#5: The Cantor Set and the Cantor Function

Posted on November 27, 2012 by Sean Li

The Cantor set $C$ is a fractal that is obtained by repeatedly removing the middle third of a segment. Start with the closed interval $[0,1]$ . Remove the open interval $(\frac{1}{3},\frac{2}{3})$ to obtain $[0,\frac{1}{3}] \cup [\frac{2}{3},1]$ , i.e. two disjoint closed segments. Remove the middle thirds of those two segments, and you end up with four disjoint segments. After infinitely many steps, the result is called the Cantor set. The diagram below is only an approximation after a finite number of steps.

Continue reading →

#3: Rings

Posted on November 6, 2012 by Sean Li

The point of this post is to provide examples and non-examples for the following ring classes:

Ring
Commutative Ring
Integral Domain
Integrally Closed Domain
Unique Factorization Domain (UFD)
Principal Ideal Domain (PID)
Euclidean Domain
Field

Continue reading →

#2: Epic Morphisms

Posted on November 1, 2012 by Sean Li

Given that the name of this site is Epic Math, I felt I should write about a topic where the word epic is a technical term! It is hardly surprising that such a term exists, as many otherwise non-mathematical words such as almost, simple, open, connected, regular, normal, field, ring, onto, map, twin, lucky, and even sexy have technical definitions.

Category Theory

The term epic is found in category theory, which is an extremely abstract branch of mathematics that formally deals with many other fields of math. It is so strange that some mathematicians have labeled it “abstract nonsense.”

Continue reading →