Saturday, April 27, 2013

More About Pyramid Building

Way back in the long-ago I wrote a little paper on how to build a pyramid using nothing but ancient tools and techniques. It was a thought experiment, done without real research, but every bit of archaeology I've seen since then tends to support my initial thoughts in every important detail. For instance, My hypothetical pyramid would take around 20,000 people to build, including all support staff, from doctors to potters to cooks. The core number of builders was 15,000. My hypothetical pyramid would have been completed in 20 years.

The Pyramid of Menkaure
There's a nice piece in that details how the builders of an actual pyramid (the pyramid of Menkaure) were fed. The article estimates a workforce of 10,000 builders.

Since the pyramid of Menkaure is a bit smaller than my hypothetical one, and mine is built on a more aggressive schedule, I'm quite pleased with my estimate of 15,000. I'm also still happy with my 5,000 man support staff. I'm sticking with both of those.

Re-reading my article, I note that I promised a follow-up, and I suppose I should write that. Before I do, though, I'd like to go ahead and answer a few observations and questions raised by a colleague of mine, Russell, in a theological article of his own. Then maybe I can turn my thought to those other matters I said I'd address.

Questions answered.
OBSERVATION: the units used in my calculations are purely arbitrary.
Answer: No, they're not. I used only one unit, the cubit, which was a well-attested Egyptian standard. However, the number of cubits is to a great degree, arbitrary. It always is. We're constrained by budget, ability, and desire. I started with 280 cubits in height, but I could have just as easily marked out acreage first. The end result is the same. Basically, I chose the height that I did for two reasons: My pharoah wants his pyramid to be taller than his dad's, and his architect picked a number that worked out to integer math.

Do not underestimate the value of that "integer math" requirement. Decimals were unknown back then, as place values and the zero hadn't yet been invented; and fractions and multiple units of measure are notoriously error-prone. You really don't want to deal with any of that on a large project.
QUESTION: Why did the Egyptians not roll the wheel 4 times or 6 times instead of 5? 
Answer: Experience. This isn't the first pyramid they built. These things didn't just jump into existence... there is a solid record of the evolution of Egyptian pyramids, which I'm about to show you. At each stage the Egyptians learned something that they applied to later constructions.

Build a pyramid too shallow and it's not impressive. Build it too steep and it crumbles. Shallow pyramids were the first to be built. They are now called "mastabas", but the ancient Egyptians called them by a word meaning "houses of eternity".  A mastaba resembled a bunker built over a shaft leading to the burial chamber.  The feature of a subterranean chamber was retained right up to the building of classical structures like the pyramids of Menkare and Khufu.

A mastaba and its structure
The Step Pyramid of Djoser was basically a series of mastabas sitting one atop the other. It had no "slope" per se. (BTW, step pyramids were quite common in the ancient world. They were called "ziggurats" in Babylonia).
The Step Pyramid of Djoser
The first true pyramids DID crumble... the pyramid of Djoser is surrounded by examples.
A failed pyramid.
This is what happens when you build them too steep.
In fact, we have a stable example of a case where they tried to use a different ratio... this is the "Bent Pyramid". Halfway through its construction it became evident that the slope was dangerous and unstable, and the builders corrected it.  That's the kind of permanent public embarrassment that no one ever wants to repeat.
THIS is what happens when you admit your mistakes.
The slope that was eventually used very nicely approximates the slope of a sand dune (and that at which the crumbling pyramids settled), while still using integer math. In other words, you heap stuff up and it tends to stay that way on its own, making this a very stable design.

BTW, the Egyptians weren't the only pyramid builders... the Ethiopians built many smaller pyramids with a steeper slope.
The steep pyramids of Meroe at Begarawiyah
Note the older ones crumbling in the background
Russell re-states the question a little more emphatically:
Question: Why is it NECESSARY to roll the wheel FIVE (5) times in one direction and FIVE (5) times in the opposite direction to yield the correct results?
Answer: As you can see above, it isn't, and from reading my article, you note that they did not measure multiples of five. Rather, the height was 280 and the base was half that number of turns, or 140*pi. That's 70 turns out from the center. Not five. So our 70 turns times the circumference of our one-cubit diameter odometer should give us about 220 cubits. It actually works out to 219.9 and change, which is so close to the integer answer as to not matter to our Architect.

The number five really has nothing to do with anything in this discussion. You can pick any height you want, but it's easiest if it's a multiple of four if you want this slope. That's purely because you'll be you'll be cutting the number in half for the total length of the base so it's structurally solid and aesthetically pleasing, and then in half again if you intend to measure out from the center. In other words, we're just sticking with integer math, that's all. But four fingers + one thumb gives my friend the "five" he needs for his theological discussion.

