Archive for December, 2018

Gaussian Curvature: the Theorema Egregium

ShapeOfUniverse

Surfaces of positive curvature (top), negative curvature (middle) and vanishing curvature (bottom) [image credit: NASA].

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 his work on geodesy, surveying and map-making, which involved taking measurements on the surface of the Earth. The total curvature — or Gaussian curvature — depends only on measurements within the surface and Gauss showed that its value is independent of the coordinate system used. This is his Theorema Egregium. The Gaussian curvature {K} characterizes the intrinsic geometry of a surface.

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Consider a Spherical Christmas Tree

ChristmasTreeLights

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 occur throughout applied mathematics [TM153 or search for “thatsmaths” at irishtimes.com].

Trees are approximately conical in shape and we may assume that the lights are confined to the surface of a cone. The peak, where the Christmas star is placed, is a mathematical singularity: all the straight lines that can be drawn on the cone, the so-called generators, pass through this point. Cones are developable surfaces: they can be flattened out into a plane without being stretched or shrunk.

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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.

Nep-Plu-Orbits-AB

Unstable (left) and stable (right) orbital configurations.

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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 sequence of any length having no discernible pattern [TM152 or search for “thatsmaths” at irishtimes.com].

Lightning-01

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