The Dual in the Crown

The beginning of topology is often traced to Euler's solution of a puzzle, the Bridges of Königsberg. The problem posed was to follow a path through the Prussian city that crossed all seven bridges exactly once. Euler proved that the problem has no solution. He drastically simplifying it by replacing the geographical context by a … Continue reading The Dual in the Crown →

The Cosmology of the Divine Comedy

If you think poetry and maths are poles apart, think again. Around the sixth century, Indian poet and mathematician Virahanka codified the structure of Sanskrit poetry, formulating rules for the patterns of long and short syllables. In this process, a sequence emerged in which each term is the sum of the preceding two. This is … Continue reading The Cosmology of the Divine Comedy →

Adding a Point to Make a Space Compact

The real line is an example of a locally compact Hausdorff space. In a Hausdorff space, two distinct points have disjoint neighbourhoods. As the old joke says, ``any two points can be housed off from each other''. We will define local compactness below. The one-point compactification is a way of embedding a locally compact Hausdorff … Continue reading Adding a Point to Make a Space Compact →

Spiric curves and phase portraits

We are very familiar with the conic sections, the curves formed from the intersection of a plane with a cone. There is another family of curves, the Spiric sections, formed by the intersections of a torus by planes parallel to its axis. Like the conics, they come in various forms, depending upon the distance of … Continue reading Spiric curves and phase portraits →

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 →

The 3-sphere: Extrinsic and Intrinsic Forms

The circle in two dimensions and the sphere in three are just two members of an infinite family of hyper-surfaces. By analogy with the circle $latex {\mathbb{S}^1}&fg=000000$ in the plane $latex {\mathbb{R}^2}&fg=000000$ and the sphere $latex {\mathbb{S}^2}&fg=000000$ in three-space $latex {\mathbb{R}^3}&fg=000000$, we can consider hyper-spheres in higher dimensional spaces. In particular, we will consider the … Continue reading The 3-sphere: Extrinsic and Intrinsic Forms →

The Chromatic Number of the Plane

To introduce the problem in the title, we begin with a quotation from the Foreword, written by Branko Grünbaum, to the book by Alexander Soifer (2009): The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators: If each point of the plane is to be given a color, how many colors … Continue reading The Chromatic Number of the Plane →

Chiral and Achiral Knots

An object is chiral if it differs from its mirror image. The favourite example is a hand: our right hands are reflections of our left ones. The two hands cannot be superimposed. The term chiral comes from $latex {\chi\epsilon\rho\iota}&fg=000000$, Greek for hand. If chirality is absent, we have an achiral object. According to Wikipedia, it … Continue reading Chiral and Achiral Knots →

Embedding: Reconstructing Solutions from a Delay Map

M In mechanical systems described by a set of differential equations, we normally specify a complete set of initial conditions to determine the motion. In many dynamical systems, some variables may easily be observed whilst others are hidden from view. For example, in astronomy, it is usual that angles between celestial bodies can be measured … Continue reading Embedding: Reconstructing Solutions from a Delay Map →

The Signum Function may be Continuous

Abstract: Continuity is defined relative to a topology. For two distinct topological spaces $latex {(X,\mathcal{O}_1)}&fg=000000$ and $latex {(X,\mathcal{O}_2)}&fg=000000$ having the same underlying set $latex {X}&fg=000000$ but different families of open sets, a function may be continuous in one but discontinuous in the other. The signum function is defined on the real line as follows: $latex … Continue reading The Signum Function may be Continuous →

The Dimension of a Point that isn’t there

A slice of Swiss cheese has one-dimensional holes; a block of Swiss cheese has two-dimensional holes. What is the dimension of a point? From classical geometry we have the definition ``A point is that which has no parts'' --- also sprach Euclid. A point has dimension zero, a line has dimension one, a plane has … Continue reading The Dimension of a Point that isn’t there →

