ThatsMaths is a blog by Peter Lynch, an emeritus professor of the University College Dublin’s School of Mathematics and Statistics. Many of the posts on the blog are articles that Lynch has written for the Irish Times. Please join me on a tour of some exciting ThatsMaths posts from the last several months.
In the early 1900’s, Dr. Muriel Bristol told two statisticians — Ronald Fisher and William Roach — that she could taste the order in which ingredients were added to tea.
“Shortly after Fisher had moved to Rothamstead Research Station in 1919, he poured a cup of tea and offered it to Bristol. She declined, saying that she preferred the milk to be poured first. The arrogant young Fisher scoffed at this, insisting that it could not possibly make any difference, but Bristol maintained her stance, assuring him that she would always know the difference. Overhearing this exchange, another scientist, William Roach said, ‘Let’s test her,'” Lynch wrote.
He describes how statistics and combinatorics were used to determine if Bistol could actually taste the order in which ingredients were added to her cup of tea.
This post covers the Jordan Curve Theorem (including an extension of the theorem to higher dimensions), the traveling salesman problem and their intersections with art.
“The Jordan Curve Theorem states that every simple closed curve, no matter how complicated or convoluted, divides the plane into two regions, an inside and an outside. The theorem appears so trivial that it does not require a proof. But results like this can be much more profound than a first glance might suggest and, on occasions, things that appear obvious can turn out to be false,” Lynch wrote.
He then discusses Bernhard Bolzano’s work. (Lynch has also written a separate post focusing on Bolzano’s life and work.) “He claimed that, for a closed loop in a plane, a line connecting a point enclosed by the loop (inside) to a point distant from it (outside) must intersect the loop. This seems obvious enough, but Bolzano realized that it was a non-trivial problem,” Lynch wrote.
“For general curves it is quite difficult to prove since “simple” curves can have some bizarre properties, such as being jagged everywhere with no definite direction, or as being fractal in nature like the boundary of a snowflake. This makes it difficult to distinguish which points are inside and which are outside. The proof uses advanced ideas from the branch of mathematics known as topology,” he added.
In this post, Lynch wrote about “The Great Wave off Kanagawa” woodcut by Katsushika Hokusai, rogue waves, non-linear modeling and the study of rogue waves in laboratory tanks.
“In recent decades, many enormous sea waves have been observed, removing all doubt about the existence of rogue waves. These waves have heights more than double the surrounding waves. In January 2014, the height of a wave off Killard Point in Co. Clare was measured at almost 30 meters. Although they are quite rare, rogue waves are part of the normal behaviour of the oceans,” Lynch wrote.
Lab studies on rogue waves utilize “mathematical theory, computer simulations, wave-tank experiments and observations” to “determine the critical factors for the formation of rogue waves. Mariners’ lives depend on their ability to avoid them, and new theoretical descriptions may enable us to anticipate their likely occurrence. Despite progress, many questions about rogue waves remain unanswered and research continues. The pay-offs include greater accuracy of wave predictions and saving of money and of lives,” Lynch noted.
This piece describes the mathematical study of pursuit problems, beginning with the work of Pierre Bouguer, who, around the year 1730, produced “the first comprehensive treatment” of the subject, according to Lynch.
“From cheetahs chasing gazelles, through coastguards saving shipwrecked sailors, to missiles launched at enemy aircraft, strategies of pursuit and evasion play a role in many areas of life (and death). From pre-historic times we have been solving such pursuit problems. The survival of our early ancestors depended on their ability to acquire food. This involved chasing and killing animals, and success depended on an understanding of relative speeds and optimal pursuit paths,” he wrote.
The rest of the piece focuses on cyclic pursuit problems (more specifically, the N-bug problem).
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