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
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
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
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
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
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
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
Low-pass Filtering and the Remarkable Integrals of Borwein and Borwein
In last week's post we looked at aspects of puzzles of the form ``What is the next number''. We are presented with a short list of numbers, for example $latex {1, 3, 5, 7, 9}&fg=000000$ and asked for the next number in the sequence. Arguments were given indicating why any number might be regarded as … Continue reading Low-pass Filtering and the Remarkable Integrals of Borwein and Borwein
What’s the Next Number?
We are all familiar with simple mathematical puzzles that give a short sequence and ask ``What is the next number in the sequence''. Simple examples would be $latex \displaystyle \begin{array}{rcl} && 1, 3, 5, 7, 9, 11, \dots \\ && 1, 4, 9, 16, 25, \dots \\ && 1, 1, 2, 3, 5, 8, \dots … Continue reading What’s the Next Number?
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
Mercury’s Mercurial Orbit
The tiny deviation of the orbit of Mercury from a pure ellipse might seem to be of no consequence. Yet the minute precession of this planet was one of the factors leading to a revolution in our world view. Attempts to explain the anomaly in the context of Newtonian mechanics were unsatisfactory. It was only … Continue reading Mercury’s Mercurial Orbit
The Power of the 2-gon: Extrapolation to Evaluate Pi
Richardson's extrapolation procedure yields a significant increase in the accuracy of numerical solutions of differential equations. We consider his elegant illustration of the technique, the evaluation of $latex {\pi}&fg=000000$, and show how the estimates improve dramatically with higher order extrapolation. [This post is a condensed version of a paper in Mathematics Today (Lynch, 2003).] … Continue reading The Power of the 2-gon: Extrapolation to Evaluate Pi
Dropping Pebbles down a Mine-shaft
If you drop a pebble down a mine-shaft, it will not fall vertically, but will be deflected slightly to the East by the Coriolis force, an effect of the Earth's rotation. We can solve the equations to calculate the amount of deflection; for a ten-second drop, the pebble falls about 500 metres (air resistance is … Continue reading Dropping Pebbles down a Mine-shaft
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
Finding the Area of a Field
It is a tricky matter to find the area of a field that has irregular or meandering boundaries. The standard method is to divide the field into triangular parts. If the boundaries are linear, this is simple. If they twist and turn, then a large number of triangles may be required. When we have the … Continue reading Finding the Area of a Field
CND Functions: Curves that are Continuous but Nowhere Differentiable
A function $latex {f(x)}&fg=000000$ that is differentiable at a point $latex {x}&fg=000000$ is continuous there, and if differentiable on an interval $latex {[a, b]}&fg=000000$, is continuous on that interval. However, the converse is not necessarily true: the continuity of a function at a point or on an interval does not guarantee that it is differentiable … Continue reading CND Functions: Curves that are Continuous but Nowhere Differentiable
Topological Calculus: away with those nasty epsilons and deltas
A new approach to calculus has recently been developed by Peter Olver of the University of Minnesota. He calls it ``Continuous Calculus'' but indicates that the name ``Topological Calculus'' is also appropriate. He has provided an extensive set of notes, which are available online (Olver, 2022a)]. Motivation Students embarking on a university programme in mathematics … Continue reading Topological Calculus: away with those nasty epsilons and deltas
The 3-sphere: Extrinsic and Intrinsic Forms
The circle in two dimensions and the sphere in three are just two members of an infinite family of hyper-surfaces. By analogy with the circle $latex {\mathbb{S}^1}&fg=000000$ in the plane $latex {\mathbb{R}^2}&fg=000000$ and the sphere $latex {\mathbb{S}^2}&fg=000000$ in three-space $latex {\mathbb{R}^3}&fg=000000$, we can consider hyper-spheres in higher dimensional spaces. In particular, we will consider the … Continue reading The 3-sphere: Extrinsic and Intrinsic Forms
