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The Never-ending Quest for Enormous Prime Numbers – ThatsMaths
The ongoing search for ever-larger prime numbers continues apace. Primes are the atoms of arithmetic: every whole number is a unique product of primes. For example, 21 is the product of primes 3 and 7, while 23 is itself prime. The primes play a central role in pure mathematics, in the field of number theory,…
Joining Forces to Improve Weather Forecasts – ThatsMaths
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…
Hamilton’s Dynamics: a Prescient Framework for Quantum Mechanics – ThatsMaths
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…
Progress towards a Grand Unified Theory of Mathematics – ThatsMaths
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…
The Doppler Effect: Simple but Remarkably Useful – ThatsMaths
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…
To See a World in a Grain of Sand – ThatsMaths
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…
Mathematical Fun and Games at Maison Poincaré – ThatsMaths
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…
Resonant Vibrations from Atoms to the Far Horizons of the Cosmos – ThatsMaths
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…
Pythagorean Tuning and the Spiral of Theodorus – ThatsMaths
Most of us are familiar with the piano keyboard. There are twelve distinct notes in each octave, or thirteen if we include the note completing the chromatic scale. The illustration above shows a complete scale from middle C (or C$latex {_4}&fg=000000$) to the C above (or C$latex {_5}&fg=000000$). Eight of the notes --- C, D,…
The Many Schools of Mathematical Thought – ThatsMaths
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…
Breaking a Stick to Form a Triangle – ThatsMaths
We consider a simple problem in probability: A thin rod is broken at random into three pieces. What is the probability that these three pieces can be used to form a triangle? This problem is solved without difficulty. If the rod is of unit length, aligned along the interval $latex {0\le x \le 1}&fg=000000$, we…
Mileva Marić and the Special Theory of Relativity – ThatsMaths
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…
ENSO: The Oscillating Atmosphere and Ocean – ThatsMaths
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”…
Post600: New Year’s Greetings, 2024 – ThatsMaths
The That’s Maths blog has been active since July 2012. This is post number 600. For the past eleven and a half years, there has been a post every Thursday. For a complete list in chronological order, just press the “Contents” button, or click here. The posts have covered a wide range of topics in…
Rubik’s Cube Solvable in 20 Moves – ThatsMaths
In 1974, the brilliant Hungarian professor of architecture, Ernö Rubik, invented the puzzle that has made his name familiar all over the world. When it was mass-produced, from 1980 onwards, the cube became an international craze; vast numbers were made and they brought both great fun and great frustration to millions of children of all…
The Decline of the Mayans: a Warning Signal for Us – ThatsMaths
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…
The Sieve of Eratosthenes and a Partition of the Natural Numbers | ThatsMaths
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…
The Logistic Map is hiding in the Mandelbrot Set | ThatsMaths
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.…
The Golden Key to Riemann’s Hypothesis | ThatsMaths
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…
Sharkovsky’s Theorem | ThatsMaths
This post is an extension and elaboration of two recent posts, with more technical details ``The reasonable man adapts himself to the world: the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.'' …
Oleksandr Sharkovsky and Chaos Theory | ThatsMaths
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…
The Logistic Map: a Simple Model with Rich Dynamics | ThatsMaths
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…
Yin and Yang — and East and West | ThatsMaths
The duality encapsulated in the concept of yin-yang is at the origin of many aspects of classical Chinese science and philosophy. Many dualities in the natural world --- light and dark, fire and water, order and chaos --- are regarded as physical manifestations of this duality. Yin is the receptive and yang the active principle.…
The Axiom of Choice: Shoes & Socks and Non-constructive Proofs | ThatsMaths
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…
Earth’s Digital Twins can help us to avert Disaster | ThatsMaths
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…
A Remarkable Sequence in OEIS | ThatsMaths
The On-Line Encyclopedia of Integer Sequences (OEIS), launched in 1996, now contains 360,000 entries. It attracts a million visits a day, and has been cited about 10,000 times. It is now possible for anyone in the world to propose a new sequence for inclusion in OEIS. The goal of the database is to include all…
Elusive Transcendentals | ThatsMaths
The numbers are usually studied in layers of increasing subtlety and intricacy. We start with the natural, or counting, numbers $latex {\mathbb{N} = \{ 1, 2, 3, \dots \}}&fg=000000$. Then come the whole numbers or integers, $latex {\mathbb{Z} = \{ \dots, -2, -1, 0, 1, 2, \dots \}}&fg=000000$. All the ratios of these (avoiding division…
Digital Signatures using Edwards Curves | ThatsMaths
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…
Maths in Action at the Rugby World Cup | ThatsMaths
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…
A Memorable Memo: Responding to Over-assiduous Administrators | ThatsMaths
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…
Hamilton’s Semaphore Code and Signalling System | ThatsMaths
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…
Sixth Irish History of Mathematics (IHoM) Conference | ThatsMaths
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…
Retroreflectors: Right Angles Save Lives | ThatsMaths
As everyone knows, left and right are swapped in a mirror image. Or are they? It is really front and back that are reversed, but that's a story for another day. During a visit to the Science and Industry Museum in Paris some years ago, I stood facing a spinning mirror. Lifting one arm, I…
Music and Maths from Bach to Bacherach | ThatsMaths
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…
Proofs without Words | ThatsMaths
The sum of the first $latex {n}&fg=000000$ odd numbers is equal to the square of $latex {n}&fg=000000$: $latex \displaystyle 1 + 3 + 5 + \cdots + (2n-1) = n^2 \,. &fg=000000$ We can check this for the first few: $latex {1 = 1^2,\ \ 1+3=2^2,\ \ 1+3+5 = 3^2}&fg=000000$. But how do we prove…
Maths in the Time of the Pharaohs | ThatsMaths
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…
Margules’ Tendency Equation and Richardson’s Forecast | ThatsMaths
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…
Dynamic Similarity and the Reynolds Number | ThatsMaths
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 —…
The Sizes of Sets | ThatsMaths
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…
The Dual in the Crown | ThatsMaths
The beginning of topology is often traced to Euler's solution of a puzzle, the Bridges of Königsberg. The problem posed was to follow a path through the Prussian city that crossed all seven bridges exactly once. Euler proved that the problem has no solution. He drastically simplifying it by replacing the geographical context by a…