Alligators, Dinosaurs, and Scaling

I was walking around a book store in Fremont and found an elementary math education book.  While flipping through the book, I found the following neat problem.

Suppose you find a dinosaur skeleton that looks like an alligator, except that it is twice as big.  A typical alligator weighs 500 pounds.  Estimate how much the dinosaur weighed?

The answer is not 1000 pounds!  This is not how things scale.  If I double a length, areas increase by 4, and volumes increase 8 times (imagine doubling the side length of a square or cube).  In \(d\) dimensions, if lengths are scaled by \(s\), volumes will scale by \(s^d\).

Weight is proportional to volume, and the dinosaur had \(2^3 = 8\) times the volume as the alligator.  Thus the dinosaur weighed 4000 pounds.  Another way to see why 1000 pounds is incorrect is to consider the following: if I scale the alligator by 2, not only is the length doubled, but the width and height are also doubled.  Three dimensions doubled, which leads to an 8-fold increase in volume.

We can use these ideas to determine the dimension of a dataset.  High-dimensional data often lies in a low-dimensional manifold.  The dimension of this manifold is the “true” dimension of the data.  Imagine I make a ball around one of the data points, and count the number of points in the ball.  If I double the radius of the ball, the number of points will increase by \(2^d\), where \(d\) is the “true” dimension of the data.  We can estimate \(d\) with

\[\hat{d} = \log_2 \left( \frac{\text{# points in big ball}}{\text{# points in small ball}} \right).\] Written on April 13th, 2018 by Scott Roy