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One lovely product

November 1, 2012

I see, when I look at a product, say of X-axis and Y-axis, a plane P.  So, maybe mistaken, I go looking for planes in search of products–products of switches turning lights on and off.

A light can be on or off.  I can think of four switches to mess with this light.

a(off) = off, a(on) = off

b(off) = on, b(on) = off

c(off) = off, c(on) = on

d(off) = on, d(on) = on

I’m not talking about the ones at your house.  How would I know–did you invite…

Four is way more than you need to kindle eharmony:

a x a

a x b

a x c

a x d

b x b

b x c

b x d

c x c

c x d

d x d

We already have 10 matches to screen through.  (I have, as you can see, nothing against homocoupling.)  Let’s do it.

Plane 1. a x a

On X-axis we have, to begin with, positioned ‘off’ and after some distance ‘on’.  Since a turns both ‘on’ and ‘off’ into ‘off’ (which, by the way, is not that exotic; it’s like a switch that blows the fuse: if you press the switch when the light is off, then you notice nothing, but if the light is on and you press the switch, then the light goes off), we have a loop going from ‘off’ to ‘off’ and an arrow going from ‘on’ to ‘off’.  Same goes with Y-axis.  What goes on P = XY is worth a penny.  Forgetting the arrows, just for a moment, there’s a dot (off, off) on the left-bottom corner of the plane P corresponding to the dot ‘off’ on X-axis and to the dot ‘off’ on Y-axis.  Enumerating the rest, I say there are four dots.  If each one of the four dots on the plane P corresponds to a dot on X and to a dot on Y, it wouldn’t be nice of me to ask any less of arrows.

There is a loop on the plane P going from (off, off) to (off, off) corresponding to the loop going from ‘off’ to ‘off’ on X-axis and to the loop going from ‘off’ to ‘off’ on Y-axis.  There is an arrow going from (on, off) to (off, off).  There is an arrow going from (off, on) to (off, off).  There is an arrow going from (on, on) to (off, off).  All these arrows descending into a dot reminding me of a phase portrait (of cell death) I was proud of for a moment or two.

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From → Note to self

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