## It’s better with ‘theory’

It seems to me that adding the word theory tends to increase my interest in a subject.  Consider puddles, they’re fun…but ‘Puddle Theory’ what’s that?  Sounds intriguing.

Definition: An n-puddle is a n-dimensional manifold with boundary such that the boundary is the union of a sky and ground.  The ground is a n-1 manifold with boundary equal to the n-2 boundary of the sky.  The sky boundary resides within a hyperplane of dimension n-1.

We can use standard geometric structures on $\mathbb{R}^n$ because in general an n-puddle will not be smooth.

Definition: Define a splash to be a family of functions $F_{\alpha}:P\to P$ indexed by $[0,1]$.  $F_{\alpha}$ need not be continuous, in fact will seldom be so.

It’s possible for $P$ to become disconnected during the splash.  Splashes can be divided into volume-preserving and non-volume-preserving splashes.  These are called dry and wet respectably.

Definition: A splash is dry if $\lim_{\alpha\to 1} F_{\alpha}\to id_P$.

It’s possible to compose a series of splashes, $F^1_{\alpha_1}, F^2_{\alpha_2},...F^k_{\alpha_k},...$ $k\in\mathbb{N}$   such that $F_1^k = F_0^{k+1}$.  It was discovered by A. Child that for any such sequence there exists an integer N such that for all $n>N$ $F^n_{\alpha}$ is the empty splash. This is known as Child’s Theorem.  We leave this as an exercise.

Fascinated by the way mathematics and physics interact, captivated by visual and tactile mathematics and hoping to become a better expositor of these things is why I blog...occasionally...when I remember.
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### 2 Responses to It’s better with ‘theory’

1. sheabrowne says:

I love this! I’d like to see A. Child’s theory of Quantum Gravity…might be a Nobel Prize in there somewhere! BTW, I’ve pick up a book on basic Algebra (rings, groups etc.) for undergraduates … I’ve missed out on a lot in my Physics education, but it’s never too late to catch up…

• Thanks Shea! I agree all we can do is continue to learn.