Published October 8, 2015
Tags: Analysis, Geometry, Mechanics
“A bicycle, certainly, but not the bicycle,” said Holmes.
In Conan-Doyle’s short story The Adventure of the Priory School Sherlock Holmes solved a mystery by deducing the direction of travel of a bicycle. His logic has been minutely examined in many studies, and it seems that in this case his reasoning fell below its normal level of brilliance.
As front wheel moves along the positive x-axis the back wheel, initially at (0,a), follows a tractrix curve (see below).
Continue reading ‘Which Way did the Bicycle Go?’
The readable surface of a Compact Disc has a spiral track over 5 km in length.
The quality of music recordings on compact discs or CDs is excellent. In the age of vinyl records, irritating clicks resulting from surface scratches were almost impossible to avoid. Modern recording media are largely free from this shortcoming. But this is curious: there are many reasons why CD music can be contaminated: dirt on the disc surface, flaws in the plastic substrate, errors in burning on the recording, scratches and fingerprints, and so on [TM077; or search for “thatsmaths” at irishtimes.com ]
Continue reading ‘New Tricks: No Clicks’
Published September 24, 2015
Tags: Algorithms, Computer Science
Richard Hamming (1915 – 1998)
In the late 1940s, Richard Hamming, working at Bell Labs, was exasperated with the high level of errors occurring in the electro-mechanical computing equipment he was using. Punched card machines were constantly misreading, forcing him to restart his programs. He decided to do something about it. This was when error-correcting codes were invented.
A simple way to detect errors is to send a message twice. If both versions agree, they are probably correct; if not, there is an error somewhere. But the discrepancy gives us no clue where the error lies. Sensing the message three times is better: if two versions agree, we assume they are correct and ignore the third version. But there is a serious overhead: the total data transmitted is three times the data volume; the information factor is 1/3.
Continue reading ‘Hamming’s Smart Error-correcting Codes’
Published September 17, 2015
Tags: Geometry, History, Puzzles
Puzzle: However fast a train is travelling, part of it is moving backwards. Which part?
For the answer, see the end of this post.
Timelapse image of bike with lights on the wheel-rims. [Photo from Website of Alexandre Wagemakers, with thanks]
Imagine a small light fixed to the rim of a bicycle wheel. As the bike moves, the light rises and falls in a series of arches. A long-exposure nocturnal photograph would show a cycloid, the curve traced out by a point on a circle as it rolls along a straight line. A light at the wheel-hub traces out a straight line. If the light is at the mid-point of a spoke, the curve it follows is a curtate cycloid. A point outside the rim traces out a prolate cycloid, with a backward loop. [TM076; or search for “thatsmaths” at irishtimes.com ]
Continue reading ‘The Ubiquitous Cycloid’
Published September 10, 2015
Tags: Geometry, History
Hans Holbein the Younger, court painter during the reign of Henry VIII, produced some spectacular works. Amongst the most celebrated is a double portrait of Jean de Dinteville, French Ambassador to Henry’s court, and Georges de Selve, Bishop of Lavaur. Painted by Holbein in 1533, the picture, known as The Ambassadors, hangs in the National Gallery, London.
Double Portrait of Jean de Dinteville and Georges de Selve (“The Ambassadors”),
Hans Holbein the Younger, 1533. Oil and tempera on oak, National Gallery, London
Continue reading ‘Holbein’s Anamorphic Skull’
Published September 3, 2015
Tags: Algebra, History
James Joseph Sylvester (1814-1897) as a graduate of Trinity College Dublin.
James Joseph Sylvester was born in London to Jewish parents in 1814, just 201 years ago today. The family name was Joseph but, for reasons unclear, Sylvester – the name of an anti-Semitic Pope from the Roman period – was adopted later. [TM075; or search for “thatsmaths” at irishtimes.com ]
Sylvester’s mathematical talents became evident at an early age. He entered Cambridge in 1831, aged just seventeen and came second in the notorious examinations known as the Mathematical Tripos; the man who beat him achieved nothing further in mathematics!
Continue reading ‘James Joseph Sylvester’
Published August 27, 2015
Tags: Astronomy, History
Sir Walter Raleigh, adventurer, explorer and privateer, was among most colourful characters of Tudor times. He acquired extensive estates in Waterford and Cork, including Molana Abbey near Youghal, which he gave to his friend and advisor, the brilliant mathematician and astronomer Thomas Harriot.
Left: Sir Walter Raleigh (1552-1618). Right: Thomas Harriot (1560?-1621)
Continue reading ‘Thomas Harriot: Mathematician, Astronomer and Navigator’