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Building a Cathedral without Science or Mathematics: The Engineering Method Explained – Transcript
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- Göbekli Tepe is an ancient ruin in southeastern Turkey featuring stone structures and T-shaped pillars erected in a purposefully designed layout by applying the engineering method.
- Medieval engineers designed buildings like Sainte-Chapelle and utilized pointed arches instead of semicircular Roman arches to save stone and costs.
- The engineering method relies on rules of thumb, or heuristics, which are imprecise methods used as shortcuts to solve problems without full scientific knowledge.
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This structure built some 12,000 years ago — that’s 6,000 years before Stonehenge, 5,000 years before the first pyramid — highlights that for millennia, humans have used the engineering method to solve problems long before the rise of science.
In this video I define and illustrate the essence of the engineering method. This is Göbekli Tepe, an ancient ruin in southeastern Turkey. Stone structures cover nine hectares, about eighteen football fields. Filled with circular rooms, its purpose remains a mystery. Embedded in the stone walls of the rooms are decorative T-shaped pillars. Carved into the pillars are
lions . . . bears . . . wild boars . . . lizards . . . spiders, and surprisingly . . . . ducks. Not only are the stones beautifully carved, they are placed with precision. To see the degree of precision look at the circular room I showed you initially. At the center are the remains of two T-pillars (the tops fell off long ago). If we look at this room from above — keep you
eye on the two central pillars highlighted in red — you see two circular rooms nearby, which each have the same arrangement of pillars at their center. The pillars in the two rooms at the bottom sit on a line — with little deviation. If we locate the exact centers between the pillars in the two lower circles and connect them to the center of pillars in the circle above, we inscribe an equilateral triangle; its sides deviate less that one point five percent from
perfection — this is unlikely to be coincidental, nor the result of some self-organization. My point is that these stones are erected in a purposefully designed layout, not haphazard, that responds to societal needs, which means that Göbekli Tepe is an engineered object. The people of Göbekli Tepe created all this by applying the engineering method. The method we still use today.
Yet, this powerful method, the “engineering method,” is an all but hidden process that few of us have heard of—let alone understand — even though it influences every aspect of our lives. In this video series I explore the essence of that engineering method. The age of Göbekli Tepe highlights that the engineering method long predates science, which conflicts with the commonplace notion that engineering somehow arises from the scientific method.
So, I start by asking: do engineers need science? And here’s the answer. This is a model of Sainte-Chapelle, a stunning thirteenth century building built nearly twelve thousand years after Göebkli Tepe. Sainte-Chapelle was designed and constructed by a team who had never learned basic arithmetic or the geometry taught today in third grade,
let alone anything we’d call “science.” In fact, they had no standardized length because a foot varied from region to region! The medieval engineers aspired to create a building that celebrated a spiritual world by streaming light through the structure’s walls. Sunlight was transformed by the stained-glass windows into a diffuse red, blue, and gold that glimmers on the chapel’s sculptures and gilded arches. To create these stunning displays of light, to house these
giant stained glass windows, they built vast walls by constructing spectacularly high ceilings. How did these engineers, called masons, know how to crave a cathedral’s thousand of building blocks to create, when assembled, not only beauty, but also stability? Like all great engineers they drew on past knowledge. They used an architectural element that originated in present-day Pakistan. Notice
that Sainte-Chapelle has, as do all cathedrals, pointed arches — some six on the front, at least ten on each side, and on the back six more. And of course many pointed arches support the ceiling.
They chose this pointed arch over the arch used in the Roman’s extraordinary Pantheon built about fourteen hundred years before Sainte-Chapelle. The Pantheon, typical of all Imperial Roman architecture, features a semicircular arch. It is exactly half of a circle.
If a medieval engineer used a semicircular Roman arch to construct a soaring cathedral it would cover entire city blocks, and use so much stone that the cost would skyrocket. The savings from using a pointed arch instead of a semicircular arch can be clearly seen by comparing Sainte-Chappelle with the Pantheon. Note how Sainte-Chappelle is nearly the same height as the Pantheon yet just a little over a third as wide. This Roman semicircular arch was rejected by
medieval engineers because its width has to grow proportionally to its height. To see the problem consider a semicircular arch that is ten meters in width. It is five meters in height because it is half of a circle — so its height is half of the diameter. Now, if I double the span covered — from ten meters to twenty meters — you see that the height is now ten meters: it has doubled.
