Who Invented the Digital Computer? There’s No Simple Answer.

The simple question of who invented the digital computer has many possible answers. Several strands of development — both theoretical and practical — converged around 1950, and no single inventor can be identified. Before that time, numerous mechanical calculating machines were devised and constructed. These greatly facilitated the preparation of mathematical tables, crucial for engineering … Continue reading Who Invented the Digital Computer? There’s No Simple Answer. →

The Hindu-Arabic Numerals: a Blessing for Modern Civilisation

Directly opposite the RDS in Dublin, on the site of the former AIB headquarters, stand two impressive new office blocks separated by a plaza, Fibonacci Place. Leonardo Bonacci — known as Fibonacci, the son of Bonaccio — lived from 1170 to about 1245 AD. It is he who we must thank for bringing us the … Continue reading The Hindu-Arabic Numerals: a Blessing for Modern Civilisation →

Music and Maths are Inextricably Intertwined.

A crucial link between mathematics and music is nicely illustrated by the Tonnetz, a geometric diagram representing the harmonic relationships between the notes of the musical scale. An early version of the Tonnetz appeared in Leonhard Euler’s book Tentamen novae theoriae musicae (A new theory of music), published in 1739 [TM274 or search for “thatsmaths” at irishtimes.com]. A modern … Continue reading Music and Maths are Inextricably Intertwined. →

John von Neumann: Computer Wizard and AI Visionary

With unique and outstanding achievements in mathematics and science, John von Neumann was on an intellectual level far above us; the oft-misused term genius truly applies to him. Von Neumann, brilliant in both pure and applied mathematics, was a towering figure of 20th century science. As a child, he demonstrated a prodigious memory and computational … Continue reading John von Neumann: Computer Wizard and AI Visionary →

Finding a Horseshoe on the Beaches of Rio

In 1966, American mathematician Steve Smale was awarded a Fields Medal, a kind of Nobel Prize for mathematics. At a press conference at the International Congress of Mathematicians in Moscow, Smale attacked both US and Russian foreign policy. He was vehemently opposed to military aggression by the two super-powers. His pacifist position resulted in serious … Continue reading Finding a Horseshoe on the Beaches of Rio →

Going for Gold: Henri Poincaré and Solar System Stability

In his classical textbook on mechanics, Edmund Whittaker described the three-body problem as “the most celebrated of all dynamical problems”. From 1906 to 1912, Whittaker was Andrews Professor at Trinity College Dublin and Royal Astronomer of Ireland. The three-body problem is to determine the motion of three massive bodies moving in space under mutual gravitational … Continue reading Going for Gold: Henri Poincaré and Solar System Stability →

The Future of Physics may be Surreal

Imagine the Earth were to shrink to the size of a marble. We might be in trouble, but the planet would continue its smooth course around the Sun while the Moon would maintain its orbit, circling Earth once a month. Isaac Newton proved that the Earth’s gravitational pull would be the same even if all … Continue reading The Future of Physics may be Surreal →

The Emergence of Spring

At 9.01AM today, the Sun crosses the Earth’s equator, marking the vernal equinox, the date when the direction of sunrise is due east and sunset due west. The Earth’s axis is at right angles to the line from Sun to Earth, the Sun is directly over the equator and the lengths of day and night … Continue reading The Emergence of Spring →

Quantum Physics, a Century Old, still Passes Understanding

This is the International Year of the Quantum, celebrating quantum science and its many applications. This year marks the centenary of Werner Heisenberg’s seminal work in modern physics. In recent years, there have been dramatic developments in quantum communications, cryptography and computing, and quantum science is key to solving pressing problems in clean energy, climate … Continue reading Quantum Physics, a Century Old, still Passes Understanding →

Surprising Discoveries in the Family Tree of Mathematics

The German word Doktorvater for a doctoral advisor indicates the close relationship between a PhD student and his or her supervisor. The relationship is often pivotal in determining the future career of the student, and the advisor also gains much from the interaction. Just as a genealogical tree can reveal fascinating information, a mathematical family … Continue reading Surprising Discoveries in the Family Tree of Mathematics →

The Surprising Utility of Square Wheels

The London Eye, a vast observation wheel with a diameter of 120 metres, on the south bank of the River Thames, opened in 2000. Resembling an enormous bicycle wheel, it has 32 passenger capsules, each weighing ten tons, attached to its circumference. Each capsule holds 25 people, so up to 800 may be on the … Continue reading The Surprising Utility of Square Wheels →

Fermi Resonance and Climate Change

Carbon dioxide (CO2) is the main culprit in causing our climate to change, but it is only recently that a key property of CO2 has been elucidated. This is linked to a surprising interaction between the molecular vibrations of carbon dioxide: the various modes of vibration of the molecules resonate with each other in a … Continue reading Fermi Resonance and Climate Change →

