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 ( C ).

The problem is to connect each appliance to the correct socket, avoiding any crossings of the connecting wires.

electricplugs-01

Fig. 1: Positions of appliances and sockets for Problem 1.

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The Spire of Light

 

Towering over O’Connell Street in Dublin, the Spire of Light, at 120 metres, is about three times the height of its predecessor [TM109 or search for “thatsmaths” at irishtimes.com]. The Spire was erected in 2003, filling the void left by the destruction in 1966 of Nelson’s Pillar. The needle-like structure is a slender cone of stainless steel, the diameter tapering from 3 metres at the base to 15 cm at its apex. The illumination from the top section shines like a beacon throughout the city.

spire-nightscape

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Metallic Means

goldenmean-pentagram

The golden mean occurs repeatedly in the pentagram [image Wikimedia Commons]

Everyone knows about the golden mean. It must be one of the most written-about numbers, certainly in recreational mathematics. It is usually denoted by {\phi} and is the positive root of the quadratic equation

\displaystyle x^2 - x - 1 = 0 \ \ \ \ \ (1)

with the value

{\phi = (1+\sqrt{5})/2 \approx 1.618}.

There is no doubt that {\phi} is significant in many biological contexts and has also been an inspiration for artists. Called the Divine Proportion, it  was described in a book of that name by Luca Pacioli, a contemporary and friend of Leonardo da Vinci.

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Voronoi Diagrams: Simple but Powerful

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

Voronoi diagram drawn using the applet of Paul Chew (see Sources below).

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The Beginning of Modern Mathematics

The late fifteenth century was an exciting time in Europe. Western civilization woke with a start after the slumbers of the medieval age. Johannes Gutenberg’s printing press arrived in 1450 and changed everything. Universities in Bologna, Oxford, Salamanca, Paris and elsewhere began to flourish. Leonardo da Vinci was in his prime and Christopher Columbus was discovering a new world.

davinci-dodecahedron

Illustrations by Leonardo da Vinci in Pacioli’s De Divina Proportione.

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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 [TM107 or search for “thatsmaths” at irishtimes.com].

libraryofbabel

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.

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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.

intuition-dscf0061-lores

“Intuition”. A sculpture piece in front of the Isaac Newton Institute [Photograph courtesy of S J Wilkinson].

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