There are five sweet spots where a spacecraft can keep pace with Earth as both orbit the Sun. They are called the Lagrange points, after the brilliant French mathematician Joseph-Louis Lagrange who found special solutions to what is called the “three-body problem”. To locate the second Lagrange point, L2, draw a line 150 million km … Continue reading The James Webb Telescope: Viewing the Universe from Lagrange Point L2
Category: Occasional
Euler’s Identity: the Most Beautiful Equation in Mathematics
A recent entry in the visitor’s book of the James Joyce Tower & Museum in Sandycove, Dublin perplexed the Friends of Joyce’s Tower, the volunteers who run the museum. The entry, reproduced as the boxed equation in the photograph, seemed as impenetrable as a passage of Finnegans Wake and quite beyond decryption. When invited to … Continue reading Euler’s Identity: the Most Beautiful Equation in Mathematics
The Never-ending Quest for Enormous Prime Numbers
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, … Continue reading The Never-ending Quest for Enormous Prime Numbers
Pythagorean Tuning and the Spiral of Theodorus
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, … Continue reading Pythagorean Tuning and the Spiral of Theodorus
Breaking a Stick to Form a Triangle
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 … Continue reading Breaking a Stick to Form a Triangle
Post600: New Year’s Greetings, 2024
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 … Continue reading Post600: New Year’s Greetings, 2024
Rubik’s Cube Solvable in 20 Moves
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 … Continue reading Rubik’s Cube Solvable in 20 Moves
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
Sharkovsky’s Theorem
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.'' … Continue reading Sharkovsky’s Theorem
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
Yin and Yang — and East and West
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. … Continue reading Yin and Yang — and East and West
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
Elusive Transcendentals
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 … Continue reading Elusive Transcendentals
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 Remarkable Sequence in OEIS
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 … Continue reading A Remarkable Sequence in OEIS
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
Retroreflectors: Right Angles Save Lives
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 … Continue reading Retroreflectors: Right Angles Save Lives
Proofs without Words
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 … Continue reading Proofs without Words
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
The Dual in the Crown
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 … Continue reading The Dual in the Crown
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.
A Simple Formula for the Weekday
People skilled in mental arithmetic sometimes amaze friends and colleagues by calculating the day of the week on which a given date falls. Thus, given a date, say D-Day, which was on 6 June 1944, they quickly announce that it was a Tuesday. Techniques for calculating the day of the week for a given date … Continue reading A Simple Formula for the Weekday
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
Wonky Wheels on Wacky Roads
Imagine trying to cycle along a road with a wavy surface. Could anything be done to minimise the ups-and-downs? In general, this would be very difficult, but in ideal cases a simple solution might be possible. Elliptic Wheels We suppose that the road runs along the $latex {x}&fg=000000$-axis, with its height varying like a sine … Continue reading Wonky Wheels on Wacky Roads
A Topological Proof of Euclid’s Theorem
Theorem (Euclid): There are infinitely many prime numbers. Euclid's proof of this result is a classic. It is often described as a proof by contradiction but, in fact, Euclid shows how, given a list of primes up to any point, we can find, by a finite process, another prime number; so, the proof is constructive. … Continue reading A Topological Proof of Euclid’s Theorem
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
Numbers Without Ones: Chorisenic Sets
There is no end to the variety of sets of natural numbers. Sets having all sorts of properties have been studied and many more remain to be discovered. In this note we study the set of natural numbers for which the decimal digit 1 does not occur. Google Translate on my mobile phone gives the … Continue reading Numbers Without Ones: Chorisenic Sets
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
Limits of Sequences, Limits of Sets
In undergraduate mathematics, we are confronted at an early stage with "Epsilon-Delta" definitions. For example, given a function $latex {f(x)}&fg=000000$ of a real variable, we may ask what is the value of the function for a particular value $latex {x=a}&fg=000000$. Maybe this is an easy question or maybe it is not. The epsilon-delta concept can … Continue reading Limits of Sequences, Limits of Sets
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
