Walking the Tree.

It’s just fun playing around with Stern-Brocot trees and a bit of Haskell…! For any fraction on the Stern-Brocot tree there is just one path to it from the root. That path can written as a sequence of left or right ‘turns’ at each node depending on whether the target node is smaller or larger than the current node. This leads to the Stern Path For example in this small tree the path to, say, 4/5 is, from 1/1, LRRR. The fraction 5/2 has path RRL. To get to 5/8 we go LRLR… and so on. (If you want to Read More


Ford and his Circles.

      A Ford circle is a circle derived from a pair of numbers that are co-prime, i.e. they have no common factors. For a pair of co-prime integers p and q the Ford circle has radius r and centre at a point P(x, y) where r = 1/(2q^2) and P = (p/q, r) No matter what co-prime numbers, p and q, are used to create Ford circles the circles never intersect and they are all tangential to the horizontal axis. Now, we could generate ‘random’ Ford circles by picking any old co-prime pair (p, q). However, the Farey Read More


Count the Fractions.

This is the first in a series of posts about sequences of fractions, circles, trees of fractions, binary search trees and ways of representing rational numbers as paths through these trees . The idea is to enjoy a bit of recreational mathematics and to use Haskell to express some of the notions in its own succinct way. One Lot Of Fractions First we’ll look at all unique and simplified fractions between 0 and 1 with a denominator no larger than a given value. For example, with a maximum denominator of 3, 4 and then 5 we have     For Read More


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