The solution of many problems requires us to compute derivatives. Complex step differentiation is a method of computing the first derivative of a real function, which circumvents the problem of roundoff error found with typical finite difference approximations. For finite difference approximations, the choice of step size $latex {h}&fg=000000$ is crucial: if $latex {h}&fg=000000$ is … Continue reading Real Derivatives from Imaginary Increments
Author: thatsmaths
Changing Views on the Age of the Earth
In 1650, the Earth was 4654 years old. In 1864 it was 100 million years old. In 1897, the upper limit was revised to 40 million years. Currently, we believe the age to be about 4.5 billion years. What will be the best guess in the year 2050? [TM217 or search for “thatsmaths” at irishtimes.com]. Ussher's Chronology … Continue reading Changing Views on the Age of the Earth
Carnival of Mathematics
The Aperiodical is described on its `About' page as ``a meeting-place for people who already know they like maths and would like to know more''. The Aperiodical coordinates the Carnival of Mathematics (CoM), a monthly blogging roundup hosted on a different blog each month. Generally, the posts describe a collection of interesting recent items on … Continue reading Carnival of Mathematics
Phantom traffic-jams are all too real
Driving along the motorway on a busy day, you see brake-lights ahead and slow down until the flow grinds to a halt. The traffic stutters forward for five minutes or so until, mysteriously, the way ahead is clear again. But, before long, you arrive at the back of another stagnant queue. Hold-ups like this, with … Continue reading Phantom traffic-jams are all too real
Simple Models of Atmospheric Vortices
Atmospheric circulation systems have a wide variety of structures and there is no single mechanistic model that describes all their characteristics. However, we can construct simple kinematic models that capture some primary aspects of the flow. For simplicity, we will concentrate on idealized extra-tropical depressions. We will not consider hurricanes and tropical storms in any … Continue reading Simple Models of Atmospheric Vortices
Finding Fixed Points
An isometry on a metric space is a one-to-one distance-preserving transformation on the space. The Euclidean group $latex {E(n)}&fg=000000$ is the group of isometries of $latex {n}&fg=000000$-dimensional Euclidean space. These are all the transformations that preserve the distance between any two points. The group depends on the dimension of the space. For the Euclidean plane … Continue reading Finding Fixed Points
All Numbers Great and Small
Is space continuous or discrete? Is it smooth, without gaps or discontinuities, or granular with a limit on how small a distance can be? What about time? Can time be repeatedly divided into smaller periods without any limit, or is there a shortest interval of time? We don’t know the answers. There is much we … Continue reading All Numbers Great and Small
Approximating the Circumference of an Ellipse
The realization that the circumference of a circle is related in a simple way to the diameter came at an early stage in the development of mathematics. But who was first to prove that all circles are similar, with the ratio of circumference $latex {C}&fg=000000$ to diameter $latex {D}&fg=000000$ the same for all? Searching in … Continue reading Approximating the Circumference of an Ellipse
Kalman Filters: from the Moon to the Motorway
Before too long, we will be relieved of the burden of long-distance driving. Given the desired destination and access to a mapping system, electronic algorithms will select the best route and control the autonomous vehicle, constantly monitoring and adjusting its direction and speed of travel. The origins of the methods used for autonomous navigation lie … Continue reading Kalman Filters: from the Moon to the Motorway
Gauss Predicts the Orbit of Ceres
On the first day of a new century, January 1, 1801, astronomer Giuseppe Piazzi discovered a new celestial object, the minor planet Ceres. He made about 20 observations from his observatory in Palermo before the object was lost in the glare of the Sun in early February. Later in the year, several astronomers tried without … Continue reading Gauss Predicts the Orbit of Ceres
Seeing beyond the Horizon
From a hilltop, the horizon lies below the horizontal level at an angle called the “dip”. Around AD 1020, the brilliant Persian scholar al-Biruni used a measurement of the dip, from a mountain of known height, to get an accurate estimate of the size of the Earth. It is claimed that his estimate was within … Continue reading Seeing beyond the Horizon
Al Biruni and the Size of the Earth
