### 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] Euclid’s Theorem I-47, also known as Pythagoras’ Theorem, in Oliver Byrne’s colourful text, The Elements of Euclid.

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 Mathematical Society in Georgia. It has been widely believed that no proof based on trigonometry was possible but we now know that to be false. Continue reading ‘The Waffle Cone and a new Proof of Pythagoras’ Theorem’

### Wonky Wheels on Wacky Roads Tricycles with three square wheels, each a different size. Image from the Museum of Mathematics, New York.

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. Continue reading ‘Wonky Wheels on Wacky Roads’

### The Potency of Pattern: Mind the Gap

In his book A Mathematician’s Apology, leading British mathematician G H Hardy wrote “A mathematician, like a painter or poet, is a maker of patterns.” He observed that the mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; beauty is the acid test  [TM245 or search for “thatsmaths” at irishtimes.com]. Continue reading ‘The Potency of Pattern: Mind the Gap’

### 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.

### 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.

### 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 many aspects and examples of broken symmetry.

In this article (and the following one) we look at some consequences of broken symmetry in atmospheric dynamics. In particular, we see how mountains (which are stationary!) can generate waves in the atmosphere that propagate towards the west. We will look at this unexpected breaking of symmetry and try to explain it.

### 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 would follow from a given state. The results were undependable for two main reasons: the data were sparse, and the empirical rules were unreliable  [TM244 or search for “thatsmaths” at irishtimes.com].

For the past sixty years or so, forecasts have been based on computer models that numerically solve the mathematical equations expressing the physical laws. This approach is radically different but, after a shaky start, the numerical weather prediction (NWP) models have become remarkably accurate, with forecasting skill increasing by about one day each decade. Now there are signs of a return to the analogue, data-driven methods.

### Numbers Without Ones: Chorisenic Sets Left: Count of elements of set ${\mathbf{Xe}_{10}}$. Right: partial density of set ${\mathbf{Xe}_{10}}$.

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 Greek for “without ones” as ${\chi\omega\rho\iota'\varsigma}$ ${\varepsilon'\nu\alpha}$ or choris ena, so let us call a set of “oneless numbers” a chorisenic set. 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, was selected for the 2014 edition of “The Best Writing on Mathematics,” published by Princeton University Press [Pitici, 2014]. Amusical Permutations was an attempt to find a simple sequence whose behaviour was undecidable.

### 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 ${f(x)}$ of a real variable, we may ask what is the value of the function for a particular value ${x=a}$. Maybe this is an easy question or maybe it is not.

The epsilon-delta concept can be subtle, and is sufficiently difficult that it has been used as a filter to weed out students who may not be considered smart enough to continue in maths (I know this from personal experience). The formulation of the epsilon-delta definitions is usually attributed to the German mathematician Karl Weierstrass. They must have caused him many sleepless nights.

### 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.

### 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 invited him to try out his collection of pianofortes. As they went from room to room, Bach improvised a new piece of music on each instrument  [TM243 or search for “thatsmaths” at irishtimes.com].

### 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. Continue reading ‘Sets that are Elements of Themselves: Verboten’

### Benford’s Law Revisited {Probability for the first decimal digit ${D_1}$ of a number to take values from 1 to 9.

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, ${D_1=1}$ occurs more than ${30\%}$ of the time while ${D_1=9}$ is found in less than ${5\%}$ of cases (see Figure above). Specifically, the probability distribution is $\displaystyle \mathsf{P}(D_1 = d) = \log_{10} \left( 1 + \frac{1}{d} \right) \,, \quad \mbox{ for\ } d = 1, 2, \dots , 9 \,. \ \ \ \ \ (1)$

A more complete form of the law gives the probabilities for the second and subsequent digits. A full discussion of Benford’s Law is given in Berger and Hill (2015).

### 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.

Can you derive a formula for this probability distribution?

Can you generalise it to the range from 1 to 10^n?

Can you relate this problem to Benford’s Law [described here]?

Solution, and more on Benford’s Law, next week.

