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3D Geometry Problem

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Imagine four rays emerge from a common point in 3D space. All the four rays are making equal angle with one another. Find this angle.

The main motive of the problem is to prove that least repulsion occurs among bond pairs when angle made by two C-H bonds in CH4 molecule 109.5°.

(Can a formula for n such rays be derived making equal angle with each other in space?)

Thanks for trying :)

Note by Lokesh Sharma
3 years, 9 months ago

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The exact angle is \(arccos(\frac{-1}{3})\) (for proof see http://math.stackexchange.com/questions/56847/angle-between-lines-joining-tetrahedron-center-to-vertices).

I don't think it's always possible to have equal angles for a given value of \(n\), you may wish to read this page: http://en.wikipedia.org/wiki/Thomson_problem Clifford Wilmot · 3 years, 9 months ago

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@Clifford Wilmot Thanks for replying. Its a nice and clean proof. I appreciate it. Although, I was wondering if there could be more general proof without making a tetrahedral because we don't know for sure that tetrahedral angle is what we are seeking.

I don't fully understand what do you mean by "I don't think it's always possible to have equal angles for a given value of n" as the link you provided gives angle for any n from 2 to 470.

For n = 4, angle is 109.471

For n = 5, angle is 90

and so on... Lokesh Sharma · 3 years, 9 months ago

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@Lokesh Sharma To see that the shape must be a tetrahedron, we simply note that there are 4 particles on the outside, the 4 points must form the vertices of a (possibly irregular) tetrahedron.

To prove that this tetrahedron must be regular seems much harder, from searching the internet, it seems that these papers prove the result:

http://arxiv.org/pdf/math/0607446v2.pdf

http://www.degruyter.com/view/j/dma.1993.3.issue-1/dma.1993.3.1.75/dma.1993.3.1.75.xml

(I don't understand the maths in them, so I can't really judge whether they answer your question, or if there is a simpler way to prove that the tetrahedron must be regular. Unless there is a simpler way to prove it, I guess for now we'll have to rely on intuition to tell us that the tetrahedron must be regular.)

In the link I gave, the angle \(90^\circ\) for \(n=5\) is the smallest angle (it says "\(\theta_1\) is the smallest angle between any two points."), so there is a pair of points with an angle of \(90^\circ\), however there are also pair of points with different angles between them. For example, in phosphorus pentafluoride (http://en.wikipedia.org/wiki/PF5), some of the angles are \(90^\circ\), others are \(120^\circ\). Clifford Wilmot · 3 years, 9 months ago

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