Abu Rayhan al-Biruni (AD 973--1048) The 11th century Persian mathematician Abu Rayhan al-Biruni used simple trigonometric results to estimate the radius and circumference of the Earth. His estimate has been quoted as 6,340 km, which is within 1% of the mean radius of 6,371 km. While al-Biruni's method was brilliant and, for its era, spectacular, … Continue reading Al Biruni and the Size of the Earth
The Simple Arithmetic Triangle is full of Surprises
Pascal’s triangle is one of the most famous of all mathematical diagrams, simple to construct and yet rich in mathematical patterns. These can be found by a web search, but their discovery by study of the diagram is vastly more satisfying, and there is always a chance of finding something never seen before [TM212 or search … Continue reading The Simple Arithmetic Triangle is full of Surprises
Hanoi Graphs and Sierpinski’s Triangle
The Tower of Hanoi is a famous mathematical puzzle. A set of disks of different sizes are stacked like a cone on one of three rods, and the challenge is to move them onto another rod while respecting strict constraints: Only one disk can be moved at a time. No disk can be placed upon … Continue reading Hanoi Graphs and Sierpinski’s Triangle
Multi-faceted aspects of Euclid’s Elements
Euclid’s Elements was the first major work to organise mathematics as an axiomatic system. Starting from a set of clearly-stated and self-evident truths called axioms, a large collection of theorems is constructed by logical reasoning. For some, the Elements is a magnificent triumph of human thought; for others, it is a tedious tome, painfully prolix … Continue reading Multi-faceted aspects of Euclid’s Elements
A Model for Elliptic Geometry
For many centuries, mathematicians struggled to derive Euclid's fifth postulate as a theorem following from the other axioms. All attempts failed and, in the early nineteenth century, three mathematicians, working independently, found that consistent geometries could be constructed without the fifth postulate. Carl Friedrich Gauss (c. 1813) was first, but he published nothing on the … Continue reading A Model for Elliptic Geometry
Improving Weather Forecasts by Reducing Precision
Weather forecasting relies on supercomputers, used to solve the mathematical equations that describe atmospheric flow. The accuracy of the forecasts is constrained by available computing power. Processor speeds have not increased much in recent years and speed-ups are achieved by running many processes in parallel. Energy costs have risen rapidly: there is a multimillion Euro … Continue reading Improving Weather Forecasts by Reducing Precision
Can You Believe Your Eyes?
Remember the old cowboy movies? As the stage-coach comes to a halt, the wheels appear to spin backwards, then forwards, then backwards again, until the coach stops. How can this be explained? The illusion is known as the stroboscopic effect. We don't see it in real life, where time flows smoothly and continuously. But, in … Continue reading Can You Believe Your Eyes?
The Size of Things
In Euclidean geometry, all lengths, areas and volumes are relative. Once a unit of length is chosen, all other lengths are given in terms of this unit. Classical geometry could determine the lengths of straight lines, the areas of polygons and the volumes of simple solids. However, the lengths of curved lines, areas bounded by … Continue reading The Size of Things
Entropy and the Relentless Drift from Order to Chaos
In a famous lecture in 1959, scientist and author C P Snow spoke of a gulf of comprehension between science and the humanities, which had become split into “two cultures”. Many people in each group had a lack of appreciation of the concerns of the other group, causing grave misunderstandings and making the world's problems … Continue reading Entropy and the Relentless Drift from Order to Chaos
Circles, polygons and the Kepler-Bouwkamp constant
If circles are drawn in and around an equilateral triangle (a regular trigon), the ratio of the radii is $latex {\cos \pi/3 = 0.5}&fg=000000$. More generally, for an N-gon the ratio is easily shown to be $latex {\cos \pi/N}&fg=000000$. Johannes Kepler, in developing his amazing polyhedral model of the solar system, started by considering circular … Continue reading Circles, polygons and the Kepler-Bouwkamp constant
Was Space Weather the cause of the Titanic Disaster?
Space weather, first studied in the 1950's, has grown in importance with recent technological advances. It concerns the influence on the Earth's magnetic field and upper atmosphere of events on the Sun. Such disturbances can enhance the solar wind, which interacts with the magnetosphere, with grave consequences for navigation. Space weather affects the satellites of … Continue reading Was Space Weather the cause of the Titanic Disaster?
