Where Music, Patterns, Math and Chemistry Meet on Teddy Rook Book

Scales: the ladders music climbs

Mon, 01 Jan 0001 00:00:00 +0000

A pattern of steps

On a piano, the distance from one key to the very next one, black or white, is a half step. Two half steps make a whole step. A scale is a set of notes chosen by a pattern of these steps. The major scale goes whole, whole, half, whole, whole, whole, half, and lands on the same letter one octave up.

Times tables on a circle

Mon, 01 Jan 0001 00:00:00 +0000

Curves hiding in a times table

Space 200 points evenly around a circle and number them 0 to 199. Now join each point to its double: 1 to 2, 2 to 4, 3 to 6, and so on. When the numbers pass 199, keep counting around the circle, so 150 joins 300, which is point 100. That is clock arithmetic, or arithmetic mod 200.

Star patterns from simple tiles

Mon, 01 Jan 0001 00:00:00 +0000

Stars from polygons

For more than a thousand years, artists across the Islamic world have covered walls, ceilings and doors with interlaced stars. Many of these designs can be rebuilt from a hidden grid of ordinary polygons. In the 1920s the British scientist E. H. Hankin, studying patterns in India, described a simple way to do it, often called polygons in contact:

The periodic table, by block

Mon, 01 Jan 0001 00:00:00 +0000

Four blocks

The periodic table has a strange shape: tall towers on the sides, a dip in the middle and two loose rows below. Color each element by the kind of subshell its newest electron goes into, and the shape falls into four neat blocks:

Ratios, clocks and rhythms

Mon, 01 Jan 0001 00:00:00 +0000

Strings and simple ratios

Pluck a string, then stop it at half its length and pluck again. The note jumps up an octave: the shorter string vibrates twice as fast. Stop it at two thirds of its length and you get a fifth, the next-sweetest sound. Three quarters gives a fourth.

Sand that draws sound

Mon, 01 Jan 0001 00:00:00 +0000

Chladni figures

In 1787 the German scientist Ernst Chladni published a book of strange drawings. He had clamped thin metal plates at the center, sprinkled sand on them, and stroked the edge with a violin bow. Each time the plate rang with a clear note, the sand leapt off the shaking parts and piled up along lines that stayed still. A higher note made a busier pattern.

Octaves of elements, notes of molecules

Mon, 01 Jan 0001 00:00:00 +0000

Newlands’ law of octaves

By the 1860s chemists knew about sixty elements and their atomic weights, but no one had a good way to arrange them. In 1865 the English chemist John Newlands listed the elements in order of atomic weight and noticed that similar ones turned up at regular intervals. Lithium, sodium and potassium, all soft metals that fizz in water, came seven places apart, and so did fluorine and chlorine. Counting lithium as the first, sodium is the eighth.

Tilings and rose windows

Mon, 01 Jan 0001 00:00:00 +0000

Only 17 ways to repeat

Wallpaper, floor tiles and woven cloth repeat in two directions. Look closely and each design also has some mix of turns, mirrors and glides (a mirror followed by a slide, like footprints in snow) that leave it looking the same.

The numbers behind the periodic table

Mon, 01 Jan 0001 00:00:00 +0000

Why 2, 8, 18, 32?

An electron in an atom is described by whole numbers. The shell number $n$ is 1, 2, 3, … Inside shell $n$, the subshell number $l$ runs from 0 to $n-1$ (chemists call these s, p, d and f). Subshell $l$ has $2l + 1$ orbitals: 1, 3, 5, 7. Each orbital holds two electrons, one of each spin.

Crystals, snowflakes and stained glass

Mon, 01 Jan 0001 00:00:00 +0000

Seven shapes of a crystal

A crystal is atoms stacked in a pattern that repeats in three directions, like wallpaper in 3D. The smallest box that repeats is the unit cell. Its shape is set by three edge lengths, $a$, $b$ and $c$, and the three angles between them.

A sound built from harmonics

Mon, 01 Jan 0001 00:00:00 +0000

Building a wave from harmonics

In 1807 the French mathematician Joseph Fourier made a bold claim while studying how heat spreads: any repeating pattern, however jagged, can be built by adding up simple sine waves whose frequencies are whole-number multiples of one base frequency. Mathematicians argued about it for decades, but it holds for every wave a musician cares about.

Hydrogen's whole-number lines

Mon, 01 Jan 0001 00:00:00 +0000

Balmer’s formula

Heat hydrogen gas until it glows and pass the light through a prism. Instead of a rainbow you see four thin lines: red, blue-green, blue and violet. The Swedish physicist Anders Ångström had measured their wavelengths very precisely. In 1885 Johann Jakob Balmer, a Swiss mathematics teacher, found a simple rule that fit all four:

Vibrating plates and orbital shapes

Mon, 01 Jan 0001 00:00:00 +0000

Sand on the still lines, electrons off them

On a ringing plate, sand collects on the nodes, the lines and circles that never move. An electron in an atom is also described by a standing wave, found by Erwin Schrödinger’s equation in 1926. Where that wave is zero, there is a node too, and the electron is never found there. So the two pictures are opposites in one way: the sand marks the nodes, while the electron’s dots avoid them. But the patterns of nodes are close cousins.

The symmetry of molecules

Mon, 01 Jan 0001 00:00:00 +0000

Moves that change nothing

A symmetry operation is a move that leaves a molecule looking exactly as it did: a turn about an axis, a reflection in a mirror plane, or an inversion through the center (every atom goes straight through the middle to the opposite side). The atoms do move, but each lands where an identical atom used to be.

Waves and symmetry

Mon, 01 Jan 0001 00:00:00 +0000

An atom is a tiny drum

Hit a drum and its skin vibrates in a mix of patterns, or modes. Each mode is a standing wave with its own still lines (nodes), its own symmetry and its own frequency. Unlike a string, a drum’s modes are not whole-number multiples of each other: their frequencies follow the zeros of special functions called Bessel functions, in ratios like 1 : 1.59 : 2.14 : 2.30. That is why most drums sound like a thump rather than a clear note.