Saturday, June 30, 2007

I got a new chair

Here's the thing about having kids; stuff in your house mysteriously breaks. I have an older son who's rarely in the house proper, and twin 11-year olds. With the twins, things break more often and nobody -- and I mean nobody -- ever cops to having done the breaking.

So it was with my office chair. I spent a day at work and came home to find that my chair had gone wubbly. You heard me... wubbly. The seat, which should have been firmly planted on the pedestal, was just sort of balancing precariously like a plate on a rod held by one of those Chinese circus performers. Not only that, but the pneumatic pedestal sort of stopped working so it's constantly in the up position when you get up, and sinks rapidly to the floor when you sit in it. According to the entire family, "it was just like that" without any action on their part. Sigh. I'm not looking to blame anybody; I just want to know wtf happened to the chair. It's not too much to ask.

So I bought a new chair. I went to where I bought my old one (Sam's Club), but no dice. The chair had been discontinued. So I bought this instead. I've really only got one major criterion for a chair, and that's that the armrests must attach firmly to both the seat and the back. It strengthens the whole thing. I looked at a bunch of chairs. This upscale model was rather nice (Despite the fact that the website says it's available on-line only, it actually was in the Spartanburg store). It's nice, but not nice enough to recommend it over the one I bought. I also didn't like the fact that the wood made it quite a bit more heavy. I also looked at a nice leather version of the same one I bought, which, for only $15 more than I eventually paid is a steal. However, the micro-fiber chair I got was simply more comfortable, and it doesn't make you sweat like the leather does.

I'm having to get used to the fact that it's a little narrower than my old seat, but now that I have been sitting in it a couple of days, I can tell you it's just a really, really good chair. Ahhhh.

Friday, June 29, 2007

How to Build a Pyramid

[9 Oct 2011: I originally wrote this article in 2007 and it was on its own web page. Recently I saw another television special about the pyramids, and my frustration with the astounding ignorance of the Ph.D. who wrote it moved me to re-post it here, update it and fix the formatting.]

How to Build a Pyramid - Giza Style
by David F. Leigh

"Architect! Build me a monument! It must be 280 cubits high and outlast the ages!"
So the Pharoah has appointed you to build his next pyramid. Whe're to start? You live in the Bronze Age, in the year 2580 B.C. Is it even possible to carry out the order with great precision given the tools that you have available to you? Let's find out. We'll start with an inventory.

Your resources 
  • Aproximately 20,000 highly motivated manual laborers, stonecutters, foremen, and and support staff. 
  • Adequate food and shelter for these workers (money is no object, since it doesn't exist anyway) 
  • The natural resources of the land on which you live, and those of surrounding countries for which you can trade. 
  • Your intelligence and common sense 

Your tools 
  • Wooden mallets 
  • Measuring rods and ropes in cubits (about the length of you elbow to your outstretched fingertip) 
  • Bronze, copper, and stone chisels, adzes, axes, choppers and pounders. 
  • Wooden Levers 
  • Rope. Lots of rope 
  • Wooden sledges 
  • Oil and water. 
  • Whatever you can dream up that doesn't involve steam, motors, iron, plastic, or other modern materials. Copper, bronze, stone, wood, leather, papyrus reed, ivory, bone, rope, lead, glue, and shell are all acceptable materials. 
  • You can use techniques whose discovery is credited to people living centuries later, so long as they are simply accomplished with Bronze Age technology. For instance, you can bisect an angle with a compass and straightedge. After all, improper assignment of credit happens all the time, even today. 

Making a blueprint 
There are plenty of reasons for selecting a pyramid to be our monument. First, Pharoah's father and grandfather built pyramids.... it's a family tradition. And pyramids echo the timelessness of mountains. And with its sloping sides, a pyramid looks much taller to someone standing at its base than it really is.

But there are practical reasons, too. Two hundred eighty cubits is really, really tall. So tall that we're not going to be able to build an obelisk that high. It would fall over or break with the stress of trying to raise it. We can't build in clay or brick either, since that will crack and buckle under the weight that's placed on it. Besides, bricks crumble over time. So we're going to be building in stone. And we're going to need a wide base. A pyramid is extremely stable.

Nonetheless, we have some limits. Build it too steep and the weight will crack the base. Too shallow and it won't be impressive. This needs to be extremely well built and precise, both because we take pride in our work and to celebrate the perfection of nature and our faith in the infallibility of our ruler.