Joining Forces to Improve Weather Forecasts

Since 1950, there has been a quiet but steady revolution in meteorology, and especially in numerical weather prediction (NWP). The growth in accuracy, range and scope of weather forecasts over the past half-century has been spectacular. As late as the mid-1970s, forecasts were seriously unreliable. The diagram illustrates the inexorable increase in skill of the … Continue reading Joining Forces to Improve Weather Forecasts →

Hamilton’s Dynamics: a Prescient Framework for Quantum Mechanics

While mathematics may be viewed as an abstract creation, its origins lie in the physical world. The need to count animals and share food supplies led to the development of the concept of numbers. With five-fingered hands, we naturally tended to count in tens. Arithmetic methods were needed to allocate land, organize armies and calculate … Continue reading Hamilton’s Dynamics: a Prescient Framework for Quantum Mechanics →

Progress towards a Grand Unified Theory of Mathematics

Science advances by overturning theories, replacing them by better ones. Sometimes, the old theories continue to serve as valuable approximations, as with Newton’s laws of motion [TM260 or search for “thatsmaths” at irishtimes.com]. Sometimes, the older theories become redundant and are forgotten. The theory of phlogiston, a fire-like element released during combustion, and the luminiferous … Continue reading Progress towards a Grand Unified Theory of Mathematics →

The Doppler Effect: Simple but Remarkably Useful

We have all noticed how the horn of a speeding car changes as it approaches: each wave-peak is emitted from a closer point, so the wave is “squeezed” and the pitch increases. As the car recedes, the reverse effect stretches the wave, making it sound lower. The changing pitch of the note is called the … Continue reading The Doppler Effect: Simple but Remarkably Useful →

To See a World in a Grain of Sand

Gaze up at the night sky and you catch a glimpse of infinity. The stars seem numberless; the immensity of the universe is profoundly impressive, leading us to wonder about the nature of our cosmic home. Is space finite or unlimited in extent? If it is bounded, what lies outside? The notion of infinity is … Continue reading To See a World in a Grain of Sand →

Mathematical Fun and Games at Maison Poincaré

This article comes from the Institut Henri-Poincaré (IHP), part of Sorbonne University, in the Latin Quarter of Paris. IHP has no permanent researchers but serves as a venue for mathematical collaborations and organises a series of programmes, seminars, lectures and training courses, welcoming more than 10,000 mathematicians each year. The institute also publishes four international … Continue reading Mathematical Fun and Games at Maison Poincaré →

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 →

Oleksandr Sharkovsky and Chaos Theory

We all know how a simple action at a critical moment can change our lives. Over the past half-century, with the growing evidence of how small changes can lead to dramatic developments, there has been a paradigm shift in science. Earlier attempts to predict the future as if it were determined with certainty have given … Continue reading Oleksandr Sharkovsky and Chaos Theory →

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 →

Maths in Action at the Rugby World Cup

On September 8, I opened The Irish Times to find an A2 Poster with the programme for the Rugby World Cup. The plan showed the twenty teams who must do battle in which ultimate triumph requires survival through the preliminary rounds and victory in quarter-finals, semi-finals and the final climax. We have reached the quarter-finals … Continue reading Maths in Action at the Rugby World Cup →

Music and Maths from Bach to Bacherach

At a fundamental level, music may be described as a train of vibrations in the air. It can be further reduced to numbers — a string of binary digits (bits) that can be stored on a CD or sent around the world in a split second. But music carries enormous emotional content and can stir … Continue reading Music and Maths from Bach to Bacherach →

Dynamic Similarity and the Reynolds Number

Mathematics deals with pure numbers: 1, 2, 3, fractions and more exotic numbers like π. Since π depends on lengths, we might think its value depends on our units. But it is the ratio of the circumference of a circle to its diameter and, as long as both are measured in the same units — … Continue reading Dynamic Similarity and the Reynolds Number →

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 →

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 →

Weather Warnings in Glorious Technicolor

Severe weather affects us all and we need to know when to take action to protect ourselves and our property. We have become familiar with the colourful spectrum of warnings issued by Met Éireann. For several years, Met Éireann has issued warnings of extreme weather. These depend on the severity of the meteorological event and … Continue reading Weather Warnings in Glorious Technicolor →

The Cosmology of the Divine Comedy

If you think poetry and maths are poles apart, think again. Around the sixth century, Indian poet and mathematician Virahanka codified the structure of Sanskrit poetry, formulating rules for the patterns of long and short syllables. In this process, a sequence emerged in which each term is the sum of the preceding two. This is … Continue reading The Cosmology of the Divine Comedy →

Convergence of mathematics and physics

The connexions between mathematics and physics are manifold, and each enriches the other. But the relationship between the disciplines fluctuates between intimate harmony and cool indifference. Numerous examples show how mathematics, developed for its inherent interest in beauty, later played a central role in physical theory. A well-known case is the multi-dimensional geometry formulated by … Continue reading Convergence of mathematics and physics →

The Rich Legacy of Indian Mathematics

For more than three thousand years, mathematics has played an important role in Indian culture. Sometimes it was studied for practical reasons and sometimes for pure intellectual delight. The earliest traces of mathematics are found in the Indus Valley, around 3000 BC. There is clear evidence of a structured system of weights and measures and … Continue reading The Rich Legacy of Indian Mathematics →

