Mathematics & Measurement
 

Mathematics

The history of the relationship between colour and mathematics has been long and curious, contrived by many different thinkers, some practical, others Pythagorian and spiritual.

In recent times, the teaching of mathematics in schools has been greatly aided by a system called Colour-Factor. This makes use of modular rods, invented in 1806 by the mathematician, Tillich. Today, colour has been added, to bring the rods 'alive', and speed up the comprehension of number by making the activity attractive and visually exciting.




(Left) Colour-Factor Mathematics. (Below) Colours in Relation to Numbers (Cuisenaire).


The idea of colouring the rods was arrived at independantly by two teachers, Dr Catherine Stern (whose colours were chosen aesthetically and tastefully stained onto wood) and Georges Cuisenaire, who used plastic rods in colour 'families'. The Stern rods were very attractive to look at and the Cuisenaire rods were very durable, but another system was introduced which was based on a purely mathematical concept. In this there were good mathematical reasons for the colour-number attributions. This was the Colour Factor system devised by Seton Pollock in 1961.


Colour Factor actually used colour factorially, the warm colours being for even numbers and the cold for odd numbers. The rods took the form of sticks built on a cubic module 1-12. The white, red, blue and yellow, being primary colours were related in that order to the first four prime numbers, 1, 2, 3 and 5. Black was related to the prime 7 and all primes greater than 7. White was the base upon which all colours could be displayed. White cannot be made more white, just as 1 multiplied by itself any number of times remains 1. Red has a potent effect on the eye, and two is the most potent of the prime numbers. Its colour red is readily distinguishable in at least three intensities. Thus 2 x 1 is pink, 2 x 2 is scarlet, and 2 x2 x 2 is crimson. These numbers 2, 4 and 8 grade from the white end of the series towards the black end. Blue is associated with the group of numbers based on 3 in the series from 1 to 12, having only 2 members , namely 3 x 1 and 3 x 3. Thus 3 is light blue and 9 is royal blue. Yellow is related to 5 and orange to 10. Six is violet and seven is light grey and eleven dark grey. Further explanations are given by the teacher Seton Pollock in his book (see below).

There are some problems involving colour that have puzzled mathematicians for over a century. One of these is the famous map-colouring question. How few colours do you need to colour-in a map of fields so that no two fields of the same colour are contingent ? The most elegnat solutions are perhaps those of two Russian professors of mathematical logic at Moscow State University (see below).

Another aspect of colour and mathematics is that of the mathematical crystallographer, concerned with colour symmetries and classifying all symmetrical subdivisions of the Euclidean plane. This is likely to be of great interest in the field of design, as well as visual and evironmental studies and architecture. With reflection symmetry all seventeen diperiodic and seven monoperiodic ribbon configurations need to be generated by a logical process of mutual implications. An authority on the subject is Arthur Loeb of Harvard Visual and Environmental Studies. The history of colour-number ideas goes back to Babylon and the sacred book of numbers, the Kabbala, said to embody the inner and mystical aspect of Judaism. The original Kabbala does make several abstruse remarks refering to colour, using such terms as 'the yellow of all yellows that includeth and concealeth all other yellows' or the 'backness of black' and the shining of 'white brilliance'. The occultist, Sepharial, writing at the beginning of the 20th century, explained that :

The number One - symbolised the fundamental unity of all things, the Logos,. Its colour was the brilliance of the sun, &c.
The number Two - symbolised the duality of life, the law of alternation, the binomial, subject and ofbject, plus and minus, and creation, &c.
The number Three - symbolised all trilogies and trinities, the three dimensions, and past present and future &c., &c.


Sepharial gave colours to the numbers or Sephiroth as these powers are called, relating them to the colours of the planets. His attributions, however, are not consistant (see below).

Another occultist, a principal of the International Order of Kabbalists, wrote at length on colours and the sephiroth, though he explains that the relationship should be re-invented by each person using the system :

Kether (the first) had brilliance.
Chocma (the second) had blue.
Binah (the third) had black, and so on (see below).


This Kabbalist academic, James Sturzaker, also explained that a popular outcrop from the Kabbala was the Tarot, in which there is a suit for each World (the Merchants, the Army, the Common People and the Church), composed of ten cards plus page or jack, knight , queen and king to indicate four sub-levels. The twenty-two cards relate to the paths of the Kabbala tree.

A particularly popular coming together of mathematics and colour can be found in the form of fractals. Whilst the colour in these is of little relevance to the mathematics itself, they nonetheless demonstrate a brilliance and intricacy seldom seen elsewhere.

By far the most famous fractal is the Mandelbrot Set. This can be fully explored online by visiting the following website:
http://aleph0.clarku.edu/~djoyce/julia/explorer.html

DP

Bibliography

CUISENAIRE, G. and GATTEGNO, C. (1954) Numbers in Colour. London: Heinemann. Colour in school mathematics.
DYNKIN, E.B. and Uspenskii, V.A (1963) Multi-color Problems. Chicago: Un. Chicago. Field-colouring problems.
LOEB, A.L. (1970) Color and Symmetry. N.Y.: Wiley. Relates colour to the mathematics of crystallography.
PLEUGER, W.H. (1959) Discovering Arithmetic. London: Educational Supply Association. The Stern system of coloured mathematical sticks for schools.
POLLOCK, S. (1962) Colour-Factor Mathematics. London: Heinemann. Colour in school mathematics.
SEPHARIAL (1911) The Kabala of Numbers. London: William Rider. Inconsistencies can be seen in the colour attributions p.60 vol.1, p.169 vol 2.
STURZAKER, D. and J. (1975) Colour and the Kabbalah. Wellingborough: Thorsons. Occult colour in relation to the ancient book of numbers. Has no bibliography. Sturzaker's colour scales may have been taken from Aleister Crowley's four colour scales, the Emperor's, Empress', King's and Queen's scales from his book 777, London: Neptune, 1955. Crowley also quotes the researches of Dr Jellinek on the colours of the Kabbala. However it is difficult to take seriously a colour such as 'white-red-whitish red-reddish-white'.
THOMPSON, H.A. (1962) Colour-Factor Mathematics. Reading : Colour-Factor.
Colour in school mathematics.


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