Pi Approximation

  Pi approximation (Archimedes, 3rd century BCE):    ²²³⁄₇₁ < π <  ²²⁄₇

Eric Weisstein's World of Science

Eric Weisstein's World of Science contains entries on Astronomy, Chemistry, Physics, Mathematics, and Biographies.

Georg Cantor

Mathematics: German mathematician Georg Cantor (1845-1918) invented set theory (Mengenlehre). He also realized that there are different types of infinity. Cantor suffered from manic-depressive illness and ended up in mental institution.

Measuring the World: Gauß and Humboldt

Audiobook (CD): Daniel Kehlmann: Die Vermessung der Welt (2005, engl.: Measuring the World). Read in German by Ulrich Matthes. 5 CDs, ca. 345 min. Narrates and compares the biographies of two contemporaries, the natural philosopher Alexander von Humboldt and the mathematician Carl Friedrich von Gauß. "This is the history of two scientists who could not be more different, but who both have the goal to understand the world". Not comprehensive biographies, but nicely narrated with some hidden treasures. Overall rating 7 out of 10. 

Fermat's Last Theorem

Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than two. The equation is an expansion of the Pythagorean theorem where n = 2 and certain solutions can be found (e.g. a = 3, b = 4, c = 5). In 1637, Pierre de Fermat conjectured that this equation was not true for any positive integer greater than two. In 1995, British mathematician Andrew Wiles presented a final proof for the theorem. His story has been described by the BBC (video below) and is also discussed among experts in BBC radio show In Our Time.

Ulam Spiral

The Ulam spiral is a method of visualizing the distribution of prime numbers. Discovered by Stanislaw Ulam in 1963. Construct by writing down a rectangular grid of numbers, starting with 1 at the center and spiraling out. Highlight prime numbers. Yields, in part, diagonal line patterns.

Ulam spiral. Image source: Wikipedia.