Volumes of n-balls: what is so special about n=5?

  1. The volume of an n-dimensional ball of radius 1 is given by the classical formula
    V_n=\frac{\pi^{n/2}}{\Gamma(n/2+1)}.
    For small values of n, we have
    V_1=2\qquad
    V_2\approx 3.14
    V_3\approx 4.18
    V_4\approx 4.93
    V_5\approx 5.26
    V_6\approx 5.16
    V_7\approx 4.72
    It is not difficult to prove that V_n assumes its maximal value when n=5.

    Question. Is there any non-analytic (i.e. geometric, probabilistic, combinatorial…) demonstration of this fact? What is so special about n=5?

  2. I also have a similar question concerning the n-dimensional volume S_n (“surface area”) of a unit n-sphere. Why is the maximum of S_n attained at n=7 from a geometric point of view?

note: the question has also been asked on MathOverflow for those curious to other answers.

Answer

If you compare the volume of the sphere to that of its enclosing hyper-cube you will find that this ratio continually diminishes. The enclosing hyper-cube is 2 units in length per side if R=1. Then we have:

V_1/2=1\qquad
V_2/4\approx 0.785
V_3/8\approx 0.5225
V_4/16\approx 0.308
V_5/32\approx 0.164

The reason for this behavior is how we build hyper-spheres from low dimension to high dimensions. Think for example extending S_1 to S_2. We begin with a segment extending from -1 to +1 on the x axis. We build a 2 sphere by sweeping this sphere out along the y axis using the scaling factor \sqrt{1-y^2}. Compare this to the process of sweeping out the respective cube where the scale factor is 1. So now we only occupy approximately 3/4 of the enclosing cube (i.e. square for n=2). Likewise for n=3, we sweep the circle along the z axis using the scaling factor, loosing even more volume compared to the cylinder if we had not scaled the circle as it was swept. So as we extend S_{n-1} to get S_n we start with the diminished volume we have and loose even more as we sweep out into the n^{th} dimension.

It would be easier to explain with figures, however hopefully you can work through how this works for lower dimensions and extend to higher ones.

Attribution
Source : Link , Question Author : Andrey Rekalo , Answer Author : Tpofofn

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