The Inner Sphere Paradox: It Pokes Outside the Box in 10D

Pack unit spheres in the corners of an n-dimensional box. Drop one sphere in the middle — by dimension 10, it pokes through the walls.

Radii vs Dimension n
r_inner = √n − 1 (growing)
Corner sphere radius = 1
Box half-width = 2
2D Cross-Section (n = 2)
Gray: corner spheres (radius=1)
Red: inner sphere (radius = r_inner)
Black square: the box (side 4)
n =2
r_inner =0.414
Corner r =1.000
Box half-width =2.000
Inner sphere inside box
2
√2 − 1 ≈ 0.414