Venn binary is better

by Dr Gavin Hitchcock

Consider the above table, and then see how neatly we can code the subsets of school-pupils we looked at in the article before last on page 22:

Questions:
1.  What kind of person is 0110?
2.  What binary code represents you?
3.  What nice ways are there of ordering these 16 four-digit codes?

Now, in the new-style Venn diagrams, we get each new set (see page , and our cover picture) by weaving in and out all around the circle, and the order in which you divide old compartments to create new compartments (2ⁿ of them, all touching the circle, at the n^th stage) provides a natural (but far from obvious) ordering of the binary codes of the 2^n-1 old compartments, corresponding to ordering the 2^n-1 combinations of n-1 things.

The nice surprise is that this order is none other than the "Gray code" ordering used in combinatorial mathematics. It was invented for a very practical purpose by an American telephone engineer Elisha Gray, over a century ago. The point, in those early telecommunications days, was that you had to shuffle levers to change binary (off-on) codes, and it was good economics to know which codes were 'nearest' to which, in the sense of fewest lever changes. Look at the circular ordering given here of the 4-figure codes, determined by the route taken by the (dotted) 5th curve in the Venn diagram above; you will see that in this arrangement EACH CODE DIFFERS FROM ITS NEIGHBOURS BY CHANGING ONLY ONE DIGIT.

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