### Golden Moments

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.

### You Can Do Maths

Bragging about mathematical ineptitude is not cool. There is nothing admirable about ignorance and incompetence. Moreover, everyone thinks mathematically all the time, even if they are not aware of it. Can we all do maths? Yes, we can!  [See this week’s That’s Maths column (TM064) or search for “thatsmaths” at irishtimes.com].

When you use a map of the underground network, you are doing topology.

### A King of Infinite Space: Euclid I.

O God, I could be bounded in a nutshell, and count myself a king of infinite space …
[Hamlet]

Euclid. Left: panel from series Famous Men by Justus of Ghent. Right: Statue in the Oxford University Museum of Natural History.

### Café Mathematics in Lvov

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).

### The Birth of Functional Analysis

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.

### MGP: Tracing our Mathematical Ancestry

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).

### The Klein 4-Group

What is the common factor linking book-flips, solitaire, twelve-tone music and the solution of quartic equations?   Answer: ${K_4}$.

Symmetries of a Book — or a Brick

The four symmetric configurations of a book under 3D rotations.}