# Crinkle Crankle Calculus

The amount of material used in the wall is proportional to the product of its length and thickness.

Suppose the wall is shaped like a sine wave and consider a section of wall 2π long.

If the wall is in the shape of a sin(θ), then we need to find the arc length of this curve.

This works out to the following integral.

The parameter a is the amplitude of the sine wave.

If a = 0, we have a flat wave, i.

e.

a straight wall, as so the length of this segment is 2π = 6.

2832.

If a = 1, the integral is 7.

6404.

So a section of wall is 22% longer, but uses 50% less material per unit length as a wall two bricks thick.

The integral above cannot be computed in closed form in terms of elementary functions, so this would make a good homework exercise for a class covering numerical integration.

The integral can be computed in terms of special functions.

It equals 4 E(-a²) where E is the “complete elliptic integral of the second kind.

” This function is implemented as EllipticE in Mathematica and as scipy.

special.

ellipe in Python.

As the amplitude a increases, the arc length of a section of wall increases.

You could solve for the value of a to give you whatever arc length you like.

For example, if a = 1.

4422 then the length is twice that of a straight line.

So a crinkle crankle wall with amplitude 1.

4422 uses about as many bricks as a straight wall twice as thick.