Resonant Vibrations from Atoms to the Far Horizons of the Cosmos

Panta Rhei — everything flows — said Heraclites, describing the impermanence of the world. He might well have said “everything vibrates”. From sub-atomic particles to the farthest reaches of the cosmos we find oscillations. Vibration is key for aircraft wing, motor engine and optical system design. Ocean tides forced by the Moon and seasonal variations … Continue reading Resonant Vibrations from Atoms to the Far Horizons of the Cosmos →

The Many Schools of Mathematical Thought

Mathematics is used widely, playing a central role in science and engineering and, increasingly, in the social and biological sciences. But users seldom consider the fundamental nature of mathematics. Many cannot improve on the vacuous definition: mathematics is what is done by mathematicians. We could try harder, with something like “mathematics is the language of … Continue reading The Many Schools of Mathematical Thought →

Mileva Marić  and the Special Theory of Relativity

The year 1905 was Albert Einstein’s “miracle year”. In that year, he published four papers in the renowned scientific journal Annalen der Physik. The first, on the photoelectric effect, established the quantum nature of light, and led to the award of a Nobel Prize some 17 years later. The second, on Brownian motion, confirmed the … Continue reading Mileva Marić  and the Special Theory of Relativity →

ENSO: The Oscillating Atmosphere and Ocean

The weather of 2023 was certainly interesting, with broken records in Ireland and around the world. Newspaper articles attributed the cause of the heat waves, droughts, floods and fires to the climate pattern known as El Niño. Less restrained reports claimed that this year’s weather would be even more anomalous [TM253 or search for “thatsmaths” … Continue reading ENSO: The Oscillating Atmosphere and Ocean →

The Decline of the Mayans: a Warning Signal for Us

The rising temperatures of today’s climate are being linked to extreme weather, droughts, floods and intense storms, and global food and water supplies are coming under severe stress. While the current changes are unprecedented in their rapidity, climate variations in the past have had devastating consequences. What can we learn from them? [TM252 or search for … Continue reading The Decline of the Mayans: a Warning Signal for Us →

The Sieve of Eratosthenes and a Partition of the Natural Numbers

The sieve of Eratosthenes is a method for finding all the prime numbers less than some maximum value $latex {M}&fg=000000$ by repeatedly removing multiples of the smallest remaining prime until no composite numbers less than or equal to $latex {M}&fg=000000$ remain. The sieve provides a means of partitioning the natural numbers. We examine this partition … Continue reading The Sieve of Eratosthenes and a Partition of the Natural Numbers →

The Logistic Map is hiding in the Mandelbrot Set

The logistic map is a simple second-order function on the unit interval: $latex \displaystyle x_{n+1} = r x_n (1-x_n) \,, &fg=000000$ where $latex {x_n}&fg=000000$ is the variable value at stage $latex {n}&fg=000000$ and $latex {r}&fg=000000$ is the ``growth rate''. For $latex {1 \le r \le 4}&fg=000000$, the map sends the unit interval [0,1] into itself. … Continue reading The Logistic Map is hiding in the Mandelbrot Set →

The Golden Key to Riemann’s Hypothesis

The Riemann Hypothesis Perhaps the greatest unsolved problem in mathematics is to explain the distribution of the prime numbers. The overall ``thinning out'' of the primes less than some number $latex {N}&fg=000000$, as $latex {N}&fg=000000$ increases, is well understood, and is demonstrated by the Prime Number Theorem (PNT). In its simplest form, PNT states that … Continue reading The Golden Key to Riemann’s Hypothesis →

The Logistic Map: a Simple Model with Rich Dynamics

Suppose the population of the world $latex {P(t)}&fg=000000$ is described by the equation $latex \displaystyle \frac{\mathrm{d}P} {\mathrm{d}t} = a P \,. &fg=000000$ Then $latex {P(t)}&fg=000000$ grows exponentially: $latex {P(t) = P_0 \exp(at)}&fg=000000$. This was the nightmare prediction of Thomas Robert Malthus. Taking a value $latex {a=0.02\ \mathrm{yr}^{-1}}&fg=000000$ for the growth rate, we get a doubling … Continue reading The Logistic Map: a Simple Model with Rich Dynamics →

