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Figure 1. An extract from Einstein’s 1917 paper on cosmology.
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 
Continue reading ‘The 3-sphere: Extrinsic and Intrinsic Forms’
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 do we need if every two points at unit distance are to receive distinct colors?
About 70 years ago it was shown that the least number of colours needed for such a colouring is one of 4, 5, 6 and 7. But which of these is the correct number? Despite efforts by many very clever people, some of whom had solved problems that appeared to be much harder, no advance has been made to narrow the gap
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 
According to Wikipedia, it was William Thomson, aka Lord Kelvin, who wrote:
“I call any geometrical figure, or group of points, ‘chiral‘, and say that it has chirality if its image in a plane mirror … cannot be brought to coincide with itself.”
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 with high accuracy, while distances to these bodies are much more difficult to find and can be determined only indirectly.
Continue reading ‘Embedding: Reconstructing Solutions from a Delay Map’
Abstract: Continuity is defined relative to a topology. For two distinct topological spaces 
Continue reading ‘The Signum Function may be Continuous’
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 dimension two, and so on.
Continue reading ‘The Dimension of a Point that isn’t there’
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’
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.
Continue reading ‘Doughnuts and Dumplings are Distinct: Homopoty-101’
John H Conway in 2009
[image Denise Applewhite, Princeton University].
John Horton Conway was a charismatic character, something of a performer, always entertaining his fellow-mathematicians with clever magic tricks, memory feats and brilliant mathematics. A Liverpudlian, interested from early childhood in mathematics, he studied at Gonville & Caius College in Cambridge, earning a BA in 1959. He obtained his PhD five years later, after which he was appointed Lecturer in Pure Mathematics.
In 1986, Conway moved to Princeton University, where he was Professor of Mathematics and John Von Neumann Professor in Applied and Computational Mathematics. He was awarded numerous honours during his career. Conway enjoyed emeritus status from 2013 until his death just two weeks ago on 11 April.
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 likely to become entangled: just think of garden hoses or the wires of headphones [TM178 or search for “thatsmaths” at irishtimes.com].
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 of closed curves [TM165 or search for “thatsmaths” at irishtimes.com].
Detail from Michaelangelo’s The Creation of Adam, and a Jordan Curve representation [image courtesy of Prof Robert Bosch, Oberlin College. Downloaded from here].
Continue reading ‘Simple Curves that Perplex Mathematicians and Inspire Artists’
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 occur throughout applied mathematics [TM153 or search for “thatsmaths” at irishtimes.com].
Trees are approximately conical in shape and we may assume that the lights are confined to the surface of a cone. The peak, where the Christmas star is placed, is a mathematical singularity: all the straight lines that can be drawn on the cone, the so-called generators, pass through this point. Cones are developable surfaces: they can be flattened out into a plane without being stretched or shrunk.
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 wide range of applications. He once joked that he was “a pure mathematician who had sunk so low that his latest paper actually contained numbers with decimal points” [TM138 or search for “thatsmaths” at irishtimes.com].
Operation Castle, Bikini Atoll, 1954
Continue reading ‘Stan Ulam, a mathematician who figured how to initiate fusion’
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.
Möbius band in 3-space and a flat representation in 2-space.
Continue reading ‘Moebiquity: Ubiquity and Versitility of the Möbius Band’
Can one of these shapes be continuously distorted to produce the other?
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.
Continue reading ‘Disentangling Loops with an Ambient Isotopy’
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.
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.
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 ( C ).
The problem is to connect each appliance to the correct socket, avoiding any crossings of the connecting wires.
Fig. 1: Positions of appliances and sockets for Problem 1.
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 [TM107 or search for “thatsmaths” at irishtimes.com].
Artist’s impression of the Library of Babel [Image from Here].
Long before the Internet, the renowned Argentine writer, poet, translator and literary critic Jorge Luis Borges (1889 – 1986) envisaged the Universe as a vast information bank in the form of a library. The Library of Babel was imagined to contain every book that ever was or ever could be written.
Continue reading ‘The Library of Babel and the Information Explosion’
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.
“Intuition”. A sculpture piece in front of the Isaac Newton Institute [Photograph courtesy of S J Wilkinson].
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.
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 Venn’s paper, but he formalized and popularized them. He used them as logical diagrams: the interior of each set means the truth of a particular proposition. Unions and intersections of sets correspond to the logical operators OR and AND.
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 of nodes and links is what mathematicians call a graph [TM078; or search for “thatsmaths” at irishtimes.com ].
Detail of a Twitter communications network.
Image from: https://dhs.stanford.edu/gephi-workshop/twitter-network-gallery/
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: search for “thatsmaths” at irishtimes.com ]
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.
Continue reading ‘Brouwer’s Fixed-Point Theorem’
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.
Continue reading ‘Perelman’s Theorem: Who Wants to be a Millionaire?’
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 to the network, yielding Steiner’s minimal tree. This generally results in a reduction of the overall length of the network.
![Solution of Steiner 5-point problem with soap film [from Courant and Robbins].](https://thatsmaths.files.wordpress.com/2015/01/steiner-5point.jpg)
A solution of Steiner 5-point problem with soap film [from Courant and Robbins].
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 required to prove it. It was only in the late 1800s that a formal proof of optimality was completed by Hermann Schwarz [Schwarz, 1884].
Computer-generated double bubble
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 [Fig from Van Wijk (2006)].](https://thatsmaths.files.wordpress.com/2015/01/knots-vanwijk.jpg)
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)].
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’
Poincaré’s half-plane model for hyperbolic geometry comprises the upper half plane 
It is remarkable that the entire structure of the space 
Continue reading ‘Hyperbolic Triangles and the Gauss-Bonnet Theorem’
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 by the Dublin-born mathematician Henry Smith and, independently, by the Italian Vito Volterra. General sets of this form are now called Smith-Volterra-Cantor sets (SVC sets).
Continue reading ‘Ternary Variations’
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 logic in the twentieth century” through his work on model theory.
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 a contour chart.
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’
ThatsMaths 