Remember that my article details what the builders most probably did do. If they had used a different slope, the "correct" results would have been different. However, the "correct" results that we actually see are strongly driven by experience and physics for a structure of this size. We've seen that smaller structures have completely different slopes, and we've seen that they structurally fail when scaled up. Nevertheless we do find reasonable variation in the slopes of all pyramids, which should curtail the argument that any particular ratio is "necessary".

Also note that myriad numerical coincidences and relationships would still have existed had they chosen a different slope... they'd just be different from the ones you have now. You'd just be focusing on those instead of the one you have, drawing similar theological conclusions and concluding that they are also somehow "necessary".  For instance, if we had divided the height by 3 instead of four (despite the resulting fractions), we'd have insisted on theological comparisons resulting from the number three (such as the trinity, or past-present-future). If we'd used five instead of four then we could use the same relationships my friend does now without having to refer to the fingers + thumb kludge. If it had been six, then we could invoke the Magen David and the six days of the Creation. And so it goes. Instead of pi, we could be discussing tauPick any number whatsoever, and you can construct a series of relationships, all of which are "important", "necessary", and "beyond coincidence".

Cherry-picking the numerical relationships that necessarily exist is closely related to the "anthropic principle".  The weak anthropic principle is an unremarkable tautology while the strong anthropic principle is a logical fallacy.
Question: His Pharaoh wanted a height of 280, nothing else.  They would have had a pyramid 20 diameters in height, but no demonstration of Pi had they chosen any other model than 5 rolls of the wheel.  Is Dave saying that the builders did not understand Pi (no advanced technology) and therefore found it by accident?  
Answer: Yes, that's exactly what I'm saying. Furthermore, I'm saying they didn't even "find it" if by "find it" we mean that they realized its significance.  By the way, it doesn't matter how many rolls of the wheel we're talking about. The only thing that "encodes" pi into the design is that we're using a cubit both for the diameter of the wheel and for any other measurements we make, and that the builders prefer integer math. ONE roll of the wheel and ONE cubit height would still "encode" pi into the design. ONE roll and THREE cubits height would do the same. The only way to eliminate that "encoding" is to use some other wheel diameter, which means arbitrarily mixing units of measure, which is exactly what I did not do. A cursory reading of the Bible would give you a value of 3 for pi[*]. But there's no evidence that the ancient Egyptians understood pi as anything useful or special.

The Rhind Papyris
A primary use for pi is to calculate the area of a circle. But an ancient textbook the Rhind Mathematical Papyrus, does this (imperfectly) by a completely different method. In a compelling blog post, Jason Dyer has some more thoughts on that.  Pi is never acknowledged in the Rhind Mathematical Papyrus, but to the extent that it is implied, it would equate to 256/81, or about 3.16. Since the Egyptians' knowledge was actually written down, and we have a textbook, we know that any value more accurate than that is accidental.

Also, the "280, nothing else" isn't as dictatorial as it's made to sound. As I stated above, the pharoah would have arrive at this number through consultation with his Architect. He'd want it bigger than his predecessors', but still allowing for integer math.

Odds and Ends

A list of numerical coincidences follow Russell's questions, and most of them require the use of "pyramid inches". The problem with that is that "pyramid inches" are made up. The reason the coincidences occur is that the coincidence was assumed from the start, and the value of the pyramid inch was adjusted until its "correct" value was "deduced". There's no archaeological evidence for any such thing.

A man is entitled to his religion, so I won't discuss any of the theology on Russell's page. However, I can't resist asking a couple of questions and one observation of my own...
  1. Why would the ancient Egyptians have any interest in recording or predicting the exact dates of the birth of Adam, the Exodus, or the birth, baptism, and crucifixion of Jesus?
  2. If they call all that right in exacting detail, how did they manage to flub their entire pantheon, as well as the entirety of their cosmology and escatology?
These are far more difficult to answer than whether the builders used integer math and wheels.

As I mentioned in my original article, it is a perfectly valid rhetorical device to use a thing to illustrate an idea. We just shouldn't get so caught up in it that we mistake our illustration for the builders' intent. When we do, we run the risk of letting the minute details a carefully crafted cosmology detract from the bigger message.


(August 2013)

This 2007 issue of Archaeology magazine describes Jean-Pierre Houdin's compelling theory on construction techniques, none of which involve space aliens or super-science. Click on the magazine cover to read the article.


[*] of course, the Bible isn't a math textbook, and 3 is close enough for the narrative in 1 Kings 7:23-26 given all the talk of "hands-breadths" and fluted rims. Nevertheless there's almost a branch of apologetics devoted to this numerical "problem". Usually these involve using the "hand's breadth" thickness to take the outer diameter of the bowl and the inner circumference, these being the diameter and circumference of the mold. But the most inventive answer, sure to appeal to math geeks and gematria buffs, is this one.

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