Suppose a circle is divided by two radii and the two arcs a and b are in the golden ratio:
b / a = ( a + b ) / b = φ ≈ 1.618
Then the smaller angle formed by the radii is called the golden angle. It is equal to about 137.5° or 2.4 radians. We will denote the golden angle by γ. Its exact value, as a fraction of a complete circle, is ( 3 – √5 ) / 2 ≈ 0.382 cycles.
Continue reading ‘Golden Moments’
Published March 12, 2015
Tags: Geometry, History
O God, I could be bounded in a nutshell, and count myself a king of infinite space …
Euclid. Left: panel from series Famous Men by Justus of Ghent. Right: Statue in the Oxford University Museum of Natural History.
Continue reading ‘A King of Infinite Space: Euclid I.’
Published March 5, 2015
Tags: Analysis, History
For 150 years the city of Lvov was part of the Austro-Hungarian Empire. After Polish independence following World War I, research blossomed and between 1920 and 1940 a sparkling constellation of mathematicians flourished in Lvov [see this week’s That’s Maths column in The Irish Times (TM063, or search for “thatsmaths” at irishtimes.com).
The Scottish Café, Lvov in earlier times (left), now Hotel Atlas in Lviv (right).
Continue reading ‘Café Mathematics in Lvov’
Published February 26, 2015
Tags: Analysis, History, Set Theory
Stefan Banach (1892–1945) was amongst the most influential mathematicians of the twentieth century and the greatest that Poland has produced. Born in Krakow, he studied in Lvov, graduating in 1914 just before the outbreak of World War I. He returned to Krakow where, by chance, he met another mathematician, Hugo Steinhaus who was already well-known. Together they founded what would, in 1920, become the Polish Mathematical Society.
A coin and a postage stamp commemorating Stefan Banach.
Continue reading ‘The Birth of Functional Analysis’
Published February 19, 2015
There is great public interest in genealogy. Many of us live in hope of identifying some illustrious forebear, or enjoy the frisson of having a notorious murderer somewhere in our family tree. Academic genealogies can also be traced: see this week’s That’s Maths column in The Irish Times (TM062, or search for “thatsmaths” at irishtimes.com).
Continue reading ‘MGP: Tracing our Mathematical Ancestry’
Published February 12, 2015
Tags: Algebra, Group Theory
What is the common factor linking book-flips, solitaire, twelve-tone music and the solution of quartic equations? Answer: .
Symmetries of a Book — or a Brick
The four symmetric configurations of a book under 3D rotations.}
Continue reading ‘The Klein 4-Group’