Gaussian Curvature: the Theorema Egregium

One of greatest achievements of Carl Friedrich Gauss was a theorem so startling that he gave it the name Theorema Egregium or outstanding theorem. In 1828 he published his ``Disquisitiones generales circa superficies curvas'', or General investigation of curved surfaces. Gauss defined a quantity that measures the curvature of a two-dimensional surface. He was inspired by … Continue reading Gaussian Curvature: the Theorema Egregium →

Consider a Spherical Christmas Tree

A minor seasonal challenge is how to distribute the fairy lights evenly around the tree, with no large gaps or local clusters. Since the lights are strung on a wire, we are not free to place them individually but must weave them around the branches, attempting to achieve a pleasing arrangement. Optimization problems like this … Continue reading Consider a Spherical Christmas Tree →

The 3 : 2 Resonance between Neptune and Pluto

For every two orbits of Pluto around the Sun, Neptune completes three orbits. This 3 : 2 resonance has profound consequences for the stability of the orbit of Pluto. The Harmony of the Spheres Pythagoras based his musical analysis on two ratios: the octave 2 : 1 and the perfect fifth 3 : 2. He … Continue reading The 3 : 2 Resonance between Neptune and Pluto →

Random Numbers Plucked from the Atmosphere

Randomness is a slippery concept, defying precise definition. A simple example of a random series is provided by repeatedly tossing a coin. Assigning “1” for heads and “0” for tails, we generate a random sequence of binary digits or bits. Ten tosses might produce a sequence such as 1001110100. Continuing thus, we can generate a … Continue reading Random Numbers Plucked from the Atmosphere →