## Posts Tagged 'Geometry'

### The Tunnel of Eupalinos in Samos

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.

### Slicing Doughnuts

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.

### Squircles

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

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

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)

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.

### Franc-carreau or Fair-square

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

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

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.

Left: Archimedean spiral. Centre: Fermat spiral. Right: Hyperbolic spiral.

### Which Way did the Bicycle Go?

“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.

As front wheel moves along the positive x-axis the back wheel, initially at (0,a), follows a tractrix curve (see below).

### The Ubiquitous Cycloid

Puzzle: However fast a train is travelling, part of it is moving backwards. Which part?
For the answer, see the end of this post.

Timelapse image of bike with lights on the wheel-rims. [Photo from Website of Alexandre Wagemakers, with thanks]

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

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.

Double Portrait of Jean de Dinteville and Georges de Selve (“The Ambassadors”),
Hans Holbein the Younger, 1533. Oil and tempera on oak, National Gallery, London

### Maps on the Web

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

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].

Mercator projection of the Earth, truncated at 75 degrees North and South [Wikimedia Commons, author: Strebe].

### Brouwer’s Fixed-Point Theorem

A climber sets out at 8 a.m. from sea-level, reaching his goal, a 2,000 metre peak, ten hours later. He camps at the summit and starts his return the next morning at 8 a.m. After a leisurely descent, he is back at sea-level ten hours later.

Is there some time of day at which his altitude is identical on both days? Try to answer this before reading on.

### Mode-S: Aircraft Data improves Weather Forecasts

A simple application of vectors yields valuable new wind observations for weather forecasting [see this week’s That’s Maths column (TM065) or search for “thatsmaths” at irishtimes.com].

The air speed is A (blue), the wind speed is W (black) and the ground speed is G (red). Since the ground speed is the resultant (vector sum) of air speed and wind speed, a simple vector subtraction gives the wind speed: W= G – A.

### A King of Infinite Space: Euclid I.

O God, I could be bounded in a nutshell, and count myself a king of infinite space …
[Hamlet]

Euclid. Left: panel from series Famous Men by Justus of Ghent. Right: Statue in the Oxford University Museum of Natural History.

### Seifert Surfaces for Knots and Links.

We are all familiar with knots. Knots keep our boats securely moored and enable us to sail across the oceans. They also reduce the cables and wires behind our computers to a tangled mess. Many fabrics are just complicated knots of fibre and we know how they can unravel.

From left to right: Figure-of-8 knot 4_1, knot 6_3 , knot 7_1, and knot 8_5 [Drawn with SeifertView (image from Van Wijk, 2006)].

### New Curves for Old: Inversion

Special Curves

A large number of curves, called special curves, have been studied by mathematicians. A curve is the path traced out by a point moving in space. To keep things simple, we assume that the point is confined to two-dimensional Euclidean space ${\mathbb{R}^2}$ so that it generates a plane curve as it moves. This, a curve results from a mapping ${\mathbf{\gamma} : [a,b]\longrightarrow \mathbb{R}^2}$. Continue reading ‘New Curves for Old: Inversion’

### Earth’s Shape and Spin Won’t Make You Thin

Using a simple pendulum we can determine the shape of the Earth. That amazing story is told in this week’s That’s Maths column in The Irish Times (TM057 or search for “thatsmaths” at www.irishtimes.com ).

Is the Earth oblate like an orange (Newton) or prolate like a lemon (the Cassinis)?

### Curves with Singularities

Many of the curves that we study are smooth, with a well-defined tangent at every point. Points where the derivative is defined — where there is a definite slope — are called regular points. However, many curves also have exceptional points, called singularities. If the derivative is not defined at a point, or if it does not have a unique value there, the point is singular.

Slinky traces a smooth helical curve in three dimensions.

Generally, if we zoom in close to a point on a curve, the curve looks increasingly like a straight line. However, at a singularity, it may look like two lines crossing or like two lines whose slopes converge as the resolution increases. Continue reading ‘Curves with Singularities’

### Cartoon Curves

The powerful and versatile computational software program called Mathematica is widely used in science, engineering and mathematics. There is a related system called Wolfram Alpha, a computational knowledge engine, that can do Mathematica calculations and that runs on an iPad.

Yogi Bear Curve. The Mathematica command to generate this is given below.

### The Biggest Harp in Ireland

This week’s That’s Maths column in The Irish Times (TM052, or search for “thatsmaths” at irishtimes.com) is about “Samuel Beckett Playing Bridge in Dublin”.

Image from Tiger Dublin Fringe Festival website.
Photo Credit: Ciara Corrigan

### Beauty is the First Test

This week, That’s Maths in The Irish Times (TM048: Search for “thatsmaths” at irishtimes.com) is about the beauty of mathematics.

Indra’s Indigo, detail of patchwork quilt by textile artist Janice Gunner.

### The High-Power Hypar

Maths frequently shows us surprising and illuminating connections between physical systems that are not obviously related: the analysis of one system often turns out to be ideally suited for describing another. To illustrate this, we will show how a surface in three dimensional space — the hyperbolic paraboloid, or hypar — pops up in unexpected ways and in many different contexts.

Warszawa Ochota railway station, a hypar structure
[Image Wikimedia Commons].

### Clothoids Drive Us Round the Bend

The article in this week’s That’s Maths column in the Irish Times ( TM043 ) is about the mathematical curves called clothoids, used in the design of motorways.

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### The Unity of Mathematics

This week, That’s Maths in The Irish Times ( TM041  ) is about an ambitious program to unify mathematics.

