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
Tag: Geometry
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
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
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
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
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
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
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
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
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?
Buffon’s Noodle and the Mathematics of Hillwalking
In addition to some beautiful photos and maps and descriptions of upland challenges in Ireland and abroad, the November issue of The Summit, the Mountain Views Quarterly Newsletter for hikers and hillwalkers, describes a method to find the length of a walk based on ideas originating with the French naturalist and mathematician George-Louis Leclerc, Comte … Continue reading Buffon’s Noodle and the Mathematics of Hillwalking
The Square Root Spiral of Theodorus
The square-root spiral is attributed to Theodorus, a tutor of Plato. It comprises a sequence of right-angled triangles, placed edge to edge, all having a common point and having hypotenuse lengths equal to the roots of the natural numbers. The spiral is built from right-angled triangles. At the centre is an isosceles triangle of unit … Continue reading The Square Root Spiral of Theodorus
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
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
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
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
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
Jung’s Theorem: Enclosing a Set of Points
Let us imagine that we have a finite set $latex {P}&fg=000000$ of points in the plane $latex {\mathbb{R}^2}&fg=000000$ (Fig. 1a). How large a circle is required to enclose them. More specifically, what is the minimum radius of such a bounding circle? The answer is given by Jung's Theorem. Left: a set P of points in … Continue reading Jung’s Theorem: Enclosing a Set of Points
Changing the way that we look at the world
Albrecht Dürer was born in Nuremberg in 1471, third of a family of eighteen children. Were he still living, he would be celebrating his 549th birthday today. Dürer's artistic genius was clear from an early age, as evidenced by a self-portrait he painted when just thirteen [TM187; or search for “thatsmaths” at irishtimes.com ]. In 1494, … Continue reading Changing the way that we look at the world
A New Perspective on Perspective
The development of perspective in the early Italian Renaissance opened the doors of perception just a little wider. Perspective techniques enabled artists to create strikingly realistic images. Among the most notable were Piero della Francesca and Leon Battista Alberti, who invented the method of perspective drawing. For centuries, artists have painted scenes on a sheet … Continue reading A New Perspective on Perspective
John Casey: a Founder of Modern Geometry
Next Tuesday - 12th May - is the 200th anniversary of the birth of John Casey, a notable Irish geometer. Casey was born in 1820 in Kilbeheny, Co Limerick. He was educated in nearby Mitchelstown, where he showed great aptitude for mathematics and also had a gift for languages. He became a mathematics teacher, first … Continue reading John Casey: a Founder of Modern Geometry
Archimedes and the Volume of a Sphere
One of the most remarkable and important mathematical results obtained by Archimedes was the determination of the volume of a sphere. Archimedes used a technique of sub-dividing the volume into slices of known cross-sectional area and adding up, or integrating, the volumes of the slices. This was essentially an application of a technique that was … Continue reading Archimedes and the Volume of a Sphere
Some Fundamental Theorems of Maths
Every branch of mathematics has key results that are so important that they are dubbed fundamental theorems. The customary view of mathematical research is that of establishing the truth of propositions or theorems by rigorous deduction from axioms and definitions. Mathematics is founded upon axioms, basic assumptions that are taken as true. Logical reasoning is … Continue reading Some Fundamental Theorems of Maths
Symplectic Geometry
For many decades, a search has been under way to find a theory of everything, that accounts for all the fundamental physical forces, including gravity. The dictum ``physics is geometry'' is a guiding principle of modern theoretical physics. Einstein's General Theory of Relativity, which emerged just one hundred years ago, is a crowning example of … Continue reading Symplectic Geometry
Chase and Escape: Pursuit Problems
From cheetahs chasing gazelles, through coastguards saving shipwrecked sailors, to missiles launched at enemy aircraft, strategies of pursuit and evasion play a role in many areas of life (and death). From pre-historic times we have been solving such pursuit problems. The survival of our early ancestors depended on their ability to acquire food. This involved … Continue reading Chase and Escape: Pursuit Problems
Gaussian Curvature: the Theorema Egregium
One of greatest achievements of Carl Friedrich Gauss was a theorem so startling that he gave it the name Theorema Egregium or outstanding theorem. In 1828 he published his ``Disquisitiones generales circa superficies curvas'', or General investigation of curved surfaces. Gauss defined a quantity that measures the curvature of a two-dimensional surface. He was inspired by … Continue reading Gaussian Curvature: the Theorema Egregium
A Trapezoidal Prism on the Serpentine
Walking in Hyde Park recently, I spied what appeared to be a huge red pyramid in the middle of the Serpentine. On closer approach, and with a changing angle of view, it became clear that it was prismatic in shape, composed of numerous barrels in red, blue and purple. An isoceles trapezoidal prism A prism … Continue reading A Trapezoidal Prism on the Serpentine
The Miraculous Spiral on Booterstown Strand
We all know what a spiral looks like. Or do we? Ask your friends to describe one and they will probably trace out the form of a winding staircase. But that is actually a helix, a curve in three-dimensional space. A spiral is confined to a plane – it is a flat curve. In general … Continue reading The Miraculous Spiral on Booterstown Strand
