My long lost 1970s pullover, slip-over, or perhaps more properly 70s tank-top, has turned up at the back of a cupboard. Last year we had turned the house upside down looking for it.

The Missing Pullover

The design is apparently based on the colour theory of Interaction of Colour, (Albers 1963)  and illustrates the first of  The Twelve Fold Ways from Stanley’s Enumerative Combinatorics, (Stanley 1986 and 1997). The first way being n-tuples of x things with enumeration formula x to the power n.

Here there are 2 objects, an inside and an outside, and 4 colours giving 2⁴ = 16 different combinations. Ignoring the 4 same-on-same combinations gives the 12 unique combinations numbered below.

Using a spreadsheet type program each of these 12 combinations is then associated with a random number function.

The combinations are then sorted on their associated random numbers and this is repeated as often as necessary with newly generated random numbers.

Repeatedly applied selections

This gives an even mixture because every combination is used before it is used again. This ensures that there are equal numbers of each combination, and therefore that equal numbers of balls of wool are required.

Knitted Sample

An equivalent procedure would be to put, say cardboard samples, representing each of the 12 combinations in a bag and drawing them out blindfold one-by-one until none are left, then putting all the cardboard samples back in the bag and repeating the procedure.

An even mixture would not be guaranteed if each sample was drawn out blindfold and then immediately put back in the bag before making another selection, such a method would just statistically tend towards an even distribution.

Architectural applications of this techniques to follow.

Unfortunately, as perhaps the observant will have noticed, the long lost pullover was not made in accordance with the knitted sample or the procedure above, but appears to just randomly list all the 2 colour combinations of 3 colours. I think then that it had better go back in the cupboard.

Bibliography

Albers, J., 1963. Interaction of Colour, Yale University Press.

Knuth, D.E., 2005. The Art of Computer Programming, Volume 4 Fascicle 2, Generating All Tuples and Permutations .Addison-Wesley

Stanley, R.P., 1986. Enumerative Combinatorics (Volume 1),Wadsworth & Brook.

Stanley, R.P., 1997. Enumerative Combinatorics (Volume 2), Cambridge University Press.