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How does the enclosed sphere's radius changes with the number of dimensions, if the enclosing spheres are the following:

2D: 3 mutually touching 2-spheres (circles)

3D: 4 mutually touching 3-spheres (or spheres)

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This variation of the problem doesn't rely on an artificial construct of a hypercube, I wonder if this yields a similarly unintuitive result.

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If my calculations are correct, then for this variation the enclosed n-sphere's radius converges to sqrt(2)-1 from below, and remains enclosed in the bounding hyper-tetrahedron.

Buf, you may be right but I just cannot visualize it. It took me quite a while to do for the cube, imagine a tetrahedron. But you might be right.

Very interesting, I've considered doing something similar with other regular polyhedra, like the n-simplex (the one you analyzed) and n-orthoplex.

What was the side length in your calculation? did you find an equation for the size of the center n-ball?

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