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 zero on the boundary, and that lengths varied in proportion to the temperature.
Posts Tagged 'Geometry'
The Whole is Greater than the Part — Or is it?
Published April 21, 2022 Irish Times Leave a CommentTags: Cantor, Geometry, Set Theory
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, are specific assumptions that may be considered as self-evident, for example “the whole is greater than the part” [TM232 or search for “thatsmaths” at irishtimes.com].
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Buffon’s Noodle and the Mathematics of Hillwalking
Published December 2, 2021 Irish Times Leave a CommentTags: Geometry, Geophysics
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 de Buffon [TM224 or search for “thatsmaths” at irishtimes.com].
[Image from November issue of The Summit, the Mountain Views Quarterly Newsletter.]
The Square Root Spiral of Theodorus
Published October 14, 2021 Occasional Leave a CommentTags: Analysis, Geometry
Spiral of Theodorus [image Wikimedia Commons].
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 side and hypotenuse 
Seeing beyond the Horizon
Published June 17, 2021 Irish Times Leave a CommentTags: Geometry, Geophysics
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 1% of the true value but, since he was not aware of atmospheric refraction and made no allowance for it, this high precision must have been fortuitous [TM213 or search for “thatsmaths” at irishtimes.com].
Snowdonia photographed from the Ben of Howth, 12 January 2021. Photo: Niall O’Carroll (Instagram).
Al Biruni and the Size of the Earth
Published June 10, 2021 Occasional Leave a CommentTags: Geometry, History, Trigonometry
Abu Rayhan al-Biruni (AD 973–1048)
Al Biruni at Persian Scholars Pavilion in Vienna.
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, the accuracy claimed must be regarded with suspicion.
Al-Biruni assumed that the Earth is a perfect sphere of (unknown) radius 
Multi-faceted aspects of Euclid’s Elements
Published May 20, 2021 Irish Times Leave a CommentTags: Geometry
A truncated octahedron within the coronavirus [image from Cosico et al, 2020].
A Model for Elliptic Geometry
Published May 13, 2021 Occasional Leave a CommentTags: Geometry, Spherical Trigonometry
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 topic. Nikolai Ivanovich Lobachevsky, around 1830, and János Bolyai, in 1832, published treatises on what is now called hyperbolic geometry.
Mamikon’s Theorem and the area under a cycloid arch
Published February 25, 2021 Occasional Leave a CommentTags: Analysis, Geometry
The cycloid, the locus of a point on the rim of a rolling disk.
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
where 
* * * * *
That’s Maths II: A Ton of Wonders
by Peter Lynch now available.
Full details and links to suppliers at
http://logicpress.ie/2020-3/
>> Review in The Irish Times <<
* * * * *
Continue reading ‘Mamikon’s Theorem and the area under a cycloid arch’
Decorating Christmas Trees with the Four Colour Theorem
Published December 3, 2020 Irish Times Leave a CommentTags: Geometry, Topology
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 required to avoid such a catastrophe? [TM200 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘Decorating Christmas Trees with the Four Colour Theorem’
From Impossible Shapes to the Nobel Prize
Published October 8, 2020 Occasional 1 CommentTags: Astronomy, Geometry
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.
Impossible triangle sculpture in Perth, Western Australia [image Wikimedia Commons].
Continue reading ‘From Impossible Shapes to the Nobel Prize’
Jung’s Theorem: Enclosing a Set of Points
Published August 27, 2020 Occasional Leave a CommentTags: Geometry
Let us imagine that we have a finite set 
Left: a set P of points in the real plane. Right: The span s is the maximum distance between two points of P.
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Changing the way that we look at the world
Published May 21, 2020 Irish Times Leave a CommentTags: Geometry
Self-portrait by Dürer when aged 26.
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, Dürer visited Italy, where he travelled for a year. A novel connection between art and mathematics was emerging around that time. By using rules of perspective, artists could represent objects in three-dimensional space on a plane canvas with striking realism. Dürer was convinced that “the new art must be based upon science; in particular, upon mathematics, as the most exact, logical, and graphically constructive of the sciences”.
Continue reading ‘Changing the way that we look at the world’
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.
School of Athens, a fresco painted by Raphael in 1509-11 illustrates the power of perspective.