Mathematics expands! Results once proven to be true remain forever true. They are not displaced by subsequent results, but absorbed in an ever-growing theoretical web. Thus, it is increasingly difficult for any individual mathematician to have a comprehensive understanding of even a single field of mathematics: the web of knowledge grows so fast that no-one can master it all.

### Euclid in Technicolor

The article in this week’s That’s Maths column in the Irish Times ( TM039 ) is about Oliver Byrne’s amazing technicolor Elements of Euclid, recently re-published by Taschen.

### Pythagorean triples

The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. It can be written as an equation,

a2 + b2 = c2,

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

### Euler’s Gem

This week, That’s Maths in The Irish Times ( TM032  ) is about Euler’s Polyhedron Formula and its consequences.

Euler’s Polyhedron Formula

The highlight of the thirteenth and final book of Euclid’s Elements was the proof that there are just five “Platonic solids”. Recall that a regular polygon is a plane figure with all sides and angles equal, for example a square. By joining identical polygons together, we can form solid bodies called regular polyhedra. Continue reading ‘Euler’s Gem’

### Hyperbolic Triangles and the Gauss-Bonnet Theorem

Poincaré’s half-plane model for hyperbolic geometry comprises the upper half plane ${\mathbf{H} = \{(x,y): y>0\}}$ together with a metric

$\displaystyle d s^2 = \frac { d x^2 + d y^2 } { y^2 } \,.$

It is remarkable that the entire structure of the space ${(\mathbf{H},ds)}$ follows from the metric.
Continue reading ‘Hyperbolic Triangles and the Gauss-Bonnet Theorem’

### Poincare’s Half-plane Model (bis)

In a previous post, we considered Poincaré’s half-plane model for hyperbolic geometry in two dimensions. The half-plane model comprises the upper half plane ${H = \{(x,y): y>0\}}$ together with a metric

$\displaystyle d s^2 = \frac { d x^2 + d y^2 } { y^2 } \,.$

It is remarkable that the entire structure of the space follows from the metric.

In the earlier post, we derived the total curvature by evaluating the Riemann tensor. Here, we compute the curvature directly, using Gauss’s “Remarkable Theorem”.
Continue reading ‘Poincare’s Half-plane Model (bis)’

### Geometry in and out of this World

Hyperbolic geometry is the topic of the That’s Maths column in the Irish Times this week (TM031 or  click Irish Times and search for “thatsmaths”).

Living on a Sphere

The shortest distance between two points is a straight line. This is one of the basic principles of Euclidean geometry. But we live on a spherical Earth, and we cannot travel the straight line path from Dublin to New York: we have to stick to the surface of the globe, and the geometry we need is more complicated than the plane geometry of Euclid. Spherical geometry is central for the study of geophysics and astronomy, and vital for navigation.

### Poincaré’s Half-plane Model

For two millennia, Euclid’s geometry held sway. However, his fifth axiom, the parallel postulate, somehow wrankled: it was not natural, obvious nor comfortable like the other four.

In the first half of the nineteenth century, three mathematicians, Bolyai, Lobachevesky and Gauss, independently of each other, developed a form of geometry in which the parallel postulate no longer applied. This later bacame known as hyperbolic geometry.

### The School of Athens

That’s Maths in the Irish Times this week ( TM024: search for “thatsmaths” ) deals with perspective in art and its mathematical expression as projective geometry.

### Gauss Misses a Trick

Carl Friedrich Gauss is generally regarded as the greatest mathematician of all time. The profundity and scope of his work is remarkable. So, it is amazing that, while he studied non-Euclidian geometry and defined the curvature of surfaces in space, he overlooked a key connection between curvature and geometry.  As a consequence, decades passed before a model demonstrating the consistency of hyperbolic geometry emerged.

### Topology Underground

That’s Maths in this week’s Irish Times ( TM013 ) is about the branch of mathematics called topology, and treats the map of the London Underground network as a topological map.

Topology is the area of mathematics dealing with basic properties of space, such as continuity and connectivity. It is a powerful unifying framework for mathematics. Topology is concerned with properties that remain unchanged under continuous deformations, such as stretching or bending but not cutting or gluing. Continue reading ‘Topology Underground’

### Pons Asinorum

The fifth proposition in Book I of Euclid’s Elements states that the two base angles of an isosceles triangle are equal (in the figure below, angles B and C).

For centuries, this result has been known as Pons Asinorum, or the Bridge of Asses, apparently a metaphor for a problem that separates bright sparks from dunces. Euclid proved the proposition by extending the sides AB and AC and drawing lines to form additional triangles. His proof is quite complicated. Continue reading ‘Pons Asinorum’

### Where Circles are Square

Imagine a world where circles are square and π is equal to 4. Strange as it seems, we live in such a world: urban geometry is determined by the pattern of streets in a typical city grid and distance “as the crow flies” is not the distance that we have to travel from place to place. Continue reading ‘Where Circles are Square’

### Shackleton’s spectacular boat-trip

A little mathematics goes a long, long way; in the adventure recounted below, elementary geometry brought an intrepid band of six men 800 sea miles across the treacherous Southern Ocean, and led to the saving of 28 lives. Continue reading ‘Shackleton’s spectacular boat-trip’

### Carving up the Globe

This week, That’s Maths (TM007) describes various ways of dividing up the sphere. This is an important problem in geometry, biology, chemistry, astronomy, meteorology and climate modelling. Continue reading ‘Carving up the Globe’