. They would be contained perfectly in there with no room to "slop around". You see that the ratio of the radius of the outer sphere to the radius of the inner sphere is 3/1. If we increase the number of spheres which will fit in the container by reducing the radius of the inner spheres, are there other ratios which will also be perfectly packed with no room for any of the small spheres to "slop around"?

Thanks, Pianowow.

RTH

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