/sci/ - Science & Math

>>7496662

Especially notice here, that there are "MAJOR" diagonals, and "MINOR" diagonals (imagine planes slicing through the plane segment of this face of the octahedron). These describe general areas of possibility (or impossibility) for where the far vertex of an Euler brick may lie in R^3.

One given Euler brick endpoint (a,b,c) lies somewhere strictly inside the green region, for example. Its permutations which stay in the same octant, populate the other sectors.

In the overall scheme, the far point cannot possibly lie on the outside edge of our triangle, for then it should have an edge of zero, and be a filthy degenerate. Similarly, the far point cannot possibly lie on any of the MAJOR diagonals (let alone the center!), for the reason that this means that some two of its coordinates are equal, which is impossible. Imagine the planes x=y, x=z, y=z slicing through the principal planes at 45 degree angles (and thus also through our octahedral face).

But in the course of investigating Euler-brick point permutations and data (and in light of this geometric treatment), it became reasonable to consider these MINOR diagonals, which generally satisfy that three unequal terms are EVENLY SPACED- that is, that c-b = b-a, for example. Moreover, the author is not aware of any Euler brick which has evenly-spaced edge lengths, and he thinks he's close to dispensing with the idea (but we'll see). This leads us to the central reason for this thread's existence, a:

CONJECTURE: There does not exist an Euler brick with edge lengths a, b, c such that a<b<c, and c-b = b-a.

Regardless of what becomes of this conjecture, we are going to have a long and winding road which will take us into other parts of number theory, and the author is encouraged by the initial response.

Now it's time to present review details, previous results, and links.