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We can put 4 unit circles inside a square of length 4 units, such that all the circles will touch the boundary and also adjacent circles, and there will be a hollow in between the 4 circles. So we can also consider a 3D cube of length 4 units, and put 8 spheres, where each one will touch the wall, and still be able to put a sphere, at the center of the cube. Now, if we can calculate the volume of the mid-spheres (for 2D -> Circle, for 3D -> Sphere, and from then on, Hypershperes), and we find that at the 9th dimension, the 9D-mid-hypershpere just touches the hyper-cube-walls and in the 10th dimension, the mid-sphere, becomes larger than even the 10 Dimensional cube itself. Now my question, how can these be imagined physically, is there any way out?

Thanks in advance.

prikarsartam
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    This seems more suitable for Math SE. – G. Smith Apr 13 '20 at 19:22
  • No, this is also a good problem in Statistical Mechanics. These hypersphere concepts are very useful in that topic. – prikarsartam Apr 14 '20 at 07:17
  • Can you explain how nine and ten dimensions apply to statistical mechanics? – G. Smith Apr 14 '20 at 16:08
  • No, not directly that comes, you might know better. I;m learning. I just recently learned the concepts of microstates and it's connection with the hypersphere constructed in the n-dimensional phase space. And then when i started learning about the hyperspheres and it's properties, i came to know about this problem. It might not have direct co-relation with Stat Mech. Don't misunderstand please ! – prikarsartam Apr 15 '20 at 08:10

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