John Casey: a Founder of Modern Geometry
Published May 7, 2020 Irish Times Leave a CommentTags: Geometry, Ireland
John Casey (1820-1891).
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 in Tipperary Town and later in Kilkenny [TM186; or search for “thatsmaths” at irishtimes.com ].
Archimedes and the Volume of a Sphere
Published November 28, 2019 Occasional 1 CommentTags: Archimedes, Geometry
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 — close to two thousand years later — formulated as integral calculus.
Cone, sphere and cylinder on the same base. The volumes are in the ratios 1 : 2 : 3 [image from mathigon.org].
Some Fundamental Theorems of Maths
Published October 24, 2019 Occasional Leave a CommentTags: Algebra, Analysis, Geometry
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 then used to deduce the consequences of those axioms with each major result designated as a theorem.
As each new theorem is proved, it provides a basis for the establishment of further results. The most important and fruitful theorem in each area of maths is often named as the fundamental theorem of that area. Thus, we have the fundamental theorems of arithmetic, algebra and so on. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus.
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 this synergy. He showed how matter distorts the geometry of space and this geometry determines the motion of matter. The central idea is encapsulated in an epigram of John A Wheeler:
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 chasing and killing animals, and success depended on an understanding of relative speeds and optimal pursuit paths.
Gaussian Curvature: the Theorema Egregium
Published December 27, 2018 Occasional 9 CommentsTags: Gauss, Geometry
Surfaces of positive curvature (top), negative curvature (middle) and vanishing curvature (bottom) [image credit: NASA].
characterizes the intrinsic geometry of a surface.
Continue reading ‘Gaussian Curvature: the Theorema Egregium‘
A Trapezoidal Prism on the Serpentine
Published September 13, 2018 Occasional Leave a CommentTags: Geometry
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.
Changing perspective on approach to the Mastaba
The Miraculous Spiral on Booterstown Strand
Published August 16, 2018 Irish Times Leave a CommentTags: biology, Geometry, Geophysics
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 terms, a spiral is formed by a point moving around a fixed centre while its distance increases or decreases as it revolves [see TM145, or search for “thatsmaths” at irishtimes.com].
The spiral sandbank on Booterstown strand (satellite image digitally enhanced by Andrew Lynch).
Continue reading ‘The Miraculous Spiral on Booterstown Strand’
Optical Refinements at the Parthenon
Published June 21, 2018 Irish Times Leave a CommentTags: Geometry, History
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 just 15 years, between 447 and 432 BC. [TM141 or search for “thatsmaths” at irishtimes.com].
A Glowing Geometric Proof that Root-2 is Irrational
Published May 24, 2018 Occasional 1 CommentTags: Geometry
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 meant that there was no number to describe the diagonal of a unit square.
Continue reading ‘A Glowing Geometric Proof that Root-2 is Irrational’
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.
Cubic with roots at x=1, x=2 and x=3.
Geodesics on the Spheroidal Earth-II
Published April 19, 2018 Irish Times Leave a CommentTags: Geometry, Geophysics, Navigation, Pythagoras
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 astronomical phenomena, convinced savants of its spherical shape. In the third century BC, Eratosthenes accurately estimated the circumference of the Earth [TM137 or search for “thatsmaths” at irishtimes.com].
Geodesic at bearing of 60 degrees from Singapore. Passes close to Quito, Ecuador. Note that it is not a closed curve: it does not return to Singapore.
Geodesics on the Spheroidal Earth – I
Published April 12, 2018 Occasional Leave a CommentTags: Geometry, Geophysics, Navigation, Pythagoras
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?
A drastically flattened spheroid. Clearly, the equatorial route between the blue and red points is not the shortest path.
The Evolute: Envelope of Normals
Published February 22, 2018 Occasional Leave a CommentTags: Analysis, Geometry
Every curve in the plane has several other curves associated with it. One of the most interesting and important of these is the evolute.
Sin t (blue) and its evolute (red).
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 Tonnetz diagram (note that the arrangement here is inverted relative to that used in the text. It appears that there is no rigid standard, and several arrangements are in use) [Image from WikimediaCommons].
Spherical ball contained within a cubic region
[Image from https://grabcad.com ].
We all know that the area of a disk — the interior of a circle — is 
A Symbol for Global Circulation
Published November 23, 2017 Occasional Leave a CommentTags: Geometry, Topology
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.
