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 →

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. →

Literomathic Synergy for Bloomstime

Bloomsday approaches, so let us re-Joyce once more. In Ulysses, Joyce makes numerous allusions to mathematical matters. On previous Bloomsdays, ThatsMaths has written about non-Euclidean geometry, Bloom’s efforts to square the circle, a possible encounter between Joyce and the famed logician Ernst Zermelo and the fractal complexity of Finnegans Wake  [TM246 or search for “thatsmaths” at irishtimes.com]. … Continue reading Literomathic Synergy for Bloomstime →

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 →

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]. Everyone is familiar with the concept of … Continue reading The Potency of Pattern: Mind the Gap →

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 →

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 … Continue reading DLWP: A New Age of Weather Forecasting →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →