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

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May 30

Deriving the square root of seven from a unit polycube

Distances between vertices of a double unit cube are square roots of the first six natural numbers, including the reference desk/archives/mathematics/2023 may 30 (√7 is not possible due to Legendre's three-square theorem)

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 √8 and √9=3, whereas adding it in-line gives √9, √10 and √11.

Is there any way to get √7?

Thanks,
cmɢʟee⎆τaʟκ 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, cmɢʟee⎆τaʟκ 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)