## Turing and commutators

http://en.wikipedia.org/wiki/Commutator_subgroup

if you want to know what a commutator is, don’t read above.

But, the pictures I added do tell the core story.

As we discussed yesterday, the commutator is about those missing feet (or the extra feet) that occur when having travelled around the two sets of long and short sides of a soccer pitch you don’t end up back at the corner you started from. Rather, you are a few feet short (or a few feet to far on). This is diagrammed above in blue (by me). I.E. The lines don’t match up (if straight). Yes Turing was not straight, too; but that’s too obvious a joke.

So what do we do.

Well we force them to match up. The shorter wire is now made of something bendy (but relatively ridged, like spring steel). Thus it can bend out (blue or black bends) and the geometry of the bend takes up the extra feet (at the cost of putting tension into the spring) allowing now there to be a “continuous line”. Its not exactly straight, but you can travel along it, and it will “take-up” the “extra few feet” so you do at least end up where you started.

if you do more work and push the blue bend rightwards, the spring will “trigger” and flip over into the black bend – and then stay there. This is course is the principle of the light switch (and you can feel the flip point in your finger, as you push against the spring).

The old name for a switch is a commutator (– still used inside car engine “points” to select which cylinder gets a firing signal, in sequence).

We can add a couple of current sources now:

http://en.wikipedia.org/wiki/Commutator_subgroup

pointing out how the “bulge” of the spring together with a well positioned terminal with electricity (one at –7v, and another at –8v say) can be “switched through” to the output”. Or, for your light switch, all volts or none (if its an on/off switch).

Now why did Mr Crypto (Turing) care?

Well it turns out that mathematics has commutators in abundance. Since the heart of ciphering is permuting the components of information (once encoded as bits, or enigma characters, say), its useful to know that there is a relationship between permuting (changing the order of the bits/chars) and commutators. So we need a source of them. That’s hard if you, in “exceptional” American high math schooling, don’t even know what to look  for (not even (pun) knowing what one is)!

One good source is the alternating group (which, durr, “alternates” between positive and negative – or +1 and –1 if you want symbols). Of the elements in a symmetrical group that have positive parity, this set  forms the alternating group since there are an even number – and here comes the WWII crypto bit – of the pairs of engima points that showcase a certain type of difference between the two characters found on one side of the engima wheel and the difference found on the other side, after mapping. If you count how many “in” side pairs start out with two (e.g. a and b) pads such that a is before to be before b alphabetically, then you count +1 if then the mapped.a and the mapped.b on the other side of the enigma wheel are still in the same order (with mapped.a before mapped.b, alphabetically). Once you have counted all pairs, you see if your total count is odd or even…

Now why did we want commutators again, for ciphering?