So we start with an architect's drawing on papyrus. This will be a scale drawing we'll use to get approval from Pharoah, and it will be important for other reasons to be seen later. We start with our height: two hundred eighty cubits. That's what Pharoah ordered, that's what we'll deliver.
[The height here is completely arbitrary. We could have as easily said that we're going to fill up a certain area, and the calculations will work out exactly the same. However, there's no actual problem with the fact that it's arbitrary: the Pharoah wants something bigger than has ever been built before, and it should be divisible by two for the simple reason that we want, in this early age, to avoid fractions. As you'll see, even though the value of pi will be inherent in the pyramid, all of the numbers we'll manipulate are integers.]
Now for the base. We know that it'll be big, big, big, so we're going to need to be able to measure it exactly when we scale it up. Measuring with rope is no good... it stretches. Measuring rods have to many opportunities for slippage. Pacing it off is a joke. I choose to roll a wheel because an odometer is the best low-tech way of measuring the long distance exactly. So, in laying out my blueprint, I measure the base with a wheel, a little scaled-down odometer, one unit in diameter. It looks a little like a pizza cutter. As it happens, 140 turns of the wheel gives us a pretty pleasing shape, just shy of 440 cubits wide at the base. The pyramid is broad but imposing. It's stable, with a slope that's not too steep or shallow. And the math is pretty easy, too. All integers. I'll be able to pace off 70 turns in either direction from the center to find the edges.

I have no intention of sticking with just a papyrus blueprint. We'll build a number of scale models of wood or stone. Heck, we'll build a limestone model for Pharoah to gaze upon while we're building the real thing.
[Now, with a real pyramid we'd include a bunch of other features... a gallery, tombs, conduits... I'll describe those at a later date. But for right now, for the purposes of this web page, I'm building a featureless mountain.] 
Our project plan is to finish the work in about 20 years, before Pharoah dies. To do this we're going to have to shift over 2 million stone blocks into place. Working 360 days a year for 20 years (I'm leaving off a few days for contingency and festivals, or whatever), we're going to need to move about 300 blocks per day. With an average of maybe 2.5 tons per block we're going to need teams of 20 to 25 men per block.... let's say 25. That means 7,500 unskilled men at a time on the gangs if each gang only delivered one block per day. More likely you'd be able to shift a number of blocks per day. Add rampbuilders, quarrymen, and more workers to place the stones, then some masons to finish the blocks. I'm going to estimate 15,000 people working on the pyramid at any one time, but I'll call it 20,000 to include support workers... cooks, potters, doctors, etc. And I think that's conservative. This is doable. We don't have hundreds of thousands of mouths to feed, and given our population, we can do it with part-time conscripted labor.... sort of like a term in the army. At least we won't need slaves. And we can scale up or back for certain portions of the construction.
[Just after I first wrote this I saw a TV special about the Pyramids on the National Geographic channel, narrated by Avery Brooks (talk about timing!). They came to the same reckoning regarding total personnel, but for different reasons. The producers figured that 2,000 laborers would be sufficient to the task. If so, they'd each be shifting an awful lot of blocks each day. As I've personally engaged in labor, and I calculate the length that these blocks would be pulled, I conclude that you'd be doing good to walk the distance in the time allotted per day, much less pull a 2.5 ton weight. I think we'd need at least double the number of laborers they're suggesting. Likewise, there are too many support staff. People worked harder and longer... we're not talking about a 40-hour week with weekends off. I think they were extremely light on common labor and extremely heavy on support staff, but their total figure of 25,000 was reasonably close to mine.] 

Selecting a location 
The criteria for the location our monument are going to be critical. We're going to need a sturdy, level rock base, near a river and limestone quarry. We need limestone instead of granite because we're using Bronze Age tools, and granite's too hard for us to work. The quarry will be the source of our raw materials, but the river will bring us wood and supplies, and (very importantly) and unlimited supply of water. Proximity to the river means we'll have farmland on which to grow the food for our workers. Good climate is a plus so we can get in as much work as possible per year. Look at this satellite image of the Giza Plateau (29°58'51"N 31°09'00"E). Zoom out and see how it meets every one of our requirements. It's a limestone plateau on the edge of a fertile river valley cutting through a desert with a stable climate. Furthermore, we already own it. Isn't that a great location? We'll use it.

Leveling the foundation 
This is a fun task! How do we level a rock the size of many professional football fields? Well, I'd do it with a bucket of water.

Have you ever poured water on a concrete floor? It fills every dip and shows off every bump. So we're going to take some of that awesome manpower we have and splash buckets of water all over the plateau. Then it's scrape... level... scrape with our adzes, gavels, and granite blocks until the water doesn't run or pool. It's just a lot of hard work. However, there's nothing technically challenging to a Bronze Age engineer. I think the scrubbed and level plateau would have been one heck of a sight, impressive in its own right.
[The TV dramatization had the Pharoah pounding a stake into the thick sand to indicate True North, as he's surrounded by scrub and creosote bushes, or whatever it is that grows in Egypt. It's a wonder they could they didn't trip over all the plants. The Architect dangled a pendulum from his outstretched hand to site the circumpolar stars as the pendulum swayed to and fro. Why lay a stake that's going to be swept away when the site is subsequently cleared? Or did they simply plan on dropping multi-ton rocks on the plants and dunes, squashing them flat? I don't ask for perfection, just common sense.]

