/sci/ - Science & Math

Now, some thoughts about AIRAs, as they relate to Euler bricks and similar forms, and the missive >>7497736 .

What is an Euler brick? We of course have our geometric definition >>7496579 , but consequently we can think of "a brick" as a system of three coupled diophantine equations, or as a point in R^3 whose coordinates are simply the edge lengths. As an aside, coordinates in R^3 generally follow a signing-permuting convention, sketched out in pic related. But by way of generality, how could AIRAs conceivably fit into an Euler brick (or similar forms) at this point?

If the "thing" is a cube, then it's no Euler brick as we already know (all edge lengths are equal). But it's worth noting the triviality that for a cube, it's also impossible that any of its (one with multiplicity three) associated pythagorean formula(e) can be an AIRA -none of the legs differ by a unit of course, so that's one degenerate case.

Another degenerate case is the situation (a,a,b) (or, for that matter, (a,b,b) ) with nonzero a and b, This has two equations (one with multiplicity two), of which one (the square one) is definitely not an AIRA, but the other two could be: imagine either of (3,3,4) or (3,4,4), say. But then of course, these are not Euler bricks, having some two edges equal (and thus entailing some √2 term along that hypotenuse).

So we meander back towards the meat of the subject, but some other degeneracy is first entertained: for an Euler brick (a,b,c) with no two of nonzero a-c equal, it certainly could /not/ be the case that all three of its equations are AIRAs, as this would entail the absurdity that c-a = c-b = b-a = 1 (you can't have three unequal naturals all having a distance of one from each other, lines don't work that way); this is simply due to the way in which the equations (and the attendant legs, and the natural number line itself) are coupled.

On the other hand, just look at some primitive known bricks! (cont.)