# “The Crinkle Crankle Wall”

A crinkle crankle wall, also called a serpentine wall, is a curvy wall which at first glance seems to be built completely for the looks. We all know that the shortest distance between two points is a straight line, so obviously it will take more materials to build a wavy wall. These walls were primarily built in East Anglia, England, where the Dutch engineers drained the marshes of The Fens in the mid-1600s. The original Dutch engineers called the walls slangenmuur, meaning snake wall. The term crinkle crankle was first used to describe these walls in the 18th century from the old English term for zigzag.

However, the Dutch were not the first to build the wavy walls as these sinusoidal walls were featured extensively in the Egyptian architecture of Aten around 1386-1353 BC. The Egyptian walls were thought to be built as the lower parts of fortifications to force invading troops to break ranks and become exposed to defensive assaults.

The most famous of these walls in the United States is at the University of Virginia, founded in 1819. The walls were designed by Thomas Jefferson and flank both sides of the university’s rotunda.

So why use these seemingly costly wavy walls instead of the traditional straight walls? The answer comes when you begin to calculate the cost of construction. The University of Virginia has a document in Thomas Jefferson’s own handwriting showing the cost savings of building the curved walls along the rotunda. This leaves the open question, how does a wavy wall use less material than a straight wall? To figure that out we have to break out the calculus book and learn a few terms.

First you need to learn what the term sinusoidal means, and the benefits of using a sinusoidal curve in construction. A sinusoidal curve is a smooth curve that waves in and out, with a set distance between the far and near points. These curves create extremely strong structures that can withstand greater forces from wind, water and soil pressure. The difference is roughly such that a straight wall needs to be two times thicker than a sinusoidal wall to withstand the same forces.

The second thing you need to learn is how to calculate the amount of material needed to build a straight wall. For this we do not need calculus as it is a simple volume calculation. You simply multiply the length, height and thickness of the wall together. To simplify the math we will use a wall that is two times pi long, or 6.2832 feet, 1 foot thick and 1 foot high. So we will need 6.2832 cubic feet of material to build the wall and close the gap.

If we use a sinusoidal wall to close the same 6.2832 foot gap, there is a complex mathematical formula for calculating the length of this curved wall. It requires calculating a complex integral, and the length of the curved wall grows with the amplitude of the sinusoidal wave the wall follows (in other words, the distance between the nearest segment of the wall and the furthest). The simplest to calculate is when this amplitude is one, which basically means when the wall weaves in and out one foot every 6.2832 feet. In this simplest of cases the wavy wall is 7.6404 feet long, but only has to be 6 inches thick. It turns out 22 percent longer than the straight wall, but uses exactly 50 percent less material. If you increase the amplitude to 1.4422 feet, the length of the wavy wall is twice as long as the straight wall and you will use the same amount of material.

Calculating the cost savings of a wavy wall is not a simple calculation and is an excellent exercise for a calculus student studying numerical integration, but it was very surprising to me to learn that you can build a wavy wall with less material than a straight wall. This also happens to be the reason why silos are round instead of square. The curved walls of a grain silo can hold more weight in grain with less materials than a square silo of the same volume, for exactly the same reason. If you want to see the math behind the crinkle crankle wall, check out John D. Cook’s consulting blog at https://www.johndcook.com/blog/2019/11/19/crinkle-crankle-calculus/. Until next week, stay safe and learn something new.

*Scott Hamilton is an Expert in Emerging Technologies at ATOS and can be reached with questions and comments via email to shamilton@techshepherd.org or through his website at https://www.techshepherd.org.*