This bit is miscellany about continued fractions theres a link at the foot of the page to some elementary descriptions of how to make a continued fraction - other than that I'm assuming you're either acquainted with them or are bright enough to find out something about them first

Numbers in art is intended to be a collection of ideas on their use based on some ancient programs written by Mike Burr.

# Continued fractions

I first started using the continued fractions approximations to make a diminishing perspective using both their result and sequence of approximation I experimented with a couple of watercolour landscapes by placing trees example
cf result --> 13*( 180**2 ) - ( 649**2) = -1
approximation to 13 by continued fraction [approximated from the upper bound !!!]
3(3) , 8(8) , 13(0) , 18(5) , 23(10) , 28(2) , 33(7) , 71(6) , 180(11) , 649(12)
the bits in brackets are the approximations modulo (13)
i drew a coastal landscape with receding trees so i could try this out .. OK if you're mad enough to have read this far then you're going to love this bit .. add up all the sequence above i.e. 649 + 180 + ...or write a program to do it as i did.
the result is 1026 i then divided the paper [about 30 inches across] and divided it by 1026 [not much point doing the first tree but they fell on about .. 1/3 inch in 11/1029 of the page by the time you get to the 5th tree ..65/1025 you're about 2 inches in from the edge and the remainder are sort of inch and a bit steps getting wider ..
now imagine you did the opposite so the trees got closer as they go into the distance and then have a look at this picture by Bruegel called Hunters in the Snow .. the smaller trees are a bit difficult to make out maybe but hes probably using some sort of ratio to do much the same ...
anyway you can find a large file of continued fraction data for