Original (Moebius) and a variation (3-twist) of the universal recycling symbol.
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, each one tangent to two sides and to the other two circles (above figure, left). This is certainly the most symmetric arrangement possible. However, it turns out not to be the optimal solution. There is another arrangement (above figure, right) for which the three circles have greater total area.
Drawing Multi-focal Ellipses: The Gardener’s Method
Published August 31, 2017 Occasional Leave a CommentTags: Algorithms, Geometry
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, 
Gardener’s method of drawing an ellipse [Image Wikimedia].
Locating the HQ with Multi-focal Ellipses
Published August 24, 2017 Occasional Leave a CommentTags: Algorithms, Geometry
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 and travel times?
One possibility is to find the location with the smallest distance sum:
where 
Continue reading ‘Locating the HQ with Multi-focal Ellipses’
It’s as Easy as Pi
Published August 3, 2017 Irish Times Leave a CommentTags: Archimedes, Geometry, Number Theory, 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 might expect to find a proof in Euclid’s Elements of Geometry, he could not prove it, and he made no mention of the ratio (see last week’s post).
Who First Proved that C / D is Constant?
Published July 27, 2017 Occasional Leave a CommentTags: Archimedes, Geometry, Pi
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 C to diameter D has the same value for all?
Slicing a disk to estimate pi (Image Wikimedia).
Quadrivium: The Noble Fourfold Way
Published July 20, 2017 Irish Times Leave a CommentTags: Arithmetic, Astronomy, Geometry, History, Music
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 for “thatsmaths” at irishtimes.com].
Image from here.
It is said that an inscription over the entrance to Plato’s Academy read “Let None But Geometers Enter Here”. This indicated that the Quadrivium was a prerequisite for the study of philosophy in ancient Greece.
Musical notes that are simply related to each other have a pleasing effect when sounded together. Each tone has a characteristic rate of oscillation, or frequency. For example, Middle C on the piano oscillates 264 times per second or has a frequency of 264 Hz (Hertz). If the frequencies of two notes have a ratio of two small whole numbers, the notes are harmonically related and sound pleasant when played together.
Beats from two notes close in pitch.
Torricelli’s Trumpet & the Painter’s Paradox
Published April 13, 2017 Occasional 1 CommentTags: Analysis, Geometry, Recreational Maths
Torricelli’s Trumpet
Evangelista Torricelli, a student of Galileo, is remembered as the inventor of the barometer. He was also a talented mathematician and he discovered the remarkable properties of a simple geometric surface, now often called Torricelli’s Trumpet. It is the surface generated when the curve 
Continue reading ‘Torricelli’s Trumpet & the Painter’s Paradox’
Voronoi Diagrams: Simple but Powerful
Published February 2, 2017 Irish Times 2 CommentsTags: Algorithms, Geometry
We frequently need to find the nearest hospital, surgery or supermarket. A map divided into cells, each cell covering the region closest to a particular centre, can assist us in our quest. Such a map is called a Voronoi diagram, named for Georgy Voronoi, a mathematician born in Ukraine in 1868. He is remembered today mostly for his diagram, also known as a Voronoi tessellation, decomposition, or partition. [TM108 or search for “thatsmaths” at irishtimes.com].
Voronoi diagram drawn using the applet of Paul Chew (see Sources below).
Unsolved: the Square Peg Problem
Published December 29, 2016 Occasional Leave a CommentTags: Geometry, Topology
The idiom “square peg in a round hole” expresses a mismatch or misfit, often referring to somebody in the wrong profession. It may also indicate a difficult or impossible task but, of course, it is quite simple to fit a square peg in a round hole, hammering it in until the corners are tight against the circular boundary of the hole. Since the peg may be oriented at any angle, there are an infinite number of ways to fit a square within a circle. In contract, for a boomerang-shaped hole, there is just one way to draw a square with its vertices on the curve.
Kepler’s Magnificent Mysterium Cosmographicum
Published October 13, 2016 Occasional 2 CommentsTags: Astronomy, Geometry, History
Johannes Kepler’s amazing book, Mysterium Cosmographicum, was published in 1596. Kepler’s central idea was that the distance relationships between the six planets (only six were known at that time) could be represented by six spheres separated by the five Platonic solids. For each of these regular polyhedra, there is an inner and an outer sphere. The inner sphere is tangent to the centre of each face and the outer sphere contains all the vertices of the polyhedron.