Bach and Euler chat in Frederick’s Court
Frederick the Great of Prussia, a devoted patron of the arts, had a particular interest in music, and admired the music of Johann Sebastian Bach. In 1747, Bach visited Potsdam, where his son Carl Philipp Emanuel was the Kapellmeister in Frederick’s court. When Frederick learned of this, he summoned ‘Old Bach’ to the palace and … Continue reading Bach and Euler chat in Frederick’s Court
Sets that are Elements of Themselves: Verboten
Can a set be an element of itself? A simple example will provide an answer to this question. Let us define a set to be small if it has less than 100 elements. There are clearly an enormous number of small sets. For example, The set of continents. The set of Platonic solids. The set … Continue reading Sets that are Elements of Themselves: Verboten
Benford’s Law Revisited
Several researchers have observed that, in a wide variety of collections of numerical data, the leading --- or most significant --- decimal digits are not uniformly distributed, but conform to a logarithmic distribution. Of the nine possible values, $latex {D_1=1}&fg=000000$ occurs more than $latex {30\%}&fg=000000$ of the time while $latex {D_1=9}&fg=000000$ is found in less … Continue reading Benford’s Law Revisited
A Puzzle: Two-step Selection of a Digit
Here is a simple problem in probability. (1) Pick a number k between 1 and 9. Assume all digits are equally likely. (2) Pick a number m in the range from 1 to k. What is the probability distribution for the number m? A graph of the probability distribution is shown in the figure here. … Continue reading A Puzzle: Two-step Selection of a Digit
Ford Circles & Farey Series
American mathematician Lester Randolph Ford Sr. (1886--1967) was President of the Mathematical Association of America from 1947 to 1948 and editor of the American Mathematical Monthly during World War II. He is remembered today for the system of circles named in his honour. For any rational number $latex {p/q}&fg=000000$ in reduced form ($latex {p}&fg=000000$ and … Continue reading Ford Circles & Farey Series
From Wave Equations to Modern Telecoms
Mathematics has an amazing capacity to help us to understand the physical world. Just consider the profound implications of Einstein's simple equation $latex {E = m c^2}&fg=000000$. Another example is the wave equation derived by Scottish mathematical physicist James Clerk Maxwell. Our modern world would not exist without the knowledge encapsulated in Maxwell's equations. Observation … Continue reading From Wave Equations to Modern Telecoms
Curvature and the Osculating Circle
Curvature is of critical importance in numerous contexts. An example is shown in the figure above, a map of the Silverstone Formula 1 racetrack. The sharp bends (high curvature) force drivers to reduct speed drastically. The Concept of Curvature Curvature is a fundamental concept in differential geometry. The curvature of a plane curve is a … Continue reading Curvature and the Osculating Circle
Adding a Point to Make a Space Compact
The real line is an example of a locally compact Hausdorff space. In a Hausdorff space, two distinct points have disjoint neighbourhoods. As the old joke says, ``any two points can be housed off from each other''. We will define local compactness below. The one-point compactification is a way of embedding a locally compact Hausdorff … Continue reading Adding a Point to Make a Space Compact
Summing the Fibonacci Sequence
The Fibonacci sequence must be familiar to anyone reading this. We define it by means of a second-order recurrence relation, $latex \displaystyle F_{n+1} = F_{n-1} + F_n \,. \ \ \ \ \ (1)&fg=000000$ and two initial values, $latex {F_0 = 0}&fg=000000$ and $latex {F_1 = 1}&fg=000000$. This immediately yields the well-known sequence $latex \displaystyle … Continue reading Summing the Fibonacci Sequence
Spiric curves and phase portraits
We are very familiar with the conic sections, the curves formed from the intersection of a plane with a cone. There is another family of curves, the Spiric sections, formed by the intersections of a torus by planes parallel to its axis. Like the conics, they come in various forms, depending upon the distance of … Continue reading Spiric curves and phase portraits
Closeness in the 2-Adic Metric
When is 144 closer to 8 than to 143? The usual definition of the norm of a real number $latex {x}&fg=000000$ is its modulus or absolute value $latex {|x|}&fg=000000$. We measure the ``distance'' between two real numbers by means of the absolute value of their difference. This gives the Euclidean metric $latex {\rho(x,y) = |x-y|}&fg=000000$ … Continue reading Closeness in the 2-Adic Metric
Curvature and Geodesics on a Torus
We take a look at the curvature on a torus, and the various forms that geodesics can have. These are compared to the geodesics on a ``flat torus''. Toroidal-Poloidal Coordinates The position on a torus may be specified by the toroidal and poloidal coordinates. The toroidal component ($latex {\lambda}&fg=000000$) is the angle following a large … Continue reading Curvature and Geodesics on a Torus
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