Your author tried a different method of structuring the paintings almost every time he created a new picture.
Many of the methods here are already known and there are a few new ideas too but more relevant is that this
shows you how it can be done and any maths you need to do it.
I was originally very interested in 2 aspects of constructing
an image. These were the structure and what the structures might
One of the first things I found out was that painters such as Rembrandt
had experimented with structures. A traditional way of constructing an
area within an image was to draw diagional lines
from one corner to the other
Then, using a set square, place the set square on the diagional and make a
line at right angles to the diagional up to the corner of the image
If you do this from the two diagonals to each of the corners
you end up with a total of four points, two on each diagonal.
Join then up and it makes an inner square.
I did a bit of simple maths which shows that if the image size is
A cms wide and B cms high
then the smaller box you've just made is in the ratio
(A**2 - B**2) / (A**2 + B**2) [ where **2 = squared i.e times itself]
If I have an image 200 cms wide by 100 cms high
then this method will create a box
(200**2 - 100**2)/(200**2 +100**2) = (40000 - 10000)/40000 + 10000)
= 3/5 the size of the original
i.e it will be 200*3/5 = 120 cms wide and 100*3/5 = 60 high
You can easily create smaller and smaller boxes inside the next.
The image of the single horse Jupiter is loosely based on this method
and a linear perspective which I think gives it a more archaic look.
I decided to approach this using some of the maths skills I have
I started by calculating approximations to a point
or a plane on a surface using techniques like continued fractions.
There is a separate section on continued fractions which outlines the
three different approaches I used.
Heres an example:-
The continued fraction approximation to phi could be used as follows
to give successive terms
2/3 5/8 13/21 34/55 89/144 233/137
Now supposing I multiply the first by say 4
4*(2/3) = 2.67 and if i then multiply 2.67*(5/8) I get 1.67
1.67*(13/21) = 1.03 , 1.03*(34/55) = .637 and so on
Now look at the placement of the trees in Bruegel's "Hunters in the snow"
make the image about 20 cms wide and measure their positions
and you'll see he's done something fairly similar.
We could have used the ratios as they stood and divided the height
of the image by 2/3 5/8 etc so that if the image was say 100 cms high
we would have ended up with lines at
66.67cms , 62.5cms , 61.9cms , 61.8 cms
these could be construed as localised perspective lines within a narrow
area of the image
Obviously we need not restrict ourselves to lines
We could have made circles of decreasing diameter
or calculated say squares or rectangles of changing sizes
I wrote several programs to do just this and then fit the
blocks onto an image area
The image of 2 horses Silent Rapport on the main web page is an example
of this although I used a calculator to size the areas, divided across
the stable by the uprights. The image is complicated slightly
by the use of a curved perspective which is more natural to the eye.
[I hope to be able to put some of these facilities on a website
, mine isn't suitable, so that you can use them]
The picture of 2 cats in a window was painted for my [now ex-]
mother in law who was a skilled biologist and her husband from
an engineering background. Many of the structures in biology
and engineering are logarithmically based and so I used
logarithmic continued fraction approximation
to structure the image
If we had used the square root of 2 in the example above
we would have the following
6/4 20/14 68/48 232/164 792/560 etc
dividing our 100 cm image by these ratios would result in lines at
66.667cms 70.000cms 70.588cms 70.690cms 70.707cms
You can see that both these series converge fairly rapidly although
I could have spread them out by say doubling the gaps between them
If I multiplied them by say 10 .. I would get differences of
33.3cms 5.9cms 1.0cms 0.1cms
but even this only gives me a few lines.
I then decided to use patterns resembling the spectral lines
from atomic physics
[this is what I supposedly studied in Physics at Kings in between
bouts of drinking, from experience I strongly advise against the latter]
Its simplest form atoms emit light according to a pattern
known as Balmer lines.
They are very easy to calculate as they are of the form 1/n**2
i.e 1/4 1/9 1/16 1/25 etc
I thought these would be intriguing as they should strongly mirror
the most basic of our own structures and therefore from a commercial
standing be more appealing [read ; make more money]
Unfortunately this simplistic model of the atom is only approximate
and a much better model is afforded by the solutions to wave equations
These generally have the intriguing properties that they have
what mathematicians describe as 'real' and 'imaginary' properties.
They are referring to the roots of negative numbers ..
example:- something times itself = -1 [square root of unity]
In practical terms only the real parts are 'implemented'
[An aside :- approximate solutions to the roots of imaginary numbers
may be used to define structures that look like mountains , coastlines
and more intriguingly tree branches..
I would have thought that someone with a good knowledge of these sets
would try to tie them in with tree and plant structures and thus form
an equivalent of the Linean catalogue. This might then be used with a
knowledge of the DNA. Linking the algorithm the DNA creates,
to the DNA itself, to the atomic make-up and maths behind it.
Everything Is Mathematics !]
Wave equations are examples of differential equations
The real solutions to these equations are generally exponential
and frequently oscillatory
An example of this is the closing of a swing door.
If it is well designed it will swing back quickly to a stop.
If the spring is too strong the door takes ages to close
as you will notice if its very cold outside.
If the spring is too weak it will swing back and forth
before gradually coming to a halt
Supposing in this last case that we marked the furthest point
that the door came to. We would end up with a series of marks
most of which would be around the point where the door
is properly closed.
We could use these in the same way as we used
the approximations above.
I will add a separate section like that for continued fractions
relating to the maths
The next thing that I tried were cubic equations.
The little black cat is based on a cubic equation
[An aside :- I spent a couple of days exhibiting and painting
at a shopping mall in Uxbridge. It probably took longer to
position the cat than it did to paint it even though I had
to paint it twice using a lamp black (blueish)
then an ivory black (?) (brownish and therefore an opposite)
to get a pure black.]
At this time I was experimenting with numbers trying to find a way
to differentiate[in the sense of calculus] them, this being sort of
related to some ideas I had regarding the solving of
the differential equations I've written about above.
One of the things I tried was adding up the components
of a number and you can read a much fuller discussion on this
in the separate section entitled 'Ranks of numbers'
Around 1994 I was shifting the remnants of my ex's things from the loft
among which was a small book entitled 'Number theory' by T.H Jackson.
In it (p74) is a question relating to the sum of 3 squares for multiples
of 8 with seven added [ 8k + 7]. I had already used a spreadsheet to
create tables for different number bases and from it permissible squares
are rank 1 , 4 , 1 and 8 repeated [i.e 8k , 8k + 1, 8k + 4]
combinations of these give
1+8+8 = 1 , 1+1+8 = 2 , 1+1+1 = 3 , 1+4+4 = 1 , 1+1+4 = 6 and so on
the end result is 1,2,3,4,5,6,8 are allowable but not 7
hence the sum of 3 squares CANNOT be a multiple of 8 plus 7
e.g 7 , 15 , 23 , 31 , 39 etc
This is a fairly trivial example of the power of this kind of methodology
You will notice that from this example that 1 is the same as 9 is the same
as 17 is the same as 25 ..[8k + 1] and that 2 is the same as 10 etc
and so the mathematics defines SETS of possible solutions without defining
the actual number precisely. However, if you then start to use combinations
of the solutions there is the possibility to reduce the solution set.
There is an example in the 'Ranks of numbers' section where I use just such
a technique to deduce the square root of 961 using in combination
a 10 and a 9 base. With bigger numbers the number of combinations increases
in proportion to their length, this being the limiting factor of the method
It is a simple matter to use a logarithmic base and perform the same
trick thus giving non linear solution sets