Surfaces of positive curvature (top), negative curvature (middle) and vanishing curvature (bottom) [image credit: NASA].
characterizes the intrinsic geometry of a surface.
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Archive for December, 2018
Gaussian Curvature: the Theorema Egregium
Published December 27, 2018 Occasional ClosedTags: Gauss, Geometry
characterizes the intrinsic geometry of a surface.
Consider a Spherical Christmas Tree
Published December 20, 2018 Irish Times ClosedTags: Algorithms, Topology
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.
The 3 : 2 Resonance between Neptune and Pluto
Published December 13, 2018 Occasional ClosedTags: Astronomy
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.
Unstable (left) and stable (right) orbital configurations.
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Random Numbers Plucked from the Atmosphere
Published December 6, 2018 Irish Times ClosedTags: Number Theory
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].
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ThatsMaths 