### 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 the level of confidence in the forecast. They are formulated using forecasts produced by computer, algorithms that determine the likely impacts of extreme weather, and the expertise of the forecasters [TM242 or search for “thatsmaths” at irishtimes.com]. 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 ${p/q}$ in reduced form ( ${p}$ and ${q}$ coprime), a Ford circle is a circle with center at ${(p/q,1/(2q^{2}))}$ and radius ${1/(2q^{2})}$. There is a Ford circle associated with every rational number. Every Ford circle is tangent to the horizontal axis and each two Ford circles are either tangent or disjoint from each other.

### 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 ${E = m c^2}$. 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. 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 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 precisely the sequence studied centuries later by Leonardo Bonacci of Pisa, which we now call the Fibonacci sequence [TM241 or search for “thatsmaths” at irishtimes.com].

### Adding a Point to Make a Space Compact Stereographic projection between ${\mathbb{R}}$ and ${S^1}$. There is no point on the real line corresponding to the “North Pole” in ${S^1}$. Can we add another point to ${\mathbb{R}}$?

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 space in a compact space. In particular, it is a way to “make the real line compact”.

### Summing the Fibonacci Sequence Left: Fibonacci, or Leonardo of Pisa. Right: Italian postage stamp issued on the 850th anniversary of his birth.

The Fibonacci sequence must be familiar to anyone reading this. We define it by means of a second-order recurrence relation, $\displaystyle F_{n+1} = F_{n-1} + F_n \,. \ \ \ \ \ (1)$

and two initial values, ${F_0 = 0}$ and ${F_1 = 1}$. This immediately yields the well-known sequence $\displaystyle \{F_n\} = \{ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \dots \} \,.$

### 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 the plane from the axis of the torus (see Figure above). We examine how spiric curves may be found in the phase-space of a dynamical system.

### 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 ${x}$ is its modulus or absolute value ${|x|}$. We measure the “distance” between two real numbers by means of the absolute value of their difference. This gives the Euclidean metric ${\rho(x,y) = |x-y|}$ and, using it, we can define the usual topology on the real numbers ${\mathbb{R}}$.

The standard arrangement of the real numbers on a line automatically ensures that numbers with small Euclidean difference between them are geometrically close to each other. It may come as a surprise that there are other ways to define norms and distances, which provide other topologies, leading us to a radically different concept of closeness, and to completely new number systems, the p-adic numbers.

### 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 Bernhard Riemann in the mid 19th century, which was exactly what Albert Einstein needed 50 years later for his relativity theory [TM240 or search for “thatsmaths” at irishtimes.com].

### 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”.
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 ${1, 3, 5, 7, 9}$ and asked for the next number in the sequence. Arguments were given indicating why any number might be regarded as the next number. Borwein integrals evaluated by Mathematica. The first seven integrals are all equal to ${\pi}$. The eighth is a tiny bit less than this.

In this article we consider a sequence of seven ones: ${1, 1, 1, 1, 1, 1 ,1}$. Most people would agree that the next number in the sequence is ${1}$. We will show how the number ${1 - 1.47\times10^{-11} \approx 0.999\,999\,999\,985}$ could be the “correct” answer. 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 $\displaystyle \begin{array}{rcl} && 1, 3, 5, 7, 9, 11, \dots \\ && 1, 4, 9, 16, 25, \dots \\ && 1, 1, 2, 3, 5, 8, \dots \,, \end{array}$

the sequence of odd numbers, the sequence of squares and the Fibonacci sequence.

### The Rich Legacy of Indian Mathematics Jeremy Irons and Dev Patel in The Man Who Knew Infinity, a film about the life of Srinivasa Ramanujan.

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 samples of decimal-based numeration [TM239 or search for “thatsmaths” at irishtimes.com].

### 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 with the emergence of general relativity that we were able to understand the observed phenomenon. 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 ${\pi}$, 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).]

### 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 neglected) and is deflected eastward by about 25 cm. The figure on the left shows the trajectory in the vertical xz-plane (scales are not the same).

We derive the equations after making some simplifying assumptions. We assume the mine-shaft is at the Equator; we assume the meridional or north-south motion is zero; we neglect variations in the gravitational force; we neglect the sphericity of the Earth; we neglect air resistance. We can still get accurate estimates provided the elapsed time is short. However, carrying the analysis to the extreme, we obtain results that are completely unrealistic. The equations predict that the pebble will reach a minimum altitude and then rise up again to its initial height a great distance east of its initial position. Then this up-and-down motion will recur indefinitely.