The Axiom of Choice: Shoes & Socks and Non-constructive Proofs

Recall Euclid's proof that there is no limit to the list of prime numbers. One way to show this is that, by assuming that some number $latex {p}&fg=000000$ is the largest prime, we arrive at a contradiction. The idea is simple yet powerful. A Non-constructive Proof Suppose $latex {p}&fg=000000$ is prime and there are no … Continue reading The Axiom of Choice: Shoes & Socks and Non-constructive Proofs →

Earth’s Digital Twins can help us to avert Disaster

Imagine another Earth, just like ours, but running a year ahead. Observing it, we could foretell events over the coming weeks or months, and take action to avoid catastrophes. There is no such planet! Even if there were, conditions there would diverge rapidly from ours, so it would provide no guidance on our future. But … Continue reading Earth’s Digital Twins can help us to avert Disaster →

Digital Signatures using Edwards Curves

A digital signature is a mathematical means of verifying that an e-document is authentic, that it has come from the claimed sender and that it has not been tampered with or corrupted during transit. Digital signatures are a standard component of cryptographic systems. They use asymetric cryptography that is based on key pairs, consisting of … Continue reading Digital Signatures using Edwards Curves →

A Memorable Memo: Responding to Over-assiduous Administrators

Anyone who has worked in a large organization, with an over-loaded Administration Division, will sympathise with the actions of two scientists at the Los Alamos National Laboratory (LANL) in issuing a spoof Memorandum. They had become frustrated with the large number of mimeographed notes circulated by Administration and Services, or A&S, ``to keep laboratory members … Continue reading A Memorable Memo: Responding to Over-assiduous Administrators →

Hamilton’s Semaphore Code and Signalling System

Sir William Rowan Hamilton (1805-1865) was Ireland's most ingenious mathematician. When he was just fifteen years old, Hamilton and a schoolfriend invented a semaphore-like signalling system. On 21 July 1820, Hamilton wrote in his journal how he and Tommy Fitzpatrick set up a mark on a tower in Trim and were able to view it … Continue reading Hamilton’s Semaphore Code and Signalling System →

Sixth Irish History of Mathematics (IHoM) Conference

I attended the sixth conference of the Irish History of Mathematics (IHoM) group at Maynooth University yesterday (Wednesday 30th August 2023). What follows is a personal summary of the presentations. This summary has no official status. If speakers or attendees spot any errors, please let me know and I will correct them. [1] After a … Continue reading Sixth Irish History of Mathematics (IHoM) Conference →

Maths in the Time of the Pharaohs

Why would the Ancient Egyptians have any interest in or need for mathematics? There are many reasons. They had a well-organised and developed civilisation extending over millennia. Science and maths must have played important or even essential roles in this culture. They needed measurement for land surveying and for designing irrigation canals, arithmetic for accounting … Continue reading Maths in the Time of the Pharaohs →

Margules’ Tendency Equation and Richardson’s Forecast

During World War One, long before the invention of computers, the English Quaker mathematician Lewis Fry Richardson devised a method of solving the equations and made a test forecast by hand. The forecast was a complete failure: Richardson calculated that the pressure at a particular point would rise by 145 hPa in 6 hours. This … Continue reading Margules’ Tendency Equation and Richardson’s Forecast →

The Sizes of Sets

The sizes of collections of objects may be determined with the help of one or other of two principles. Let us denote the size of a set $latex {A}&fg=000000$ by $latex {\mathfrak{Size}(A)}&fg=000000$. Then AP: Aristotle's Principle. If $latex {A}&fg=000000$ is a proper subset of $latex {B}&fg=000000$, then $latex {\mathfrak{Size}(A) < \mathfrak{Size}(B)}&fg=000000$. CP: Cantor's Principle. $latex … Continue reading The Sizes of Sets →

Vertical or Horizontal Slices? Riemann and Lebesgue Integration.

