About…
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
about 2000 prime numbers here..;
That was about 20 years ago .. because the next suite of programs i wrote attempted to do the same thing
but producing two dimensional ratios - as boxes. The first version i made was much more complicated than later
its children and did seriously weird things .. at this time of writing i havent had a look back at the program
yet but it iteratively modified the size of the boxes it produced and seemed to have a sort of personality. if
that can be said of a bit of software ???. The scheme behind it was to produce something
like a fibbonacci series of reducing boxes within the confines of a canvas but using the continued fraction
approximation idea youve seen crudely outlined above. The thing is that boxes didnt necessarily fit within the
confines of the canvas so it would modify its use of approximations on the fly and so instead of filling a space with
the same uniform items - which is what later simpler versions did its made a load of
odd shaped fill ins - i used the program which was then written in basic on an old atari
I had some quite nice photos of the horses at Whitbreads old stable in Garrett Street, London - i think in truth
the image i liked best was photographed by my ex-wife Celia. She particularly like Bach and Beethoven
and i had this in mind with the picture
Jupiter
the big white horse at the top right hand side of the page. The painting has something of the photograph in it
Except !! the all the perspective lines and components are placed - i also left out anything that fell across the boxes
so it looks rather sparse and also archaic .. classical along the lines of Bach's music or early 15th Century art ..
The picture was bought by the shire horse society and presented to their retiring president whose wife told me
that at first she didn't much like the picture. I moved away from a realist view in this image and combined a principle
that might have been used on constructing perspective a la the linear vanishing points
Leonardo's adoration of the magi sketch
and more particularly some of
Ghirlandio's and
Antonello de Messina's brilliant and convoluted painting
of St Jerome ..
hopefully i will be able to get some of these programs up on a web server so you can try them for yourself.
