## Archive for February, 2017

### 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.

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

### 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.

### Metallic Means

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.

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