Fractions constructible with n different-sized squares

A ratio, e.g. 200/117, is represented as a 200×117 rectangle. Operation of the Euclidean algorithm or the jigsaw method gives the continued fraction [1;1,2,2,3,1,3].
euclid 220 over 117

On this page, variables are positive integers. Rectangles x\times y are drawn only for x\geq y, but usually in the text, only the proper fractions with x<y are given, to save space (and these lie conveniently between 0 and 1.) But remember, the rectangle x\times y represents both x/y and y/x e.g. where 3/4 appears below, 4/3 also has the same number and size of constituent squares, it is just rotated 90^{\circ} (which here = inversion).

1 size

1 square alone is n/n or 1.
Putting 2, 3, 4… squares next to each other, we have the numbers 2, 3, 4.. or, looking from the y direction, 1/2, 1/3, 1/4
i.e. numbers of the form n and 1/n
In order of magnitude: 1,2,3,4,\dots
Or, from the y direction, \dots 1/4,1/3,1/2

N.B. From here on I will omit all mention of the half of the fractions >1, except in the general formulas, to save space. For every x/y mentioned, y/x is in the same group.

2 sizes

Using 1 large square and smaller ones along the side, we get 2/3, 3/4, 4/5 \dots
Using 2 squares and smaller ones along the side, 2/5, 3/7, 4/9 \dots
[illustrate a few]
Using 3 and smaller ones, 2/7, 3/10, 4/13\dots
[show a few]
i.e. numbers of the form \displaystyle \frac{n}{an+1} (and \displaystyle \frac{an+1}{n}), where n\geq 2.
[show diagram of general case]
In order of magnitude:
\dots 2/9,3/13,4/17,5/21\dots \frac{n}{4n+1},2/7,3/10,4/13,5/16,\dots \frac{n}{3n+1},
2/5,3/7,4/9,5/11\dots\frac{n}{2n+1},2/3, 3/4, 4/5, 5/6 \dots \frac{n}{n+1}

[picture of plot of these on number line, 0<a/b<l, each a short vertical line, maybe pic width 500px]

3 sizes
Working backwards, this group is made up of each of the previous group, with 1 or more squares stuck on to the side.
From the “2 sizes, 1 large square group”:
2/3 with 3×3 squares added makes: 3/5,3/8,3/11,3/14\dots
3/4 with 4×4 squares added makes: 4/7,4/11,4/15,4/19\dots
4/5 with 5×5 squares added makes: 5/9,5/14,5/19,5/24\dots etc.

4 sizes


\Delta(a/b)= the number of Different-sized squares needed to construct a/b.
= number of iterations of Euclid’s algorithm to find gcd(a,b)
= number of terms [a_0;a_1,a_2,a_3\dots] in the continued fraction.

N(a/b)= the total Number of squares needed to construct a/b, regardless of size.
= sum of the continued fraction terms [a_0;a_1,a_2,a_3\dots].
= total of the numbers n used as multipliers a=nb+c in Euclid’s algorithm.

If a/b is irrational, N(a/b) is infinite, e.g. N(\pi/1)=\infty. Although e.g. for [1;2,1,1,2,1,1,1,2\dots] = VALUE*#*#*#?!?, N()=\infty but \Delta()=2.

[picture of Stern-Brocot tree, 0<a/b<1, with D() and N() next to each number. width =500px?]

=== state rules for D and N for S-B tree. and for adding fractions generally.

=== make table of D and table of N for a=1…50 and b=1…50 .. or 30, whatever fits on a page.
===could do a table 100×100 with dots in each square representing D(a/b).. or colour coded.. could go higher.
=== what WOULD that look like?! 🙂 or even 500×500, 2 pixel colour in each one… 250×250 maybe better.

=== How many fractions have N()=1,2,3..? is is Catalan numbers or something?

=== Make another page for the 3D case!!

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