The Dimension of a Point that isn’t there
A slice of Swiss cheese has one-dimensional holes; a block of Swiss cheese has two-dimensional holes. What is the dimension of a point? From classical geometry we have the definition ``A point is that which has no parts'' --- also sprach Euclid. A point has dimension zero, a line has dimension one, a plane has … Continue reading The Dimension of a Point that isn’t there
Making the Best of Waiting in Line
Queueing is a bore and waiting to be served is one of life's unavoidable irritants. Whether we are hanging onto a phone, waiting for response from a web server or seeking a medical procedure, we have little choice but to join the queue and wait. It may surprise readers that there is a well-developed mathematical … Continue reading Making the Best of Waiting in Line
Differential Forms and Stokes’ Theorem
The theory of exterior calculus of differential forms was developed by the influential French mathematician Élie Cartan, who did fundamental work in the theory of differential geometry. Cartan is regarded as one of the great mathematicians of the twentieth century. The exterior calculus generalizes multivariate calculus, and allows us to integrate functions over differentiable manifolds … Continue reading Differential Forms and Stokes’ Theorem
Goldbach’s Conjecture: if it’s Unprovable, it must be True
The starting point for rigorous reasoning in maths is a system of axioms. An axiom is a statement that is assumed, without demonstration, to be true. The Greek mathematician Thales is credited with introducing the axiomatic method, in which each statement is deduced either from axioms or from previously proven statements, using the laws of … Continue reading Goldbach’s Conjecture: if it’s Unprovable, it must be True
Mamikon’s Theorem and the area under a cycloid arch
The Cycloid The cycloid is the locus of a point fixed to the rim of a circular disk that is rolling along a straight line (see figure). The parametric equations for the cycloid are $latex \displaystyle x = r (\theta - \sin\theta)\,, \qquad y = r (1 - \cos\theta ) \ \ \ \ \ … Continue reading Mamikon’s Theorem and the area under a cycloid arch
Machine Learning and Climate Change Prediction
Current climate prediction models are markedly defective, even in reproducing the changes that have already occurred. Given the great importance of climate change, we must identify the causes of model errors and reduce the uncertainty of climate predictions [TM205 or search for “thatsmaths” at irishtimes.com]. The Charney Report In 1979, a study group led by … Continue reading Machine Learning and Climate Change Prediction
Apples and Lemons in a Doughnut
A ring torus (or, simply, torus) is a surface of revolution generated by rotating a circle about a coplanar axis that does not intersect it. We let $latex {r}&fg=000000$ be the radius of the circle and $latex {R}&fg=000000$ the distance from the axis to the centre of the circle, with $latex {R>r}&fg=000000$. If the axis … Continue reading Apples and Lemons in a Doughnut
Complexity: are easily-checked problems also easily solved?
From the name of the Persian polymath Al Khwarizmi, who flourished in the early ninth century, comes the term algorithm. An algorithm is a set of simple steps that lead to the solution of a problem. An everyday example is a baking recipe, with instructions on what to do with ingredients (input) to produce a … Continue reading Complexity: are easily-checked problems also easily solved?
Euler’s Product: the Golden Key
The Basel problem was solved by Leonhard Euler in 1734 [see previous post]. His line of reasoning was ingenious, with some daring leaps of logic. The Basel series is a particular case of the much more general zeta function, which is at the core of the Riemann hypothesis, the most important unsolved problem in mathematics. … Continue reading Euler’s Product: the Golden Key
Euler: a mathematician without equal and an overall nice guy
Mathematicians are an odd bunch. Isaac Newton was decidedly unpleasant, secretive and resentful while Carl Friedrich Gauss, according to several biographies, was cold and austere, more likely to criticize than to praise. It is frequently claimed that a disproportionate number of mathematicians exhibit signs of autism and have significant difficulties with social interaction and everyday … Continue reading Euler: a mathematician without equal and an overall nice guy
The Basel Problem: Euler’s Bravura Performance
The Basel problem was first posed by Pietro Mengoli, a mathematics professor at the University of Bologna, in 1650, the same year in which he showed that the alternating harmonic series sums to $latex {\ln 2}&fg=000000$. The Basel problem asks for the sum of the reciprocals of the squares of the natural numbers, $latex \displaystyle … Continue reading The Basel Problem: Euler’s Bravura Performance
We are living at the bottom of an ocean
Anyone who lives by the sea is familiar with the regular ebb and flow of the tides. But we all live at the bottom of an ocean of air. The atmosphere, like the ocean, is a fluid envelop surrounding the Earth, and is subject to the influence of the Sun and Moon. While sea tides … Continue reading We are living at the bottom of an ocean
Derangements and Continued Fractions for e
We show in this post that an elegant continued fraction for $latex {e}&fg=000000$ can be found using derangement numbers. Recall from last week's post that we call any permutation of the elements of a set an arrangement. A derangement is an arrangement for which every element is moved from its original position. The number of … Continue reading Derangements and Continued Fractions for e
Arrangements and Derangements
Six students entering an examination hall place their cell-phones in a box. After the exam, they each grab a phone at random as they rush out. What is the likelihood that none of them gets their own phone? The surprising answer --- about 37% whatever the number of students --- emerges from the theory of … Continue reading Arrangements and Derangements
On what Weekday is Christmas? Use the Doomsday Rule
An old nursery rhyme begins “Monday's child is fair of face / Tuesday's child is full of grace”. Perhaps character and personality were determined by the weekday of birth. More likely, the rhyme was to help children learn the days of the week. But how can we determine the day on which we were born … Continue reading On what Weekday is Christmas? Use the Doomsday Rule
Will RH be Proved by a Physicist?
The Riemann Hypothesis (RH) states that all the non-trivial (non-real) zeros of the zeta function lie on a line, the critical line, $latex {\Re(s) = 1/2}&fg=000000$. By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the Hilbert-Polya … Continue reading Will RH be Proved by a Physicist?
