# Rule of twelfths

The rule of twelfths is a handy tool to approximate a sinusoidal curve. If you divide the curve's period into twelve equal intervals starting at the bottom, then approximately $1/12$ of the total elevation gain is contained in the first interval, $2/12$ in the second interval, $3/12$ in the third interval, $3/12$ in the fourth interval, $2/12$ in the fifth interval, and $1/12$ in the sixth interval to reach the top of the curve. The descent from the top back to the bottom is similarly distributed among the last six intervals according to the ratio

$1 : 2 : 3 : 3 : 2 : 1.$

I learned about the rule of twelfths from a kayaking instructor and guide, who used it to estimate the tides. In locations and seasons with a semidiurnal tide pattern, the period of the tide is roughly 12 hours so the rule of twelfths tells you what the water will be doing in each hour.

For example, if you remember that the forecast difference between low and high tide is 3 feet today, then you can expect the water to rise by 3 inches in the first hour, 6 inches in the second, 9 inches in the third, and so on.

$x$ $\Delta f(x)$ $f(x)$ $\frac{1-\cos(2πx)}{2}$ error
$0$ n/a $0/12$ $0$ $0$
$1/12$ $1/12$ $1/12$ $\frac{1-\sqrt{3}/2}{2}$ $0.0163$
$2/12$ $2/12$ $3/12$ $\frac{1}{4}$ $0$
$3/12$ $3/12$ $6/12$ $\frac{1}{2}$ $0$
$4/12$ $3/12$ $9/12$ $\frac{3}{4}$ $0$
$5/12$ $2/12$ $11/12$ $\frac{1+\sqrt{3}/2}{2}$ $-0.0163$
$6/12$ $1/12$ $12/12$ $1$ $0$
$7/12$ $-1/12$ $11/12$ $\frac{1+\sqrt{3}/2}{2}$ $-0.0163$
$8/12$ $-2/12$ $9/12$ $\frac{3}{4}$ $0$
$9/12$ $-3/12$ $6/12$ $\frac{1}{2}$ $0$
$10/12$ $-3/12$ $3/12$ $\frac{1}{4}$ $0$
$11/12$ $-2/12$ $1/12$ $\frac{1-\sqrt{3}/2}{2}$ $0.0163$
$12/12$ $-1/12$ $0/12$ $0$ $0$

1. I normally use metric, but it's super handy to use a duodecimal measurement system here. ↩︎