place the centers congruent unit hyperspheres at the d+1 vertices of a d-dimensional regular simplex with edge length 2. this forms a nice dense cluster of hyperspheres that all touch each other. previously, in 3D.
circumscribe a larger hypersphere around the cluster. can more unit hyperspheres be added inside? can they be added preserving symmetry?
start with a single unit hypersphere and place the maximum number of unit hyperspheres around it so that are all tangent to the first and don't overlap. this is the kissing number problem. for which dimensions does the known lower bound have a nice symmetric arrangement?
circumscribe a larger hypersphere around the cluster of kissing spheres. I'm pretty sure it always has radius 3. what proportion of the volume of circumscribing hypersphere is empty?
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