Sphere Packing

You need to fill spheres in packing infinitely. This is more complicated than packing oranges.


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.


https://prime2011.tumblr.com/post/179661361213


∴

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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