/sci/ - Science & Math

>>7499844

This is /very/ interesting. So interesting in fact, that it deserves inspection.

First of all, none of the edges in either brick are pairwise relatively prime. (not even 1155 and 12075 are relatively prime, the only remaining curiosity worth checking). But, each /triple/ has a gcd of 1, and are consequently relatively prime, primitive bricks.

Furthermore, notice that both bricks have components which are multiples of 252 and 275, two pairwise relatively prime numbers which are components of an interesting low brick (240,252,275):

https://en.wikipedia.org/wiki/Euler_brick#Examples

This brick is interesting as an early counterexample, because its /longest edge/ c=275 is /shorter/ than its /shortest face diagonal/ d=348, which serves as a spoiler of the global inequality, referenced in the BRIGHT RED cells in >>7496745 . (That is, opposite what the above image said, MOST early Euler bricks DISOBEY this inequality, with the brick currently under discussion being an exception, meaning that you can have either c>(=?)d or vice verse).

(240,252,275) is also interesting just for its simple geometry. It certainly seems to be a lot more "cube-like" and less eccentric than the other bricks listed.

Now compare (240,252,275) with (1008,1100,1155). The latter is NOT a simply scaled multiple of the former, of the form >>7497051 . And yet, in both cases, the long edge c is to the short edge a as in the other, that is, c/a : c/a , or

<span class="math"> \displaystyle \frac{1155}{1008} = \frac{275}{240} = \frac{55}{48} [/spoiler]

Meanwhile, the other ratios transpose. the LONG TWO edges of the SMALLER brick are to each other as the SHORT TWO edges of the LARGER brick are to each other...

<span class="math"> \displaystyle \frac{1100}{1008} = \frac{275}{252} = \frac{275}{252} [/spoiler]

and the SHORT TWO edges of the SMALLER brick are to each other as the LONG TWO edges of the LARGER brick! viz.

<span class="math"> \displaystyle \frac{1155}{1100} = \frac{252}{240} = \frac{21}{20} [/spoiler]