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 because in general an n-puddle will not be smooth.

**Definition**: Define a *splash* to be a family of functions indexed by . need not be continuous, in fact will seldom be so.

It’s possible for 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 .

It’s possible to compose a series of splashes, such that . It was discovered by A. Child that for any such sequence there exists an integer N such that for all is the *empty splash*. This is known as **Child’s Theorem**. We leave this as an exercise.

### Like this:

Like Loading...

*Related*

## About because0fbeauty

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.

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.