# sphere

## Klein Bottle

A Klein bottle is a surface with a very strange property. A surface is any object that is locally 2-dimensional;...

## Sphere Eversions

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

## High-Dimensional Spheres in Cubes

How good is your intuition in high dimensions? Take a square and divide it into its four quadrants. Inscribe a circle...

## Spherical Pythagorean Theorem

Did you know there is a version of the Pythagorean Theorem for right triangles on spheres? First, let’s define precisely what we...

## Hairy Ball Theorem

Another fun theorem from topology is the Hairy Ball Theorem. It states that given a ball with hairs all over it, it...

## Connected Sums

A surface is any object which is locally like a piece of the plane. A sphere, a projective plane, a Klein bottle,...

## Surface Area of a Sphere

The area of a disk enclosed by a circle of radius R is Pi*R2. The formula for the circumference of a...

## Spherical Geometry

Remember high school geometry? The sum of the angles of a planar triangle is always 180 degrees or Pi radians. However, triangles...