## Thursday, November 24, 2016

### The square root of Andy Pandy

You might argue that Andy Pandy isn't the kind of thing which could have a square root. You're clearly the kind of person who strongly believes in typed programming languages. I'm the last person to deny the utility of polymorphic typing - caught so many errors that way - but you'll never wean me from my love of Lisp: (sqrt AndyPandy).

Even so, trying to find an x such that x2 = Andy Pandy is a stretch. We plainly need to identify an operation 'multiply' over a set which includes Andy Pandy, who is not often known as a number.

In modern mathematics we define operations like 'plus' and 'times' axiomatically. In some sense both these operations are the same: group operations over different sets (integers; non-zero rationals being one pair of examples).

Can we instantiate the group axioms over a set involving Andy Pandy?

The operator 'marries and produces one child ' which we'll call * might do it.
Mother * Father = Child.
This is a very modern relationship permitting terms like:
• Mother * Mother
• Father * Father
• Mother * Child
• ...
combining both Greek myth and modern genetic engineering.

The identity e produces clones (e is a kind of null-person):
Mother * e = Mother.
Each individual has a unique inverse, for example,
Mother * Mother-1 = e
which implies everyone has one entity with whom they are infertile.

With this representation of the group axioms, we seek an X such that
X * X = Andy Pandy.
Here's the group table, the cyclic group of order three, with Mother as generator.

So there you have it. The square root of Andy Pandy is her parthenogenetic mother.

Who knew?*

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* "... that Andy Pandy was transgender .."

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† Andy Pandy's own procreative propensities? Let's not go there, children!

1. There is quite a lot of algebra and group theory in Genetics. But I am not sure that this paper on Andy Pandy is yet ready for publication...

1. I would like to think that I got the genetics right: X chromosomes, one copy deactivated in each somatic cell. The post is an extended bait-and-switch.

And of course there is only one group of order 3, up to isomorphism, so I think my conclusions are incontrovertible.

2. Actually there is another square root present here too. ZZ3 has a representation:
e=1,
mother = exp(2PIi/3),
mother^2 = Andy Pandy = exp (4PIi/3) = exp(-2PIi/3)
= mother^(-1)
These are three points 120 degrees apart on a unit circle in the Complex Plane. Hence we also have sqrt(-1) = i also involved. Perhaps there is a new departure for Mathematical Genetics here after all....

3. Z3 sure gets around!