From Sub-atomic to Cosmic Strings

The two great pillars of modern physics are quantum mechanics and general relativity. These theories describe small-scale and large-scale phenomena, respectively. While quantum mechanics predicts the shape of a hydrogen atom, general relativity explains the properties of the visible universe on the largest scales. A longstanding goal of physics is to construct a new theory … Continue reading From Sub-atomic to Cosmic Strings →

Making Sound Pictures to Identify Bird Songs

A trained musician can look at a musical score and imagine the sound of an entire orchestra. The score is a visual representation of the sounds. In an analogous way, we can represent birdsong by an image, and analysis of the image can tell us the species of bird singing. This is what happens with … Continue reading Making Sound Pictures to Identify Bird Songs →

The Navigational Skills of the Marshall Islanders

For thousands of years, the Marshall Islanders of Micronesia have been finding their way around a broadly dispersed group of low-lying islands, navigating apparently without effort from one atoll to another one far beyond the horizon. They had no maps or magnetic compass, no clocks, no weather forecasts and certainly no GPS or SatNav equipment … Continue reading The Navigational Skills of the Marshall Islanders →

ICM 2022 — Plans Disrupted but not Derailed

In just three weeks the largest global mathematical get-together will be under way. The opening ceremony of the 2022 International Congress of Mathematicians (ICM) opens on Wednesday 6 July and continues for nine days. Prior to the ICM, the International Mathematical Union (IMU) will host its 19th General Assembly in Helsinki on 3–4 July [TM234 … Continue reading ICM 2022 — Plans Disrupted but not Derailed →

Image Processing Emerges from the Shadows

Satellite images are of enormous importance in military contexts. A battery of mathematical and image-processing techniques allows us to extract information that can play a critical role in tactical planning and operations. The information in an image may not be immediately evident. For example, an overhead image gives no direct information about the height of … Continue reading Image Processing Emerges from the Shadows →

The Whole is Greater than the Part — Or is it?

Euclid flourished about fifty years after Aristotle and was certainly familiar with Aristotle's Logic. Euclid's organization of the work of earlier geometers was truly innovative. His results depended upon basic assumptions, called axioms and “common notions”. There are in total 23 definitions, five axioms and five common notions in The Elements. The axioms, or postulates, … Continue reading The Whole is Greater than the Part — Or is it? →

A Prescient Vision of Modern Weather Forecasting

One hundred years ago, a remarkable book was published by Cambridge University Press. It was a commercial flop: although the print run was just 750 copies, it was still in print thirty years later. Yet, it held the key to forecasting the weather by scientific means. The book, Weather Prediction by Numerical Process, was written … Continue reading A Prescient Vision of Modern Weather Forecasting →

Sources and Scenes of Mathematical Inspiration

Where does new mathematics come from? The great French mathematician Henri Poincaré, a brilliant expositor of the scientific method, described how he grappled for months with an arcane problem in function theory. Exasperated by lack of progress, he went on vacation and forgot about the problem. But, as he was boarding a bus in Caen, … Continue reading Sources and Scenes of Mathematical Inspiration →

Mathematical Equations are our Friends

In his scientific best-seller, A Brief History of Time, Stephen Hawking remarked that every equation he included would halve sales of the book, so he put only one in it, Einstein's equation relating mass and energy, E = mc2. This cynical view is a disservice to science; we should realize that, far from being inimical, … Continue reading Mathematical Equations are our Friends →

Letters to a German Princess: Euler’s Blockbuster Lives On

The great Swiss mathematician Leonhard Euler produced profound and abundant mathematical works. Publication of his Opera Omnia began in 1911 and, with close to 100 volumes in print, it is nearing completion. Although he published several successful mathematical textbooks, the book that attracted the widest readership was not a mathematical work, but a collection of … Continue reading Letters to a German Princess: Euler’s Blockbuster Lives On →

Bernoulli’s Golden Theorem and the Law of Large Numbers

Jakob Bernoulli, head of a dynasty of brilliant scholars, was one of the world’s leading mathematicians. Bernoulli's great work, Ars Conjectandi, published in 1713, included a profound result that he established “after having meditated on it for twenty years”. He called it his “golden theorem”. It is known today as the law of large numbers, … Continue reading Bernoulli’s Golden Theorem and the Law of Large Numbers →

Buffon’s Noodle and the Mathematics of Hillwalking

In addition to some beautiful photos and maps and descriptions of upland challenges in Ireland and abroad, the November issue of The Summit, the Mountain Views Quarterly Newsletter for hikers and hillwalkers, describes a method to find the length of a walk based on ideas originating with the French naturalist and mathematician George-Louis Leclerc, Comte … Continue reading Buffon’s Noodle and the Mathematics of Hillwalking →

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 … Continue reading Émilie Du Châtelet and the Conservation of Energy →