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
Dynamic Equations for Weather and Climate
``I could have done it in a much more complicated way'', said the Red Queen, immensely proud. --- Lewis Carroll. Books on dynamic meteorology and oceanography usually have a full chapter devoted to the basic dynamical equations. Since the Earth's fluid envelop is approximately a thin spherical shell, spherical coordinates $latex {(\lambda,\varphi, r)}&fg=000000$ are convenient. … Continue reading Dynamic Equations for Weather and Climate
Curl Curl Curl
Many of us have struggled with the vector differential operators, grad, div and curl. There are several ways to represent vectors and several expressions for these operators, not always easy to remember. We take another look at some of their properties here. We consider a vector $latex {\mathbf{V} = (u, v, w)^{\mathrm{T}}}&fg=000000$ which may be … Continue reading Curl Curl Curl
X+Y and the Special Triangle
How can mathematicians grapple with abstruse concepts that are, for the majority of people, beyond comprehension? What mental processes enable a small proportion of people to produce mathematical work of remarkable creativity? In particular, is there a connection between mathematical creativity and autism? We revisit a book and a film that address these questions. The … Continue reading X+Y and the Special Triangle
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
Space-Filling Curves, Part II: Computing the Limit Function
The Approximating Functions It is simple to define a mapping from the unit interval $latex {I := [0,1]}&fg=000000$ into the unit square $latex {Q:=[0,1]\times[0,1]}&fg=000000$. Georg Cantor found a one-to-one map from $latex {I}&fg=000000$ onto $latex {Q}&fg=000000$, showing that the one-dimensional interval and the two-dimensional square have the same cardinality. Cantor's map was not continuous, but … Continue reading Space-Filling Curves, Part II: Computing the Limit Function
Space-Filling Curves, Part I: “I see it, but I don’t believe it”
We are all familiar with the concept of dimension: a point is zero-dimensional, a line is one-dimensional, a plane is two-dimensional and the space around us is three-dimensional. A position on a line can be specified by a single number, such as the distance from a fixed origin. In the plane, a point can be … Continue reading Space-Filling Curves, Part I: “I see it, but I don’t believe it”
Poincare’s Square and Unbounded Gomoku
Henri Poincar'e was masterful in presenting scientific concepts and ideas in an accessible way. To explain that the Universe might be bounded and yet infinite, he imagined that the temperature of space decreased from the centre to the periphery in such a way that everything contracted with the distance from the centre. As travellers moved … Continue reading Poincare’s Square and Unbounded Gomoku
Fields Medals presented at IMC 2022
Every four years, at the International Congress of Mathematicians, the Fields Medal is awarded to two, three, or four young mathematicians. To be eligible, the awardees must be under forty years of age. For the chosen few, who came from England, France, Korea and Ukraine, the award, often described as the Nobel Prize of Mathematics, … Continue reading Fields Medals presented at IMC 2022
Goldbach’s Conjecture and Goldbach’s Variation
Goldbach's Conjecture is one of the great unresolved problems of number theory. It simply states that every even natural number greater than two is the sum of two prime numbers. It is easily confirmed for even numbers of small magnitude. The conjecture first appeared in a letter dated 1742 from German mathematician Christian Goldbach to … Continue reading Goldbach’s Conjecture and Goldbach’s Variation
The Size of Sets and the Length of Sets
Cardinals and Ordinals The cardinal number of a set is an indicator of the size of the set. It depends only on the elements of the set. Sets with the same cardinal number --- or cardinality --- are said to be equinumerate or (with unfortunate terminology) to be the same size. For finite sets there … Continue reading The Size of Sets and the Length of Sets
Can We Control the Weather?
Atmospheric motions are chaotic: a minute perturbation can lead to major changes in the subsequent evolution of the flow. How do we know this? There is just one atmosphere and, if we perturb it, we can never know how it might have evolved if left alone. We know, from simple nonlinear models that exhibit chaos, … Continue reading Can We Control the Weather?
