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An n-ball Between n-balls

There is a geometric thought experiment that is often used to demonstrate the counterintuitive shape of high-dimensional phenomena. This article is an interactive visual journey into the construct in the thought experiment, and the mathematics behind it.

As seen in the isolate plane perspective, the left balls maintain their size and only shift slightly away from the center. This is absurd; considering how enclosed it is in the two and three dimensional versions, and even if it’s mathematically clear that r>2r > 2r>2 when D>9D > 9D>9, the fact is hard to accept. πD2(D−1)DΓ(D2+1)\frac{\pi^{\frac{D}{2}}\bigl(\sqrt{D}-1\bigr)^D}{\Gamma\left(\frac{D}{2}+1\right)}Γ(2D​+1)π2D​(D​−1)D​and if we normalize the size of the green box to 1 by dividing it’s sidelength by 4, the volume of the red ball becomes:

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