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.

## Archive Page 2

### Andrey Markov’s Brilliant Ideas are still a Driving Force

Published September 21, 2017 Irish Times Leave a CommentTags: Algorithms, Statistics

Imagine examining the first 20,000 letters of a book, counting frequencies and studying patterns. This is precisely what Andrey Markov did when he analyzed the text of Alexander Pushkin’s verse novel *Eugene Onegin*. This work comprises almost 400 stanzas of iambic tetrameter and is a classic of Russian literature. Markov studied the way vowels and consonants alternate and deduced the probabilities of a vowel being followed by a another vowel, by a consonant, and so on. He was applying a statistical model that he had developed in 1906 and that we now call a Markov Process or Markov chain. [TM123 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Andrey Markov’s Brilliant Ideas are still a Driving Force’

Amongst the many joys of mathematics are the beautiful patterns that appear unexpectedly from time to time. In 1951, Alfred Moessner discovered a delightful and ingenious way of generating sequences of powers of natural numbers. It is surprisingly simple and offers great opportunities for development and generalization.

It is well-known that the sum of odd numbers yields a perfect square:

1 + 3 + 5 + … + (2*n* – 1) = *n *^{2}

This is easily demonstrated in a geometric way. We start with a unit square, and repeatedly add an additional row and column on the “east” and “north” sides and a unit square at the “north-east” corner. This amounts to adding the next odd number and, at each stage, a new square is produced.

### Euler and the Fountains of Sanssouci

Published September 7, 2017 Irish Times Leave a CommentTags: Applied Maths, Euler, Fluid Dynamics

When Frederick the Great was crowned King of Prussia in 1740 he immediately revived the Berlin Academy of Sciences and invited scholars from throughout Europe to Berlin. The most luminous of these was Leonhard Euler, who arrived at the academy in 1741. Euler was an outstanding genius, brilliant in both mathematics and physics. Yet, a myth persists that he failed spectacularly to solve a problem posed by Frederick. Euler is reputed to have bungled his mathematical analysis. In truth, there was much bungling, but the responsibility lay elsewhere. [TM122 or search for “thatsmaths” at irishtimes.com].

### Drawing Multi-focal Ellipses: The Gardener’s Method

Published August 31, 2017 Occasional Leave a CommentTags: Algorithms, Geometry

**Common-or-Garden Ellipses**

In an earlier post we saw how a gardener may set out oval flower-beds using a well-known property of ellipses: the sum of the distances from any point on the ellipse to the two foci is always the same value, , the length of the major axis. The gardener puts down two stakes and loops a piece of rope around them. Using a stick, he pulls the loop taut, marking the points around a curve. This is illustrated here.

Continue reading ‘Drawing Multi-focal Ellipses: The Gardener’s Method’### Locating the HQ with Multi-focal Ellipses

Published August 24, 2017 Occasional Leave a CommentTags: Algorithms, Geometry

**Motivation**

Ireland has four provinces, the principal city in each being the provincial capital: Belfast, Cork, Dublin and Galway. The map here shows the location of these cities. Now imagine a company that needs to visit and to deliver goods frequently to all four cities. Where might they locate their HQ to minimize transport costs and travel times?

One possibility is to find the location with the smallest distance sum:

where is the position of the HQ and are the positions of the cities.

Continue reading ‘Locating the HQ with Multi-focal Ellipses’

Next Monday, the shadow of the Moon will bring a two-minute spell of darkness as it sweeps across the United States along a path from Oregon to South Carolina. The eclipse is one of a series known as Saros 145. [TM121 or search for “thatsmaths” at irishtimes.com].