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 →

Doughnuts and Dumplings are Distinct: Homopoty-101

As everyone knows, a torus is different from a sphere. Topology is the study of properties that remain unchanged under continuous distortions. A square can be deformed into a circle or a sphere into an ellipsoid, whether flat like an orange or long like a lemon or banana. Technically, sets are topologically equivalent if there … Continue reading Doughnuts and Dumplings are Distinct: Homopoty-101 →

The knotty problem of packing DNA

Soon it will be time to pack away the fairy lights. If you wish to avoid the knotty task of disentangling them next December, don't just throw them in a box; roll them carefully around a stout stick or a paper tube. Any long and flexible string or cable, squeezed into a confined volume, is … Continue reading The knotty problem of packing DNA →

Simple Curves that Perplex Mathematicians and Inspire Artists

The preoccupations of mathematicians can seem curious and strange to normal people. They sometimes expend great energy proving results that appear glaringly obvious. One such result is called the Jordan Curve Theorem. We all know that a circle has an inside and an outside, and that this property also holds for a much larger collection … Continue reading Simple Curves that Perplex Mathematicians and Inspire Artists →

Consider a Spherical Christmas Tree

A minor seasonal challenge is how to distribute the fairy lights evenly around the tree, with no large gaps or local clusters. Since the lights are strung on a wire, we are not free to place them individually but must weave them around the branches, attempting to achieve a pleasing arrangement. Optimization problems like this … Continue reading Consider a Spherical Christmas Tree →

Stan Ulam, a mathematician who figured how to initiate fusion

Stanislaw Ulam, born in Poland in 1909, was a key member of the remarkable Lvov School of Mathematics, which flourished in that city between the two world wars. Ulam studied mathematics at the Lvov Polytechnic Institute, getting his PhD in 1933. His original research was in abstract mathematics, but he later became interested in a … Continue reading Stan Ulam, a mathematician who figured how to initiate fusion →

Moebiquity: Ubiquity and Versitility of the Möbius Band

The Möbius strip or Möbius band, with one side and one edge, has been a source of fascination since its discovery in 1858, independently by August Möbius and Johann Listing. It is easily formed from a strip of paper by giving it a half-twist before joining the ends. The Möbius band may be characterised in … Continue reading Moebiquity: Ubiquity and Versitility of the Möbius Band →

Disentangling Loops with an Ambient Isotopy

The surface in the left panel above has two linked loops. In the right hand panel, the loops are unlinked. Is it possible to continuously distort the left-hand surface so as to unlink the loops and produce the right-hand figure? This seems impossible, but intuition is not always reliable. Knot Theory A knot is an … Continue reading Disentangling Loops with an Ambient Isotopy →

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 →

Building Moebius Bands

We are all familiar with the Möbius strip or Möbius band. This topologically intriguing object with one side and one edge has fascinated children of all ages since it was discovered independently by August Möbius and Johann Listing in the same year, 1858. Building the Band It is a simple matter to make a Möbius … Continue reading Building Moebius Bands →

Topology in the Oval Office

Imagine a room – the Oval Office for example – that has three electrical appliances: • An air-conditioner ( a ) with an American plug socket ( A ), • A boiler ( b ) with a British plug socket ( B ), • A coffee-maker ( c ) with a Continental plug socket ( … Continue reading Topology in the Oval Office →

The Library of Babel and the Information Explosion

The world has been transformed by the Internet. Google, founded just 20 years ago, is a major force in online information. The company name is a misspelt version of "googol", the number one followed by one hundred zeros. This name echoes the vast quantities of information available through the search engines of the company … Continue reading The Library of Babel and the Information Explosion →

On Knots and Links

The picture below is of a sculpture piece called Intuition, which stands in front of the Isaac Newton Institute (INI) in Cambridge. It is in the form of the Borromean Rings, a set of three interlocked rings, no two of which encircle each other. Knot Theory Knot theory is an active research area today. In … Continue reading On Knots and Links →

Unsolved: the Square Peg Problem

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 … Continue reading Unsolved: the Square Peg Problem →