### 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.

### 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.

### CND Functions: Curves that are Continuous but Nowhere Differentiable Approximation ${W_{12}(x)}$ to the Weierstrass CND function.

A function ${f(x)}$ that is differentiable at a point ${x}$ is continuous there, and if differentiable on an interval ${[a, b]}$, 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 at the point or on the interval.

### 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)].

### 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 ${\mathbb{S}^1}$ in the plane ${\mathbb{R}^2}$ and the sphere ${\mathbb{S}^2}$ in three-space ${\mathbb{R}^3}$, we can consider hyper-spheres in higher dimensional spaces. In particular, we will consider the 3-sphere which can be embedded in ${\mathbb{R}^4}$ but can also be envisaged as a non-Euclidean manifold in ${\mathbb{R}^3}$.

### Making Sound Pictures to Identify Bird Songs Top: Audio signal with three chirps. Bottom: Time-Frequency spectrogram of signal.

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 Merlin Bird ID. In a recent episode of Mooney Goes Wild, Niall Hatch of Birdwatch Ireland interviewed Drew Weber of the Cornell Lab of Ornithology, a developer of Merlin Bird ID. This phone app enables a large number of birds to be identified [TM237 or search for “thatsmaths” at irishtimes.com].

### 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 ${(\lambda,\varphi, r)}$ are convenient. Here ${\lambda}$ is the longitude and ${\varphi}$ the latitude. In Figure 1 we show the momentum equations as presented in the monograph of Lorenz (1967): Fig 1. The momentum equations, as in Lorenz (1967). The metric terms are boxed.

### 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. ### 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 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  [TM236 or search for “thatsmaths” at irishtimes.com].

### Space-Filling Curves, Part II: Computing the Limit Function

It is simple to define a mapping from the unit interval ${I := [0,1]}$ into the unit square ${Q:=[0,1]\times[0,1]}$. Georg Cantor found a one-to-one map from ${I}$ onto ${Q}$, showing that the one-dimensional interval and the two-dimensional square have the same cardinality. Cantor’s map was not continuous, but Giuseppe Peano found a continuous surjection from ${I}$ onto ${Q}$, that is, a curve that fills the entire unit square. Shortly afterwards, David Hilbert found an even simpler space-filling curve, which we discussed in Part I of this post.

### 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 located by giving its Cartesian coordinates ${(x,y)}$, or its polar coordinates ${(\rho,\theta)}$. In space, we may specify the location by giving three numbers ${(x,y,z)}$.

### 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 outward from the centre, everything got smaller in such a way that it would take an infinite time to reach the boundary.

### 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, is the crowning achievement of their careers [TM235 or search for “thatsmaths” at irishtimes.com]. Clockwise from top left: Maryna Viazovska, James Maynard, June Huh and Hugo Duminil-Copin. Image Credits: Mattero Fieni, Ryan Cowan, Lance Murphy.

The congress, which ran from 6th to 14th July, was originally to take place in St Petersburg. When events made that impossible, the action shifted to Helsinki and the conference presentations were moved online. The International Mathematical Union generously allowed participants to register at no cost.

### 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 Leonhard Euler. The truth of the conjecture for all even numbers up to four million million million ( ${4\times 10^{18}}$) has been demonstrated. There is essentially no doubt about its validity, but no proof has been found.

### The Size of Sets and the Length of Sets Schematic diagram of ${\omega^2}$. Each line corresponds to an ordinal ${\omega\cdot m + n}$ where ${m}$ and ${n}$ are natural numbers [image Wikimedia Commons].

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 are no problems. Two sets, each having the same number ${n}$ of elements, both have cardinality ${n}$. But an infinite set has the definitive property that it can be put in one-to-one correspondence with a proper subset of itself.

### 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, that the flow is very sensitive to the starting conditions. We can run “identical twin” experiments, where the initial conditions for two runs are almost identical, and watch how the two solutions diverge. This — and an abundance of other evidence — leads us to the conclusion that the atmosphere behaves in a similar way.

### 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 ${e}$ in the triangle (Brothers, 2012(a),(b)). This is indeed surprising. The number ${e}$ is ubiquitous in analysis but it is far from obvious why it should turn up in the arithmetic triangle.