For simple sets, we have geometric length, area and volume. But how can we establish these dimensions for complicated curves, areas and volumes. Integral calculus provides a powerful tool for answering such questions. The area $latex {A}&fg=000000$ between the curve $latex {y=y(x)}&fg=000000$ and the $latex {x}&fg=000000$-axis is $latex \displaystyle A = \int_{x_1}^{x_2} y(x) \mathrm{dx} \,. &fg=000000$ … Continue reading Vertical or Horizontal Slices? Riemann and Lebesgue Integration. →

Literomathic Synergy for Bloomstime

Bloomsday approaches, so let us re-Joyce once more. In Ulysses, Joyce makes numerous allusions to mathematical matters. On previous Bloomsdays, ThatsMaths has written about non-Euclidean geometry, Bloom’s efforts to square the circle, a possible encounter between Joyce and the famed logician Ernst Zermelo and the fractal complexity of Finnegans Wake  [TM246 or search for “thatsmaths” at irishtimes.com]. … Continue reading Literomathic Synergy for Bloomstime →

Herman Melville and Ishmael’s Cycloid

Many authors use mathematical metaphors with great effect. A recent book, “Once Upon A Prime” by Sarah Hart, describes the wondrous connections between mathematics and literature. As a particular example, she discusses the relevance of the cycloid curve in the work of Herman Melville. The book Moby-Dick, first published in 1851, opens with the words … Continue reading Herman Melville and Ishmael’s Cycloid →

The Waffle Cone and a new Proof of Pythagoras’ Theorem

Jackson an' Johnson / Murphy an' Bronson / One by one dey come / An' one by one to dreamland dey go. [From Carmen Jones.  Lyrics: Oscar Hammerstein] Two young high-school students from New Orleans, Ne’Kiya Jackson and Calcea Johnson, recently presented a new proof of the Pythagorean theorem at a meeting of the American … Continue reading The Waffle Cone and a new Proof of Pythagoras’ Theorem →

The Potency of Pattern: Mind the Gap

In his book A Mathematician’s Apology, leading British mathematician G H Hardy wrote “A mathematician, like a painter or poet, is a maker of patterns.” He observed that the mathematician's patterns, like the painter's or the poet's, must be beautiful; beauty is the acid test  [TM245 or search for “thatsmaths” at irishtimes.com]. Everyone is familiar with the concept of … Continue reading The Potency of Pattern: Mind the Gap →

Broken Symmetry and Atmospheric Waves, 2

Part II: Stationary Mountains and Travelling Waves Atmospheric flow over mountains can generate large-scale waves that propagate upwards. Although the mountains are stationary(!), the waves may have a component that propagates towards the west. In this post, we look at a simple model that explains this curious asymmetry. Earth's Rotation and Symmetry Breaking If the … Continue reading Broken Symmetry and Atmospheric Waves, 2 →

Broken Symmetry and Atmospheric Waves, 1

Part I: Vertically propagating Waves and the Stratospheric Window Symmetry is a powerful organising principle in physics. It is a central concept in both classical and quantum mechanics and has a key role in the standard model. When symmetry is violated, interesting things happen. The book Shattered Symmetry by Pieter Thyssen and Arnout Ceulemans discusses … Continue reading Broken Symmetry and Atmospheric Waves, 1 →

DLWP: A New Age of Weather Forecasting

Before the age of computers, weather forecasters analysed observations plotted on paper charts, drew isobars and other features and — based on their previous knowledge and experience — constructed charts of conditions at a future time, often one day ahead. They combined observational data and rules of thumb based on physical principles to predict what … Continue reading DLWP: A New Age of Weather Forecasting →

Amusical Permutations and Unsettleable Problems

In a memorial tribute in the Notices of the American Mathematical Society (Ryba, et al, 2022), Dierk Schleicher wrote of how he convinced John Conway to publish a paper, ``On unsettleable arithmetical problems'', which included a discussion of the Amusical Permutations. This paper, which discusses arithmetical statements that are almost certainly true but likely unprovable, … Continue reading Amusical Permutations and Unsettleable Problems →

Christopher Wren and the Cycloid

The remarkable polymath Christopher Wren died in March 1723, just 300 years ago. Sarah Hart, Professor of Geometry at Gresham College, recently presented a lecture, The Mathematical Life of Sir Christopher Wren; a video of her presentation in available online (see sources below). The illustration above is from the Gresham College website. Christopher Wren In … Continue reading Christopher Wren and the Cycloid →