P^2+Σ2P

This is taxicab geometry. You move from X to Y. Each square is 1. Red:1+2+2+1+2+2=10 Green:5+5=10 They are the same line. This is called Manhattan distance. n is the dimension. Then, this is Euclidean norm. This is 1*1 squares. ∴ There are 25 suares, so K=25. √50≒7.07. 50=5^2+5^2. This is the green.

Higher dimensions are chaotic . However, you close to Zero, although you keep expanding. This is almost 1-1=0. k go to ∞, which is called Euclidean norm. Then you pile squares . They are almost circles, and cubes are also balls. They are closed. However, higher dimensions are opened, so they are chaotic.