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[Research] nD Cosine Rule Reflections

On the beginning of this academic year, my math teacher gave us a piece of homework to practice mathematical writing: to write a short proof of the law of cosines using overleaf. While doing some elementary research if there is a simple or elegant proof, I found that there was a 3-dimensional equivalent of the law of cosines, or an extension on a 3 dimensional surface with constant curvature. I had also learnt linear algebra through MIT OpenCourseWare and the Berkeley summer camp in the previous year. I had also recently begun to find enjoyment in writing math papers, thanks to KSEF! With these, I, newly exposed to the field of linear algebra, became intrigued whether the law of cosines could be extended to any finite dimension greater than 1.

To answer this question, I searched up if there are previous works that have already been on this area. I first discovered some relevant concepts, such as a simplex and hypervolume. The rigorous definition of a hypervolume seems complex than I expected, so I will begin reading on it, or find a way to complete the derivation of the multidimensional law of cosines without a rigorous definition. I found a research done by another high school student on generalizing the shoelace theorem to a tetrahedron, which may be some help for this research. I also found two works that are closely related to this topic--one by M A Murray-Lasso and another by Ding.

If I am correct, the link for Ding's will be redirected to this page.

I was quite surprised that this theorem has not been investigated that much. In Murray-Lasso’s work, although very resourceful, a few errors in notations were noticeable, and was slightly lacking in mathematical rigour. Ding’s proof used a different method, which used the Divergence Theorem. However, neither thoroughly defined the multidimensional cosine or how the a n-dimensional simplex’s “face” (which would be a n-1 dimensional simplex) could be represented as a vector. After some research, I found the exterior product to be a suitable explanation, but I will have to study much more about the exterior product to maintain mathematical rigour.

My objective for this research is to follow Murray-Lasso’s method to prove the theorem, but with greater mathematical rigour and an introduction of concepts such as the exterior product.


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