11 ms·

For the orthoplices, the kissing sphere in the middle pokes through the facets in dimension 12. I don't know when that happens with the simplices. I assume the

• gjm11 5y ago
  I think that isn't the case. The simplices are pointier, which means that the "corner spheres" don't go so far into their corners. If I've done my calculations right, putting n+1 n-spheres in the corners of an n-simplex with unit sides so that they're tangent to one another gives them a radius of 1 / (sqrt(2n(n+1)) + 2), and then if you put a sphere in the middle tangent to all those it has radius [sqrt(2n/(n+1)) - 1] times this. So for very large n, the "corner spheres" have radius of order 1/n, and the "centre sphere" has radius about sqrt(2)-1 times the radius of the "corner spheres", and both of these -> 0 as n -> oo. (But! "If I've done my calculations right" is an important condition there. I make a lot of mistakes. If you actually care about the answer then you should check it.)