The Arithmetic Triangle is Analytical too
Pascal's triangle is one of the most famous of all mathematical diagrams. It is simple to construct and rich in mathematical patterns. There is always a chance of finding something never seen before, and the discovery of new patterns is very satisfying. Not too long ago, Harlan Brothers found Euler's number $latex {e}&fg=000000$ in the … Continue reading The Arithmetic Triangle is Analytical too
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
Swingin’-Springin’-Twistin’-Motion
The swinging spring, or elastic pendulum, exhibits some fascinating dynamics. The bob is free to swing like a spherical pendulum, but also to bounce up and down due to the stretching action of the spring. The behaviour of the swinging spring has been described in a previous post on this blog [Reference 1 below]. A … Continue reading Swingin’-Springin’-Twistin’-Motion
Parity of the Real Numbers: Part I
In some recent posts, here and here we discussed the extension of the concept of parity (Odd v. Even) from the integers to the rational numbers. We found that it is natural to consider three parity classes, determined by the parities of the numerator and denominator of a rational number $latex {q = m / n}&fg=000000$ … Continue reading Parity of the Real Numbers: Part I
Fairy Lights on the Farey Tree
The rational numbers $latex {\mathbb{Q}}&fg=000000$ are dense in the real numbers $latex {\mathbb{R}}&fg=000000$. The cardinality of rational numbers in the interval $latex {(0,1)}&fg=000000$ is $latex {\boldsymbol{\aleph}_0}&fg=000000$. We cannot list them in ascending order, because there is no least rational number greater than $latex {0}&fg=000000$. However, there are several ways of enumerating the rational numbers. The … Continue reading Fairy Lights on the Farey Tree
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
Parity and Partition of the Rational Numbers. Part II: Density of the Three Parity Classes
In last week's post, we defined an extension of parity from the integers to the rational numbers. Three parity classes were found --- even, odd and none. This week, we show that, with an appropriate ordering or enumeration of the rationals, the three classes are not only equinumerate (having the same cardinality) but of equal … Continue reading Parity and Partition of the Rational Numbers. Part II: Density of the Three Parity Classes
Parity and Partition of the Rational Numbers. Part I: The Three Parity Classes
We define an extension of parity from the integers to the rational numbers. Three parity classes are found --- even, odd and none. Using the 2-adic valuation, we partition the rationals into subgroups with a rich algebraic structure. The natural numbers $latex {\mathbb{N}}&fg=000000$ split nicely into two subsets, the odd and even numbers $latex \displaystyle … Continue reading Parity and Partition of the Rational Numbers. Part I: The Three Parity Classes
A Finite but Unbounded Universe
Henri Poincaré described a beautiful geometric model with some intriguing properties. He envisioned a circular disk in the Euclidean plane, where distances were distorted to give it geometric properties quite different from those of Euclid's Elements. He supposed that the temperature varied linearly from a fixed value at the centre of the disk to absolute … Continue reading A Finite but Unbounded Universe
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?
Following the Money around the Eurozone
Take a fistful of euro coins and examine the obverse sides; you may be surprised at the wide variety of designs. The eurozone is a monetary union of 19 member states of the European Union that have adopted the euro as their primary currency. In addition to these countries, Andorra, Monaco, San Marino and Vatican … Continue reading Following the Money around the Eurozone
Mamikon’s Visual Calculus and Hamilton’s Hodograph
[This is a condensed version of an article [5] in Mathematics Today] A remarkable theorem, discovered in 1959 by Armenian astronomer Mamikon Mnatsakanian, allows problems in integral calculus to be solved by simple geometric reasoning, without calculus or trigonometry. Mamikon's Theorem states that `The area of a tangent sweep of a curve is equal to … Continue reading Mamikon’s Visual Calculus and Hamilton’s Hodograph
Infinitesimals: vanishingly small but not quite zero
A few weeks ago, I wrote about Hyperreals and Nonstandard Analysis , promising to revisit the topic. Here comes round two. We know that 2.999… is equal to three. But many people have a sneaking suspicion that there is “something” between the number with all those 9’s after the 2 and the number 3, that is not … Continue reading Infinitesimals: vanishingly small but not quite zero
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