Take the 2-minute tour ×




share|improve this question




37 Answers 37


share|improve this answer

Characteristic p commutative algebra leads naturally to the construction of various continuous functions on [0,1]^m that have beautiful self-similarity properties; for explication and some pictures see:

Pedro Teixeira, Syzygy gap fractals--I, arXiv 1008.0583
share|improve this answer

How about a function f: f(f(x)) = exp(x).

share|improve this answer

How about the function given by the Banach-Tarski paradox? This maps a ball into two copies of the same size ball, and is composed of isometries on subsets of $\mathbb{R}^3$.

share|improve this answer

The formula for the nth term in the Fibonacci sequence


This is interesting because it is a non-recursive expression for the Fibonacci sequence and also because it involves the golden ratio.

share|improve this answer

Just a simple construction to illustrate Nate Eldredge's answer about functions with dense graphs. Pick any $\mathbb{R}$-vector space E with a norm. On E, choose a non-continuous linear form $L: E \to \mathbb{R}$; now this can only be done if $\dim(E)=\infty$, of course.

Then, pick y such that $L(y)=1$, and let $T: E \to E$ be defined by $Tx=x-L(x)y$. Then obviously T maps E onto the kernel of L; it is not difficult to prove that $\ker (L)$ must be dense in E for any non-continuous L (the two conditions are even equivalent), and thus the graph of T must be dense in $E \times E$.

share|improve this answer

I like the beauty and mysticism of Euler's identity:

$$f(\theta) = e^{i\theta} = \cos\theta + i \sin\theta$$

share|improve this answer
