Introduction

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 constructingan image. These were the structure and what the structures might

subliminally mean.

Structures

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]

example :-

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.

Ultimately

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