Another Art page - Carleson square and trees

Yes indeed, who could fail to appreciate the beauty of this wonderful construction? This dyadic decomposition of a square is used in the Carleson Measure Theorem, an absolutely fundamental result in Hardy space theory; similar dyadic type methods are very important in the theory of BMO, wavelets, etc. etc..... (P.S. There is no mathematical meaning to the colours; it just adds prettiness.)

Of course, just by looking at it we would expect a close relationship with binary trees (below): each top half of a square sits on exactly two squares of half the size (or, alternatively, think of the rectangles instead as being the "tiles" which cover the large square exactly), just as each node of the infinite binary tree below spawns two new nodes as we move downwards.

A truly awful joke: apologies... The picture shows a Tree which, contrary to Graph Theory, does contain a Cycle. Ho-ho-ho! There is a story, almost certainly false, about some students who once stole the bike belonging to their lecturer, a famous Graph Theorist, and hung it in a tree in his garden - which is slightly amusing, even though probably not true.

Again, these were produced with Microsoft Paint in a fairly short space of time (the copy and paste feature being especially useful for the binary tree; this might well have been good for the Carleson square also, if I had thought of it in time). CLICK HERE TO RETURN TO MAIN Zen Harper PAGE, OR CLICK HERE TO RETURN TO THE ART PAGE

TRIANGLES ARE BEAUTIFUL!!