## Archive for December, 2017

### Doughnuts and Tonnetze

The circle of fifths is a remarkably useful diagram for the analysis of music. It shows the twelve notes of the chromatic scale arranged in a circle, with notes that are harmonically related (like C and G) being close together and notes that are discordant (like C and C) more distant from each other.

The Tonnetz diagram (note that the arrangement here is inverted relative to that used in the text.  It appears that there is no rigid standard, and several arrangements are in use) [Image from WikimediaCommons].

### Darker Mornings, Brighter Evenings

Today is the winter solstice, the shortest day of the year. We might expect that the latest sunrise and earliest sunset also occur today. In fact, the earliest sunset, the darkest day of the year, was on 13 December, over a week ago, and the latest sunrise is still more than a week away. This curious behaviour is due to the unsteady path of the Earth around the Sun. Our clocks, which run regularly at what is called mean time, move in and out of synchronization with solar time [TM129 or search for “thatsmaths” at irishtimes.com].

Sunrise in Newgrange on winter solstice [image from http://www.boynevalleytours.com/

### Vanishing Hyperballs

Spherical ball contained within a cubic region
We all know that the area of a disk — the interior of a circle — is ${\pi r^2}$ where ${r}$ is the radius. Some of us may also remember that the volume of a ball — the interior of a sphere — is ${\frac{4}{3}\pi r^3}$.