(Calculus of) Variations on a Vesper Theme

Calculus Level 5

$P_{1}, P_{2}, \ldots, P_{n}$ are points on the surface of the unit sphere. Define $D_{n}$ as the set of all possible distances between any two of these points.

Find

$\sum_{n=2}^{6} \min_{d \in D_{n, \sigma}} d$

where $\sigma$ is some distribution of $P_{1}, P_{2}, \ldots, P_{n}$ such that the mean distance between points is maximised.

The answer is 8.19347.

1 solution

Jake Lai
Apr 8, 2015

Let's consider VSEPR from chemistry. In this case, each point on the sphere is an electron domain.

VSEPR tells us two electron domains, minimising repulsion, will form a linear electron geometry. In the same way, three will form a trigonal planar geometry, four a tetrahedral geometry, five trigonal bipyramidal, and six octahedral.

There exists only one possible distance for $n = 2, 3, 4$ : $2, \sqrt{3}, \sqrt{8/3}$ respectively. However, for $n = 5$ , there exists 3 different distances. The minimum of these is $\sqrt{2}$ , as is the minimum of the 2 different distances for $n = 2$ .

As such, the sum of minimum distances is then

$2+\sqrt{3}+\sqrt{8/3}+\sqrt{2}+\sqrt{2} \approx \boxed{8.193}$

Thanks, chemistry. Harasho .

This is essentially equivalent to symmetry considerations. And that's what I used.

Funny thing is, I misread Vesper as Vespa, as in the scooter, and was wondering what a scooter has to do with minimal distances on a sphere :D

However, I would recommend using less math and more English because I have grave doubts as to the percentage of people who will understand the 'wording'(without words :D) of this question. Anyways, nice question. :)

Shashwat Shukla - 6 years, 2 months ago

Thanks for the tip! I actually spotted a mistake in the question as well while rewording the question; the constraint on $\sigma$ was "wrong" since the sum should have been taken over all (not necessarily distinct) distances rather than the set $D_{n, \sigma}$ (which contains $\leq \binom{n}{2}$ elements).

Jake Lai - 6 years, 2 months ago

Yes, I noticed that, but I thought that I wasn't understanding it right.

Shashwat Shukla - 6 years, 2 months ago

(Calculus of) Variations on a Vesper Theme

The answer is 8.19347.

1 solution

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