Figure generated using Mathematica Demonstration [2].
Continue reading ‘Kepler’s Magnificent Mysterium Cosmographicum’
Heron’s Theorem: a Tool for Surveyors
Published September 8, 2016 Occasional Leave a CommentTags: Geometry
Heron was one of the great Greek mathematicians of Alexandria, following in the tradition of Euclid, Archimedes, Eratosthenes and Apollonius. He lived in the first century, from about AD 10 to AD 70. His interests were in practical rather than theoretical mathematics and he wrote on measurement, mechanics and engineering. He devised a steam-powered device and a wind-wheel that operated an organ. He is regarded as the greatest experimenter of antiquity, but it is for a theorem in pure geometry that mathematicians remember him today.
Heron of Alexandria. Triangle of sides a, b and c and altitude h.
The Tunnel of Eupalinos in Samos
Published September 1, 2016 Irish Times Leave a CommentTags: Geometry, Pythagoras
The tunnel of Eupalinos on the Greek island of Samos, over one kilometre in length, is one of the greatest engineering achievements of the ancient world [TM098, or search for “thatsmaths” at irishtimes.com].
Approximate course of the tunnel of Eupalinos in Samos.
It is well-known that an ellipse is the locus of all points such that the sum of their distances from two fixed points, the foci, is constant. Thus, a gardener may map out an elliptical flower-bed by driving two stakes into the ground, looping a rope around them and pulling it taut with a pointed stick, tracing out a curve on the ground.
You can put a square peg in a round hole.
Shapes between circles and squares have proved invaluable to engineers and have also found their way onto our dinner tables. A plate in the shape of a `squircle’ is shown in this figure .
Squircular plate: holds more food and is easier to store.
Bending the Rules to Square the Circle
Published June 23, 2016 Occasional Leave a CommentTags: Geometry
Squaring the circle was one of the famous Ancient Greek mathematical problems. Although studied intensively for millennia by many brilliant scholars, no solution was ever found. The problem requires the construction of a square having area equal to that of a given circle. This must be done in a finite number of steps, using only ruler and compass.
Taking unit radius for the circle, the area is π, so the square must have a side length of √π. If we could construct a line segment of length π, we could also draw one of length √π. However, the only constructable numbers are those arising from a unit length by addition, subtraction, multiplication and division, together with the extraction of square roots.
Bloom’s attempt to Square the Circle
Published June 16, 2016 Irish Times Leave a CommentTags: Geometry, Recreational Maths
The quadrature of the circle is one of the great problems posed by the ancient Greeks. This “squaring of the circle” was also an issue of particular interest to Leopold Bloom, the central character in James Joyce’s novel Ulysses, whom we celebrate today, Bloomsday, 16 June 2016 [see TM093, or search for “thatsmaths” at irishtimes.com].
Joyce’s Tower, Sandycove, Co Dublin.
The challenge is to construct a square with area equal to that of a given circle using only the methods of classical geometry. Thus, only a ruler and compass may be used in the construction and the process must terminate in a finite number of steps.
Mathematics Everywhere (in Blackrock Station)
Published May 26, 2016 Occasional Leave a CommentTags: Geometry, Recreational Maths
Mathematics is everywhere. We are often unaware of it but, when we observe our environment consciously, we can see mathematical structures all around us.
This footbridge is a cornucopia of mathematical forms.
Continue reading ‘Mathematics Everywhere (in Blackrock Station)’
Franc-carreau or Fair-square
Published February 11, 2016 Occasional Leave a CommentTags: Games, Geometry, History, Probability
Franc-carreau is a simple game of chance, like the roll-a-penny game often seen at fairs and fêtes. A coin is tossed or rolled down a wooden chute onto a large board ruled into square segments. If the player’s coin lands completely within a square, he or she wins a coin of equal value. If the coin crosses a dividing line, it is lost.
The playing board for Franc-Carreau is shown above, together with a winning coin (red) contained within a square and a loosing one (blue) crossing a line. As the precise translation of franc-carreau appears uncertain, the name “fair square” would seem appropriate.
The question is: What size should the coin be to ensure a 50% chance of winning?
ThatsMaths 
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