4 ms·
It would be weird, and cool, if dimension n+1 _doesn't_ get less different as n increases.
by zellyn 10y ago
It would be weird, and cool, if dimension n+1 _doesn't_ get less different as n increases.
- akvadrako 10y agoNature is weird and cool. Sometimes features only reveal themselves at dimensions much larger than 4: http://mathworld.wolfram.com/BorsuksConjecture.html http://mathworld.wolfram.com/BorsuksConjecture.html > Borsuk conjectured that it is possible to cut an n-dimensional shape of generalized diameter 1 into n+1 pieces each with diameter smaller than the original. It is true for n=2, 3 and when the boundary is "smooth." However, the minimum number of pieces required has been shown to increase as ∼1.1^(sqrt(n)). Since 1.1^(sqrt(n))>n+1 at n=9162, the conjecture becomes false at high dimensions. > Kahn and Kalai (1993) found a counterexample in dimension 1326, Nilli (1994) a counterexample in dimension 946. Hinrichs and Richter (2003) showed that the conjecture is false for all n>297.