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.

## Posts Tagged 'Geometry'

### A Model for Elliptic Geometry

Published May 13, 2021 Occasional Leave a CommentTags: Geometry, Spherical Trigonometry

### Mamikon’s Theorem and the area under a cycloid arch

Published February 25, 2021 Occasional Leave a CommentTags: Analysis, Geometry

**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 is the angle through which the disk has rotated. The centre of the disk is at .

* * * * *

**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.

Penrose has also come up with some ingenious mathematical inventions. He discovered a way of defining a pseudo-inverse for matrices that are singular, he rediscovered an “impossible object”, now called the Penrose Triangle, and he discovered that the plane could be tiled in a non-periodic way using two simple polygonal shapes called kites and darts.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 of points in the plane (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.

Continue reading ‘Jung’s Theorem: Enclosing a Set of Points’

### Changing the way that we look at the world

Published May 21, 2020 Irish Times Leave a CommentTags: Geometry

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.

### John Casey: a Founder of Modern Geometry

Published May 7, 2020 Irish Times Leave a CommentTags: Geometry, Ireland

Next Tuesday – 12th May – is the 200^{th} 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.

Continue reading ‘Archimedes and the Volume of a Sphere’### 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 Comments
Tags: Gauss, Geometry

*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 his work on geodesy, surveying and map-making, which involved taking measurements on the surface of the Earth. The total curvature — or Gaussian curvature — depends only on measurements within the surface and Gauss showed that its value is independent of the coordinate system used. This is his

*Theorema Egregium*. The Gaussian curvature 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.

### 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].

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.

### 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.

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.

We all know that the area of a disk — the interior of a circle — is where is the radius. Some of us may also remember that the volume of a ball — the interior of a sphere — 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.

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, , the length of the major axis. The gardener puts down two stakes and loops a piece of rope around them. Using a stick, he pulls the loop taut, marking the points around a curve. This is illustrated here.

Continue reading ‘Drawing Multi-focal Ellipses: The Gardener’s Method’### 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 is the position of the HQ and are the positions of the cities.

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?

### 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 6^{th} century AD [see TM119 or search for “thatsmaths” at irishtimes.com].

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.

### Torricelli’s Trumpet & the Painter’s Paradox

Published April 13, 2017 Occasional 1 CommentTags: Analysis, Geometry, Recreational Maths

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 for is rotated in 3-space about the x-axis.

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].

### 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.

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.

### 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 .

### 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].

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.

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?*

### Mathematics Solving Crimes

Published November 19, 2015 Irish Times Leave a CommentTags: Applied Maths, Geometry, Statistics

What use is maths? Why should we learn it? A forensic scientist could answer that virtually all the mathematics we learn at school is used to solve crimes. Forensic science considers physical evidence relating to criminal activity and practitioners need competence in mathematics as well as in the physical, chemical and biological sciences [TM080: search for “thatsmaths” at irishtimes.com ].

Trigonometry, the measurement of triangles, is used in the analysis of blood spatter. The shape indicates the direction from which the blood has come. The most probable scenario resulting in blood spatter on walls and floor can be reconstructed using trigonometric analysis. Such analysis can also determine whether the blood originated from a single source or from multiple sources.

### Mowing the Lawn in Spirals

Published October 29, 2015 Occasional Leave a CommentTags: Archimedes, Geometry

*Like a circle in a spiral / Like a wheel within a wheel / **Never ending or beginning / On an ever-spinning reel. *The Windmills Of Your Mind

Broadly speaking, a spiral curve originates at a central point and gets further away (or closer) as it revolves around the point. Spirals abound in nature, being found at all scales from the whorls at our finger-tips to vast rotating spiral galaxies. The seeds in a sunflower are arranged in spiral segments. In the technical world, the grooves of a gramophone record and the coils of a watch balance-spring are spiral in form.

### Which Way did the Bicycle Go?

Published October 8, 2015 Occasional Leave a CommentTags: Analysis, Geometry, Mechanics

*“A bicycle, certainly, but not the bicycle,” said Holmes.*

In Conan-Doyle’s short story *The Adventure of the Priory School* Sherlock Holmes solved a mystery by deducing the direction of travel of a bicycle. His logic has been minutely examined in many studies, and it seems that in this case his reasoning fell below its normal level of brilliance.

### The Ubiquitous Cycloid

Published September 17, 2015 Irish Times Leave a CommentTags: Geometry, History, Puzzles

**Puzzle:** *However fast a train is travelling, part of it is moving backwards. Which part?*

For the answer, see the end of this post.

Imagine a small light fixed to the rim of a bicycle wheel. As the bike moves, the light rises and falls in a series of arches. A long-exposure nocturnal photograph would show a *cycloid*, the curve traced out by a point on a circle as it rolls along a straight line. A light at the wheel-hub traces out a straight line. If the light is at the mid-point of a spoke, the curve it follows is a curtate cycloid. A point outside the rim traces out a prolate cycloid, with a backward loop. [TM076; or search for “thatsmaths” at irishtimes.com ]

### Holbein’s Anamorphic Skull

Published September 10, 2015 Occasional 1 CommentTags: Geometry, History

Hans Holbein the Younger, court painter during the reign of Henry VIII, produced some spectacular works. Amongst the most celebrated is a double portrait of Jean de Dinteville, French Ambassador to Henry’s court, and Georges de Selve, Bishop of Lavaur. Painted by Holbein in 1533, the picture, known as *The Ambassadors*, hangs in the National Gallery, London.

**In a nutshell: ** In web maps, geographical coordinates are projected as if the Earth were a perfect sphere. The results are great for general use but not for high-precision applications. Continue reading ‘Maps on the Web’

### Mercator’s Marvellous Map

Published May 21, 2015 Irish Times Leave a CommentTags: Geometry, Geophysics, Spherical Trigonometry

Try to wrap a football in aluminium foil and you will discover that you have to crumple up the foil to make it fit snugly to the ball. In the same way, it is impossible to represent the curved surface of the Earth on a flat plane without some distortion.** ** [See this week’s *That’s Maths* column (TM068): search for “thatsmaths” at irishtimes.com].