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.”

Image from Lehninger Principles of Biochemistry, 2008: Nelson, Lehninger; and Cox. © W H Freeman.

Chirality is important in many sciences, most notably in chemistry. In mathematics, chirality is the property of a curve, surface or solid such that it is not identical to its mirror image. The term is found in topology and, in particular, in knot theory.

A helix may be left-handed or right-handed (see Figure). The DNA molecule is formed of right-handed helices, although it is not uncommon to see diagrams that use the left-handed versions. There are also left-handed and right-handed versions of the Möbius band.

Chiral and Achiral Knots

A knot is achiral if it can be continuously deformed into its mirror image. The unknot and the figure-of-eight knot are achiral. The unknot is essentially a continuously distorted circle (technically, via an ambient isotopy).

Transformation of a figure-of-eight knot into its mirror image.

The Figure above illustrates how a figure-of-eight knot can be continuously transformed into its mirror image, demonstrating that it is achiral.

The trefoil knot is chiral: there are left-handed and right-handed trefoils (see Figure below) and it is impossible to transform one into another by a continuous deformation.

These two trefoil knots are not equivalent; the trefoil is chiral [image from Wikimedia Commons].

A Knotty Puzzle

Imagine a bug confined to move within a thin flexible pipe closed by joining its ends. Assume that the pipe is in deep space. The bug can measure speed and acceleration; this is its only source of data.

Can the bug deduce the shape of the pipe and determine whether or not it is knotted? Comments to thatsmaths@gmail.com (Subject: Bug in pipe), preferably before the end of November 2021.

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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].

Continue reading ‘Émilie Du Châtelet and the Conservation of Energy’

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.

Continue reading ‘Cantor’s Theorem and the Unending Hierarchy of Infinities’

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].

Continue reading ‘Topsy-turvy Maths: Proving Axioms from Theorems’


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