Contrast this to the pointed arch: as the arch increases in height, the span covered can stay the same because the shape of the arch can change as its height increases. Although, medieval engineers didn’t adopt the Roman semicircular arch, from the Romans they inherited the key idea that enabled them to build their stunning cathedrals that have stood for centuries. We can see it in this arch from Sainte-Chapelle. To support this arch, the medieval engineers had
to correctly size the thickness of this wall. If this wall were too thin the weight of the arch would buckle the wall — the structure would collapse. If too thick, stone would be wasted (at a huge cost) and the desired open space inside the cathedral diminished. To size the wall, the medieval builder used a rule that created the stable semicircular arches of the Pantheon: a stable arch results when the supporting wall’s thickness is between a fourth or a fifth of the
arch’s span. If an arch’s span were this long, then you’d divide it into about four equal parts, and use one of one of these to size the wall’s thickness. But, there a problem with applying this proportional rule: the masons knew no mathematics nor, as I’ve said, did they even have a ruler! So this rule was not a calculation but was instead an action. The mason would have a life-sized
chalk drawing of the interior of this arch on the floor of the drawing room. Now, instead I’ve drawn the interior of a pointed arch on a large sheet of paper. In addition to this arch drawing or template you need a piece of string — the mason would have used a rope.
I need to drap it over the arch so I’ll insert map pins along the arch to support the string — the mason would have done this on the floor so he would simply lay the rope on top of his drawing. Now, drape the rope over the arch . . .
mark the string at the base of the arch . . . then cut the string so it’s only as long as the arch. Next, lay the string on the ground and fold it into thirds — I’ve marked the thirds with pink tape here, the mason would have used colored chalk — that thirds isn’t obvious, but it will get us to a wall thickness of about a fourth or a fifth. With the rope now marked into
three sections of equal length . . . return it to its original place along the arch template. We see that this divides the arch into three equal sections. Let’s mark one of them. Then pin one end of a new string to that marked spot.
Draw it taut and pin it to the base of the arch. We need to attach to the base a length of string exactly equal to the distance from the spot a third of the way along the arch to the base, so I just fold the rope and mark it.
And then draw that taut and in a straight line with the pinned section, then pin it to the base. The distance from that pin to the inner line is the width of the wall, which will be between a fourth and a fifth of the arch’s span. Let’s compare it to the arch section from Sainte Chapelle. If I superimpose it on the string construction you can see that it matches. It’s puzzling that we divided the arch into three sections, which yielded a width
between a fourth and a fifth, but that’s just how the geometry works out — that’s part of the rule. I’ve never worked out why geometrically that works because it doesn’t matter: the mason would never have worked it out analytically, what matters — and that’s the point here — is that it works. And to belabor the point a bit, its the same rule the Romans used to size their
supporting walls. Divide the interior of a Roman semicircular arch into three equal sized sections, draw a line from one of these “third markings” to the base of the arch, then double it … and you see the wall size of a bit less than a fourth of the interior arch’s span. With that rule we have arrived at the core of the engineering method. The proportional rule doesn’t come from some scientific analysis of stone and its properties. It comes from centuries of experience,
from trial and error. It is called a “rule of thumb,” more formally a “heuristic,” an imprecise method used as a shortcut to find the solution to a problem, often by narrowing the range of possible solutions. Here, a mason sized a wall’s width in a matter of minutes without understanding the fundamental material properties of stone and without understanding the mathematics needed
to apply the rule. That’s a paradigm for the engineering method: “Solving problems using rules of thumb that cause the best change in a poorly understood situation using available resources.” That’s a sharp contrast with the scientific method, because these rules of thumb are only guides that offers a high probability of success, but no guarantee. And, unlike a scientific theory, a rule of thumb is never, in a sense, disproved. That hundreds of cathedrals are still around
today, standing for eight or nine hundred years is proof of that! Instead of being disproved, this rule of thumb for stone became outdated, not wrong , as iron and steel I-beams replaced stone. Yet, no doubt, there’s a question in your mind: I’ve talked about buildings constructed eight or so centuries ago, so surely, you might wonder, is engineering based on rules of thumb antiquated in our scientific age? That line of thought misunderstands the purpose of the engineering
method, which is to solve practical problem before we have full scientific knowledge. In the next video I’ll explore how engineers work their way around that lack of scientific understanding, how they overcome uncertainty. I’m Bill Hammack, the engineer guy.