Decorating Christmas Trees with the Four Colour Theorem
When decorating our Christmas trees, we aim to achieve an aesthetic balance. Let's suppose that there is a plenitude of baubles, but that their colour range is limited. We could cover the tree with bright shiny balls, but to have two baubles of the same colour touching might be considered garish. How many colours are … Continue reading Decorating Christmas Trees with the Four Colour Theorem
Laczkovich Squares the Circle
The phrase `squaring the circle' generally denotes an impossible task. The original problem was one of three unsolved challenges in Greek geometry, along with trisecting an angle and duplicating a cube. The problem was to construct a square with area equal to that of a given circle, using only straightedge and compass. That's Maths … Continue reading Laczkovich Squares the Circle
Ireland’s Mapping Grid in Harmony with GPS
The earthly globe is spherical; more precisely, it is an oblate spheroid, like a ball slightly flattened at the poles. More precisely still, it is a triaxial ellipsoid that closely approximates a “geoid”, a surface of constant gravitational potential [TM199 or search for “thatsmaths” at irishtimes.com]. Mapping the Globe Positions on the globe are given by … Continue reading Ireland’s Mapping Grid in Harmony with GPS
Aleph, Beth, Continuum
Georg Cantor developed a remarkable theory of infinite sets. He was the first person to show that not all infinite sets are created equal. The number of elements in a set is indicated by its cardinality. Two sets with the same cardinal number are ``the same size''. For two finite sets, if there is a … Continue reading Aleph, Beth, Continuum
Weather Forecasts get Better and Better
Weather forecasts are getting better. Fifty years ago, predictions beyond one day ahead were of dubious utility. Now, forecasts out to a week ahead are generally reliable [TM198 or search for “thatsmaths” at irishtimes.com]. Careful measurements of forecast accuracy have shown that the range for a fixed level of skill has been increasing by one day every … Continue reading Weather Forecasts get Better and Better
The p-Adic Numbers (Part 2)
Kurt Hensel, born in Königsberg, studied mathematics in Berlin and Bonn, under Kronecker and Weierstrass; Leopold Kronecker was his doctoral supervisor. In 1901, Hensel was appointed to a full professorship at the University of Marburg. He spent the rest of his career there, retiring in 1930. Hensel is best known for his introduction of the … Continue reading The p-Adic Numbers (Part 2)
The p-Adic Numbers (Part I)
The motto of the Pythagoreans was ``All is Number''. They saw numbers as the essence and foundation of the physical universe. For them, numbers meant the positive whole numbers, or natural numbers $latex {\mathbb{N}}&fg=000000$, and ratios of these, the positive rational numbers $latex {\mathbb{Q}^{+}}&fg=000000$. It came as a great shock that the diagonal of a … Continue reading The p-Adic Numbers (Part I)
Terence Tao to deliver the Hamilton Lecture
Pick a number; if it is even, divide it by 2; if odd, triple it and add 1. Now repeat the process, each time halving or else tripling and adding 1. Here is a surprise: no matter what number you pick, you will eventually arrive at 1. Let's try 6: it is even, so we … Continue reading Terence Tao to deliver the Hamilton Lecture
From Impossible Shapes to the Nobel Prize
Roger Penrose, British mathematical physicist, mathematician and philosopher of science has just been named as one of the winners of the 2020 Nobel Prize in Physics. Penrose has made major contributions to general relativity and cosmology. Penrose has also come up with some ingenious mathematical inventions. He discovered a way of defining a pseudo-inverse for … Continue reading From Impossible Shapes to the Nobel Prize
Mathematics and the Nature of Physical Reality
Applied mathematics is the use of maths to address questions and solve problems outside maths itself. Counting money, designing rockets and vaccines, analysing internet traffic and predicting the weather all involve maths. But why does this work? Why is maths so successful in describing physical reality? How is it that the world can be understood … Continue reading Mathematics and the Nature of Physical Reality
Doughnuts and Dumplings are Distinct: Homopoty-101
As everyone knows, a torus is different from a sphere. Topology is the study of properties that remain unchanged under continuous distortions. A square can be deformed into a circle or a sphere into an ellipsoid, whether flat like an orange or long like a lemon or banana. Technically, sets are topologically equivalent if there … Continue reading Doughnuts and Dumplings are Distinct: Homopoty-101
Will mathematicians be replaced by computers?
There are ongoing rapid advances in the power and versatility of AI or artificial intelligence. Computers are now producing results in several fields that are far beyond human capability. The trend is unstoppable, and is having profound effects in many areas of our lives. Will mathematicians be replaced by computers? [TM195 or search for “thatsmaths” at irishtimes.com]. … Continue reading Will mathematicians be replaced by computers?
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