By the end of high school, most students should be familiar with the “three-dimensional” Pythagorean theorem (aka, “the distance formula”) that relates the length of the diagonal of a shoebox to the lengths of its edges. (Students should also know that this generalization extends to relate diagonals and edges in “any-dimensional” shoeboxes; always, the square of the diagonal equals the sum of the squares of the edges at a corner.)
However, view students seem to be aware of this result:
Given a “right-corner tetrahedron”, the square of the area of the face opposite the right corner is equal to the sum of the squares of the areas of the other three faces.
This result extends to “any-dimensional” space as simply as the distance formula: the square of the hypotenuse is equal to the sum of the squares of the legs. However, unlike the distance formula —which only ever relates one-dimensional lengths— the values being squared here are increasingly “dimensionally-enhanced” aspects of a figure: lengths in 2-d, areas in 3-d, volumes in 4-d, hyper-volumes in 5-d, and so forth.
I was thrilled as a high school junior when I discovered this result, and I was devastated as a college freshman when I discovered that others had beaten me to it by at least a century. Nevertheless, in the years —decades!— since, I’ve worked off and on to find related, completely new, results in the field I call “(Tetra)Hedronometry”, the dimensionally-enhanced trigonometry of tetrahedra. I’ve had a bit of success —in particular, in the realm of non-Euclidean geometry, where (so far as I can tell) absolutely nothing was previously known— and I believe my focus on tetrahedral “pseudo-faces” is a significant contribution to tetrahedral lore. We’ll see.