Two dimensions are easy to image, but spheres are three dimensions and you pile them infinitely.
This is n-dimensional Pythagorean theorem, the distance between (a,b,c,d) and (w,x,y,z) in R^n.
There are always spaces between spheres. You need to put small ones. Fraction of boundary is less than 0.99. You can find 99% spaces for spheres anytime.
∴
0.99^n
0.99^300≒0.049
0.99^500≒0.00657
0.99^800≒0.00032
Therefore, spheres are almost zero in higher dimensions.
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