# analysis

## How many Rationals?

How many rational numbers are there? Yes, infinitely many, I hear you say. But how large is that infinity? Are...

## Volume of a Ball in N Dimensions

The unit ball in Rn is defined as the set of points (x1,…,xn) such that x12 + … + xn2 <= 1. What...

## Sphere Eversions

If you take a loop of string in the plane and place an arrow along it pointing clockwise, is it...

After learning about the Taylor series for 1/(1+x) in calculus, you can find an interesting expression for Pi very easily. Start with...

Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using...

## i to the i is a Real Number

If you are familiar with complex numbers, the “imaginary” number i has the property that the square of i is -1....

## Face Derivatives and Computer Vision

One challenge in robotics is the problem of computer vision: how do you program a computer to interpret and “understand”...

## Ellipsoidal Paths

Given an ellipse, and a smaller ellipse strictly inside it, start at a point on the outer ellipse, and in a...