A Klein bottle is a surface with a very strange property. A surface is any object that is locally 2-dimensional;...
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Sphere Eversions
If you take a loop of string in the plane and place an arrow along it pointing clockwise, is it...
Continue reading...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...
Continue reading...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...
Continue reading...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...
Continue reading...Connected Sums
A surface is any object which is locally like a piece of the plane. A sphere, a projective plane, a Klein bottle,...
Continue reading...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...
Continue reading...Spherical Geometry
Remember high school geometry? The sum of the angles of a planar triangle is always 180 degrees or Pi radians. However, triangles...
Continue reading...Fundamental Group
You are abducted by space aliens and dumped, blindfolded, on a strange asteroid! Removing your blindfold, you decide to play...
Continue reading...Spherical Bands
Take any sphere and slice it up by parallel knives 1 inch apart. The areas of the spherical bands between the cuts...
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