Optical Refinements at the Parthenon
The Parthenon is a masterpiece of symmetry and proportion. This temple to the Goddess Athena was built with pure white marble quarried at Pentelikon, about 20km from Athens. It was erected without mortar or cement, the stones being carved to great accuracy and locked together by iron clamps. The building and sculptures were completed in … Continue reading Optical Refinements at the Parthenon
A Glowing Geometric Proof that Root-2 is Irrational
It was a great shock to the Pythagoreans to discover that the diagonal of a unit square could not be expressed as a ratio of whole numbers. This discovery represented a fundamental fracture between the mathematical domains of Arithmetic and Geometry: since the Greeks recognized only whole numbers and ratios of whole numbers, the result … Continue reading A Glowing Geometric Proof that Root-2 is Irrational
Marden’s Marvel
Although polynomial equations have been studied for centuries, even millennia, surprising new results continue to emerge. Marden's Theorem, published in 1945, is one such -- delightful -- result. For centuries, mathematicians have struggled to find roots of polynomials like p(x) ≡ xn + an-1 xn-1 + an-2 xn-2 + an-3 xn-3 + … a1 x … Continue reading Marden’s Marvel
Geodesics on the Spheroidal Earth-II
Geodesy is the study of the shape and size of the Earth, and of variations in its gravitational field. The Earth was originally believed to be flat, but many clues, such as the manner in which ships appear and disappear at the horizon, and the changed perspective from an elevated vantage point, as well as … Continue reading Geodesics on the Spheroidal Earth-II
Geodesics on the Spheroidal Earth – I
Both Quito in Ecuador and Singapore are on the Equator. One can fly due eastward from Singapore and reach Quito in due course. However, this is not the shortest route. The equatorial trans-Pacific route from Singapore to Quito is not a geodesic on Earth! Why not? The General Equation for Geodesics Open a typical text … Continue reading Geodesics on the Spheroidal Earth – I
The Evolute: Envelope of Normals
Every curve in the plane has several other curves associated with it. One of the most interesting and important of these is the evolute. Suppose the curve $latex {\gamma}&fg=000000$ is specified in parametric form $latex {(x(t), y(t))}&fg=000000$ for $latex {t \in [0,1]}&fg=000000$. The centre of curvature $latex {\Gamma = (X, Y)}&fg=000000$ at a particular point … Continue reading The Evolute: Envelope of Normals
Doughnuts and Tonnetze
The circle of fifths is a remarkably useful diagram for the analysis of music. It shows the twelve notes of the chromatic scale arranged in a circle, with notes that are harmonically related (like C and G) being close together and notes that are discordant (like C and C♯) more distant from each other. The … Continue reading Doughnuts and Tonnetze
Vanishing Hyperballs
We all know that the area of a disk --- the interior of a circle --- is $latex {\pi r^2}&fg=000000$ where $latex {r}&fg=000000$ is the radius. Some of us may also remember that the volume of a ball --- the interior of a sphere --- is $latex {\frac{4}{3}\pi r^3}&fg=000000$. The unit disk and ball have … Continue reading Vanishing Hyperballs
A Symbol for Global Circulation
The recycling symbol consisting of three bent arrows is found on bottles, cartons and packaging of all kinds. It originated in 1970 when the Chicago-based Container Corporation of America (CCA) held a competition for the design of a symbol suitable for printing on cartons, to encourage recycling and re-use of packaging materials. The competition for … Continue reading A Symbol for Global Circulation
Malfatti’s Circles
Sometimes the “obvious” answer to a mathematical problem is not the correct one. The case of Malfatti's circles is an example of this. In an equilateral triangle of unit side length, we must draw three non-overlapping circles such that the total area of the circles is maximal. The solution seems obvious: draw three identical circles, … Continue reading Malfatti’s Circles
Drawing Multi-focal Ellipses: The Gardener’s Method
Common-or-Garden Ellipses In an earlier post we saw how a gardener may set out oval flower-beds using a well-known property of ellipses: the sum of the distances from any point on the ellipse to the two foci is always the same value, $latex {2a}&fg=000000$, the length of the major axis. The gardener puts down two stakes … Continue reading Drawing Multi-focal Ellipses: The Gardener’s Method
Locating the HQ with Multi-focal Ellipses
Motivation Ireland has four provinces, the principal city in each being the provincial capital: Belfast, Cork, Dublin and Galway. The map here shows the location of these cities. Now imagine a company that needs to visit and to deliver goods frequently to all four cities. Where might they locate their HQ to minimize transport costs … Continue reading Locating the HQ with Multi-focal Ellipses
It’s as Easy as Pi
Every circle has the property that the distance around it is just over three times the distance across. This has been known since the earliest times [see TM120 or search for “thatsmaths” at irishtimes.com]. The constant ratio of the circumference to the diameter, denoted by the Greek letter pi, is familiar to every school-child. You … Continue reading It’s as Easy as Pi
Who First Proved that C / D is Constant?
Every circle has the property that the distance around it is just over three times the distance across. This has been “common knowledge” since the earliest times. But mathematicians do not trust common knowledge; they demand proof. Who was first to prove that all circles are similar, in the sense that the ratio of circumference … Continue reading Who First Proved that C / D is Constant?
Quadrivium: The Noble Fourfold Way
According to Plato, a core of mathematical knowledge – later known as the Quadrivium – was essential for an understanding of the Universe. The curriculum was outlined in Plato's Republic. The name Quadrivium means four ways, but this term was not used until the time of Boethius in the 6th century AD [see TM119 or search … Continue reading Quadrivium: The Noble Fourfold Way
ThatsMaths 