Archive for November, 2021

Chiral and Achiral Knots

An object is chiral if it differs from its mirror image. The favourite example is a hand: our right hands are reflections of our left ones. The two hands cannot be superimposed. The term chiral comes from {\chi\epsilon\rho\iota}, Greek for hand. If chirality is absent, we have an achiral object.

According to Wikipedia, it was William Thomson, aka Lord Kelvin, who wrote:

“I call any geometrical figure, or group of points, ‘chiral‘, and say that it has chirality if its image in a plane mirror  …  cannot be brought to coincide with itself.”

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Émilie Du Châtelet and the Conservation of Energy

A remarkable French natural philosopher and mathematician who lived in the early eighteenth century, Émilie Du Châtalet, is generally remembered for her translation of Isaac Newton’s Principia Mathematica, but her work was much more than a simple translation: she added an extensive commentary in which she included new developments in mechanics, the most important being her formulation of the principle of conservation of energy [TM223 or search for “thatsmaths” at irishtimes.com].

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Cantor’s Theorem and the Unending Hierarchy of Infinities

The power set of the set {x,y,z}, containing all its subsets, has 2^3=8 elements. Image from Wikimedia Commons.

In 1891, Georg Cantor published a seminal paper, U”ber eine elementare Frage der Mannigfaltigkeitslehren — On an elementary question of the theory of manifolds — in which his “diagonal argument” first appeared. He proved a general theorem which showed, in particular, that the set of real numbers is uncountable, that is, it has cardinality greater than that of the natural numbers. But his theorem is much more general, and it implies that the set of cardinals is without limit: there is no greatest order of infinity.

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Topsy-turvy Maths: Proving Axioms from Theorems

Mathematics is distinguished from the sciences by the freedom it enjoys in choosing basic assumptions from which consequences can be deduced by applying the laws of logic. We call the basic assumptions axioms and the consequent results theorems. But can things be done the other way around, using theorems to prove axioms? This is a central question of reverse mathematics  [TM222 or search for “thatsmaths” at irishtimes.com].

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