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].

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

**1 size**

1 square alone is or .

Putting 2, 3, 4… squares next to each other, we have the numbers 2, 3, 4.. or, looking from the direction, , , …

i.e. numbers of the form and

In order of magnitude:

Or, from the direction,

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

**2 sizes
**

Using 1 large square and smaller ones along the side, we get

[illustrate]

Using 2 squares and smaller ones along the side,

[illustrate a few]

Using 3 and smaller ones,

[show a few]

i.e. numbers of the form (and ), where .

[show diagram of general case]

In order of magnitude:

[picture of plot of these on number line, , 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”:

with 3×3 squares added makes:

with 4×4 squares added makes:

with 5×5 squares added makes: etc.

**4 sizes**

**Functions**

Define:

the number of Different-sized squares needed to construct .

number of iterations of Euclid’s algorithm to find

number of terms in the continued fraction.

the total Number of squares needed to construct , regardless of size.

sum of the continued fraction terms .

total of the numbers used as multipliers in Euclid’s algorithm.

If a/b is irrational, is infinite, e.g. . Although e.g. for = VALUE*#*#*#?!?, but .

[picture of Stern-Brocot tree, , 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 =1,2,3..? is is Catalan numbers or something?

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