/sci/ - Science & Math

>>8421260

Your insult is unwarranted in that by reading my above posts, it is clear that I am aware of not doing (contemporary, relevant) /math/ as such. This is also why I qualified my recent activity as being a study of the /history/ of math, which I have taken up again above all as a personal amusement.

However, I would go a step further and correctly observe that even your central premise is (partly) false, thus contradicting the first part just a little bit, and only to a certain modest point. Specifically, working through the derivations themselves and appreciating why they are the way they are is a properly mathematical activity, even if not "particularly useful or interesting" by modern standards. But of course the

Writing about this has reminded me of something else I was doing, which vaguely prompted the reviewing of cubics/quartics as an exercise. I proved a few simple lemmas relating to Euler bricks and perfect cuboids, and I scratched the surface of the literature just a little bit to see if anyone had made like observations, and I did not see any.

However, I did find an interesting note by Nyblom on Almost-Isosceles Right Triangles and their 1-1 correspondence to square triangular numbers, which marries up with later information in Sierpinski's "Pythogorean Triangles" which has certain implications for Euler Bricks and (conjectured) Perfect Cuboids.

The spot where I left off was to compare the logical possibilities of a given Euler Bricks' various lengths and diagonals {a-f} where a-c are the shortest-longest edge lengths, d-f the shortest-longest face diagonals and g the spatial diagonal. I was able to establish a series of inequalities by simple logic but the simple fact that it is possible that c>d or c<d (examples of both exist) left me stuck, about the incomplete white cells in pic related.

The idea of the above has been to build up a basic toolkit of knowledge about Euler Bricks/Perfect Cuboids.

>>/sci/thread/7437394

>>7496710

I lied! Now here are the BIG TRUE FACTS (lemmata) about Euler bricks, previously covered (and how to prove them):

1. No two edge lengths of an Euler brick are equal. Pf: Pythagorean thm and √2 is irrational, reductio ad absurdum (RAA).

2. An Euler brick does not have a unit (=1) edge. Pf. RAA, squares are also finite sums (series) of consecutive odd numbers.

3. The longest edge length c of an Euler brick may be shorter than its shortest face diagonal d, and vice verse. (inspect low known bricks at wiki for this: 240-252-275 & 44-117-240 for example)

4. No two face diagonals are equal. And, they "correspond" length-wise to a,b,c in the expected way (short-mid-long). Pf. Use 1. and 2, RAA, and manipulate various equalities and inequalities.

5. (INCOMPLETE!!!) "the expressions a-g^2 satisfy this big chard of inequalities". Partial proof: a long, ordered train of thought, in pic related. (help?)

6. The 48 versions of an EB in R^3 have endpoints on the faces of a regular octahedron with vertices "+- a,b,c", etc. Pf. describe planes in space, find vertices, etc.

(conjecture, I wrote: c-b = b-a)

>>7439912

An improved state...

(The 14x14 matrix's transpose implies the converse relation about the main diagonal, due to symmetry...)