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Do amateurs ever solve outstanding mathematical problems? Professional mathematicians are aware that almost every new idea they have about a mathematical problem has already occurred to others. Any really new idea must have some feature that explains why no one has thought of it before [TM155 or search for “thatsmaths” at irishtimes.com].
Pierre de Fermat and Srinivasa Ramanujan, two brilliant “amateur” mathematicians.
Continue reading ‘Discoveries by Amateurs and Distractions by Cranks’
The value of the recurring decimal 0.999999 … is a popular topic of conversation amongst amateur mathematicians of various levels of knowledge and expertise. Some of the discussions on the web are of little value or interest, but the topic touches on several subtle and deep aspects of number theory.
[Image Wikimedia Commons]
Continue reading ‘Really, 0.999999… is equal to 1. Surreally, this is not so!’
The Pythagoreans believed that the planets generate sounds as they move through the cosmos. The idea of the harmony of the spheres was brought to a high level by Johannes Kepler in his book Harmonices Mundi, where he identified many simple relationships between the orbital periods of the planets [TM154 or search for “thatsmaths” at irishtimes.com].
Artist’s impression of the Trappist-1 planetary system. Image from https://www.eso.org/public/images/eso1805b/
Kepler’s idea was not much supported by his contemporaries, but in recent times astronomers have come to realize that resonances amongst the orbits has a crucial dynamical function. Continue reading ‘Trappist-1 & the Age of Aquarius’
Surfaces of positive curvature (top), negative curvature (middle) and vanishing curvature (bottom) [image credit: NASA].
characterizes the intrinsic geometry of a surface.
Continue reading ‘Gaussian Curvature: the Theorema Egregium‘
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.
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.
Continue reading ‘The 3 : 2 Resonance between Neptune and Pluto’
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].
Continue reading ‘Random Numbers Plucked from the Atmosphere’
