Rule of twelfths

You can get a really good approximation of a sinusoidal curve from twelve equally-spaced line segments of slope 1/12, 2/12, 3/12, 3/12, 2/12, 1/12, -1/12, -2/12, -3/12, -3/12, -2/12, and -1/12, respectively.

This approximation, known as the rule of twelfths, rounds √3≈5/3​ but otherwise uses exact values along the curve.

The rule of twelfths approximates points on (1−cos(2πx))/2.

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, and the rule of twelfths tells you what the water will be doing in each hour.

For example, if you know that the difference between low and high tide is 3 feet, then you can quickly estimate that it the tide will rise by about 3 inches in the first hour, 6 inches in the second, 9 inches in the third and fourth, 6 inches in the fifth hour, and 3 inches in the last hour before high tide.