Wikipedia:Reference desk/Archives/Mathematics/2023 May 30

= May 30 =

Deriving the square root of seven from a unit polycube
Based on File:distances_between_double_cube_corners.png, I drew an illustration of ways to get square roots of 1 to 6 from a polycube. Adding a third cube to make an L shape gives &radic;8 and &radic;9=3, whereas adding it in-line gives &radic;9, &radic;10 and &radic;11.

Is there any way to get &radic;7?

Thanks, cm&#610;&#671;ee&#9094;&#964;a&#671;&#954; 00:10, 30 May 2023 (UTC)


 * The way that these lengths are obtained is by taking the square root of the sums of squares of lengths of the "bounding box" of the line segment. For example, $$\sqrt{6} = \sqrt{2^{2} + 1^{2} + 1^{2}}$$ derives from two sides of length $$1$$ and a side of length $$2$$. Since there are three sides to the bounding box, the only lengths obtainable are precisely those that can be written as the square root of the sums of three squares. $$7$$, quite famously, is not one of those numbers. GalacticShoe (talk) 01:39, 30 May 2023 (UTC)
 * Note that this implies that all $$\sqrt{n}$$ for nonnegative $$n$$ can be obtained if working with four-dimensional polycubes. GalacticShoe (talk) 01:46, 30 May 2023 (UTC)
 * Thank you very much, @GalacticShoe. Glad to learn about Legendre's three-square theorem and Lagrange's four-square theorem. Cheers, cm&#610;&#671;ee&#9094;&#964;a&#671;&#954; 08:27, 30 May 2023 (UTC)
 * Happy to have been of help :) GalacticShoe (talk) 19:06, 30 May 2023 (UTC)


 * In fact there is no ternary quadratic form which hits every positive integer. In other words there is no expression of the form ax2+bxy+cxz+dy2+eyz+fz2 with fixed a, b, c, d, e, f which takes on every positive integer value, but no negative values, as x, y and z vary over the integers. A paper here proves this, and the introduction states that it was "well known" in 1933 though they couldn't find out who originally proved it. --RDBury (talk) 20:58, 31 May 2023 (UTC)