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

Continue reading...# analysis

## 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...

Continue reading...## Sphere Eversions

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

Continue reading...## Taylor-made Pi

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

Continue reading...## Banach-Tarski Paradox

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

Continue reading...## 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....

Continue reading...## 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”...

Continue reading...## Ellipsoidal Paths

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

Continue reading...## Whiffle Ball

Does a ball take longer to come down than go up, or does it take the same amount of time...

Continue reading...## Area of a Circle or Regular Polygon

There’s a nice way to see why the formula for the area of a circle of radius R is: Pi...

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