In my last post I showed a teaser image of the shapes involved in figuring out the three angles that define refraction (see below). However, in the post I said pretty much nothing about what sort of mathematics are involved in coming up with these angles.
You might be surprised that the very basic mathematics required to figure the theory of everything out is old. Not just hundreds of years old, but thousands of years! The way to solve these angles is to represent the non-refracted dots as on a plane and knowing the distances between them, just increasing the angle connecting the three dots by θ.
While in a non-refracted state a vector drawn from the middle point of first two dots to the third dot had a length of √3r, where r is the radius of the dot, the length of the vector changes when there is refraction.
Below is an image depicting this, but also other parts to be solved as well.
So, the length of the new connecting vector can be determined using the law of cosines, which to some extent has been known for about 2300 years, or from 300 BC.
Also, when the vector connecting the second and third dot is shifted by θ, a vector can be drawn connecting the old and the new dots. The vector is comprised of two identical components, each with a length of sin(θ/2)2r, the length of the whole vector thus being 4sin(θ/2)r.
This means that we have a triangle with the length of all sides known. Thus, the size of the angle φ, or the angle of dot displacement can be figured using the law of sines. At least according to Wikipedia, has been known since 7th century. So, not as long as the law of cosines, but still well over a millennium.
However, there is a tiny caveat. The third angle γ, or the angle of displacement from the original plane means that the length 2r for the shifted vector isn’t exactly right. I think the right length is 2r cos γ, which should be almost (but not exactly) the same as 2r, as γ should be much closer to zero than either θ or φ.
But for now, we can live for a while with the assumption that the effect of γ can be ignored. In this case we get φ as the equation:
That is, if I haven’t made a mistake anywhere.
The curious thing to me is that from pure geometry, you begin with basic numbers of one, two and a single ‘tricky number’ of square root of three and you soon find yourself with numbers 2, 4, 8, 12 and 15.
In my next post, I’m going to look at the how γ can be determined. And if I have any luck in figuring out how these angles can help me come up with the vectors that define refraction, I’ll be a happy man.
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