Venn Again’s Awake

We wrote about the basic properties of Venn diagrams in an earlier post. Now we take a deeper look. John Venn, a logician and philosopher, born in Hull, Yorkshire in 1834, introduced the diagrams in a paper in 1880 and in his book Symbolic Logic, published one year later. The diagrams were used long before … Continue reading Venn Again’s Awake →

It’s a Small – Networked – World

Networks are everywhere in the modern world. They may be physical constructs, like the transport system or power grid, or more abstract entities like family trees or the World Wide Web. A network is a collection of nodes linked together, like cities connected by roads or people genetically related to each other. Such a system … Continue reading It’s a Small – Networked – World →

The Bridges of Paris

Leonhard Euler considered a problem known as The Seven Bridges of Königsberg. It involves a walk around the city now known as Kaliningrad, in the Russian exclave between Poland and Lithuania. Since Kaliningrad is out of the way for most of us, let's have a look closer to home, at the bridges of Paris. [TM073: … Continue reading The Bridges of Paris →

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 … Continue reading Brouwer’s Fixed-Point Theorem →

Perelman’s Theorem: Who Wants to be a Millionaire?

This week’s That’s Maths column in The Irish Times (TM061, or search for “thatsmaths” at irishtimes.com) is about the remarkable mathematician Grisha Perelman and his proof of a one-hundred year old conjecture. Topology During the twentieth century topology emerged as one of the pillars of mathematics, alongside algebra and analysis. Geometers consider lengths, angles and … Continue reading Perelman’s Theorem: Who Wants to be a Millionaire? →

The Steiner Minimal Tree

Steiner's minimal tree problem is this: Find the shortest possible network interconnecting a set of points in the Euclidean plane. If the points are linked directly to each other by straight line segments, we obtain the minimal spanning tree. But Steiner's problem allows for additional points – now called Steiner points – to be added … Continue reading The Steiner Minimal Tree →

Plateau’s Problem and Double Bubbles

Bubbles floating in the air strive to achieve a spherical form. Large bubbles may oscillate widely about this ideal whereas small bubbles quickly achieve their equilibrium shape. The sphere is optimal: it encloses maximum volume for any surface of a given area. This was stated by Archimedes, but he did not have the mathematical techniques … Continue reading Plateau’s Problem and Double Bubbles →

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. If the ends of a rope … Continue reading Seifert Surfaces for Knots and Links. →

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 … 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 $latex {\mathbf{H} = \{(x,y): y>0\}}&fg=000000$ together with a metric $latex \displaystyle d s^2 = \frac { d x^2 + d y^2 } { y^2 } \,. &fg=000000$ It is remarkable that the entire structure of the space $latex {(\mathbf{H},ds)}&fg=000000$ follows from the metric. The … Continue reading Hyperbolic Triangles and the Gauss-Bonnet Theorem →

Ternary Variations

Georg Cantor (1845-1918) was led, through his study of trigonometric series, to distinguish between denumerably infinite sets like the rationals and uncountable sets like the reals. He introduced a set that is an abstract form of what we now call Cantor's Ternary Set. In fact, the ternary set had been studied some ten years earlier … Continue reading Ternary Variations →

The Loaves and the Fishes

One of the most amazing and counter-intuitive results in mathematics was proved in 1924 by two Polish mathematicians, Stefan Banach and Alfred Tarski. Banach was a mathematical prodigy, and was the founder of modern functional analysis. Tarski was a logician, educated at the University of Warsaw who, according to his biographer, “changed the face of … Continue reading The Loaves and the Fishes →

Peaks, Pits & Passes

In 1859, the English mathematician Arthur Cayley published a note in the Philosophical Magazine, entitled On Contour and Slope Lines, in which he examined the structure of topographical patterns. In a follow-up article, On Hills and Dales, James Clark Maxwell continued the discussion. He derived a result relating the number of maxima and minima on … Continue reading Peaks, Pits & Passes →

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 … Continue reading Topology Underground →