Dror Bar-Natan: Classes: 2014-15: Math 1100 Algebra I:

A riddle from MathOverflow, http://mathoverflow.net/questions/5372/dimension-leaps/5483#5483: Put (2^n) unit balls centered at the points (±1, ±1,..., ±1) in (R^n); they are all bound in the cube C=(([-2,2])^n). Let B be the maximal ball centered at 0 and not intersecting with any of the previous balls. As n→∞, what's the limit of Vol(B)/Vol(C)?

Rasterize /@ {
Graphics[{
Blue, Circle[List@@#, 1]& /@ Flatten[Outer[Center, {-1,1}, {-1,1}]],
Red, Circle[{0,0}, Sqrt[2]-1],
Black, Opacity[0.2],Rectangle[{-2,-2},{2,2}]
}, ImageSize ->250],
Graphics3D[{
Blue, Opacity[0.3],Sphere[List@@#, 1]& /@ Flatten[Outer[Center, {-1,1}, {-1,1}, {-1,1}]],
Red, Opacity[1], Sphere[{0,0,0}, Sqrt[3]-1],
Black, Opacity[0.1],Cuboid[{-2,-2,-2},{2,2,2}]
}, Boxed -> False, ImageSize -> 250]
}

WolframAlpha["Volume of n-sphere", "Result"]

vr[n_] := ((2(π^(n/2))((((Sqrt[n])-1))^n))/(n Gamma[(n/2)]))/(4^n); {Table[vr[n], {n,2,10}], vr[100]} // N

An image from http://drorbn.net/AcademicPensieve/2011-12/index.html?im=MobiusStairwell.png:



Is this stairway shell a 2D manifold (with boundary)? If so, which one is it? If not, why not?

(The classification of surfaces is, for example, at http://en.wikipedia.org/wiki/Surface)

Pensieve Header: Riddiles for November 24, 2014: a ball squezed by 2^n balls, and the stairway shell.