A crinkle crankle wall, also called a serpentine wall, is a wavy wall that may seem to sacrifice some efficiency for aesthetics. The curves add visual interest, but they use more material than a straight way. Except they don’t! They use more bricks than a straight wall of the same thickness but they don’t have to be as thick.
Crinkle crankle walls resist horizontal forces, like wind, more than straight wall would. So if the alternative to a crinkle crankle wall one-brick thick is a straight wall two or more bricks thick, the former saves material. How much material?
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.