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!!