Measuring the base 
This is where we use our full-size odometer one unit (a cubit) in diameter. first we do some gross measurement to figure roughly the center of the plateau. Then we mark off right-angle centerlines North/South and East/West. The North/South line was easy to determine, as we have the clear desert air to help sight the North Star. There are a number of low-tech ways to determine a perfect right angle. The easiest is probably the compass method you learned in high school. Our "compass" is going to be pretty big, though... we can use a length of very stout rope for it. We would not use it to mark the long sides of the structure since it would stretch, but the shorter length we'll use here can be accurate enough.
[My survey would look entirely different from the TV dramatization. Imagine the site completely cleared of scrub and sand . Only bare limestone remains. This might resemble a huge deserted Air Force concrete paved storage field for B-52 bombers. There's no point in pinning down True North until the ground has been prepared to this degree. Very small people in the midst of this vast stone dance floor would survey the centerline to True North. The plumb bob would be suspended from a wooden armature and shielded from the wind. These are professionals.] 

Getting the angles right 
One of the most important things we're able to derive from the blueprint is the angles at the base. In fact, we're going to use the drawing to create some tools to help us in building. Namely, our angle gauge. It's basically like a crude protractor with a plumb bob. There are a number of variations of this tool that would work. We'll make some similar T-shaped or A-shaped tools to make sure our blocks are level.

During building we can put this thing against the side of the pyramid at any time and sight along it to ensure that our angles are proper. It doesn't matter what the angle actually is in degrees or radians... all that matters is that we exactly match the drawing. If it does, then all four sides will meet at the exact center at exactly the right height, assuming we measured the base correctly. There's no need for us to devise a method of measuring the height during building, since it's necessarily determined.

Ramping up ** (see update) 
[Cutting stone blocks isn't something that's generally disputed in a discussion of pyramid building, so we're going to take it as a given that our experienced Bronze age quarry workers can actually do the job we know they did. At a later date I'll discuss the how we determined the size of the blocks we're to use.] 
We're using a ramp to get our blocks up to the plateau and to hoist them onto the structure. Yep, that's it. No hoists, no cranes. See, there are three things we have plenty of as a result of our quarrying and location: sand, limestone rubble, and water. This ramp is going to be huge, and we really don't care, because there are far too many advantages to the ramp to give it up. First, there's ease of transportation. With a big enough ramp with a shallow enough slope, we have the the energy-saving benefits of an inclined plane. Trying to lift the blocks vertically will take far too much effort and will be much slower by comparison. Second, we can get more workers on the task of moving each block. You can't crowd very many people around a crane. Third, there's safety. There are no cranes to collapse. The block is never off of the surface; if a rope breaks, it won't fall on someone. And the shallow slope prevents a block from sliding back very far or fast. Finally, we can depend on unskilled labor to do the work. The architects may be professionals, but the laborers are just citizens doing community service. When they've done their time they'll go back to farming or fishing or herding goats.

A good portion of our crew will be engaged in a never-ending road paving project. As we build up each level of the pyramid, the ramp is extended to the next level. It becomes longer, and longer as it gets higher. At the end of the project, this ramp will be completely dismantled. We might use the material for other construction projects, roads or paving, or we might just use it to back-fill the quarry.

We're going to be moving our large stone blocks with ropes, wooden sledges, and brute force provided by gangs of workers who drag them up the ramp. We could use some other clever techniques, such as attaching rockers to the sides of a block to turn the block into a wheel. The problem here is with safety. Pushing a wheel up the ramp is ironically tougher than moving the block on a sledge. Why? Because with the sledge friction isn't just your enemy; it's your friend. You can rest, change crews, etc without expending a lot of energy to keep the sledge stationary. If a rope breaks it's not a big deal in the grand scheme of things. To overcome friction you'll be using water or oil on the sledge runners. With the wheel rockers, you'd have to expend a great deal of energy just to keep it stationary. It means constant tension on the ropes at all times. And it means that a rope breaking could be a devastating event as two tons of rock goes careening down the ramp, killing workers as it goes, causing them to release their ropes on their blocks, which go careening down... you get the picture. Ironically, for our construction purposes, dragging is the superior technique.

The work goes something like this: some workers build the ramp up. At the same time lots of quarrying is going on and stones are awaiting transport. When the ramp is finished, then the ramp-builders lay off or are reassigned while stones are dragged up. When the level is completed, it's the same thing all over again. The number of workers available for quarrying varies depending on how many are needed for transporting the stones or working on the ramp at any particular time.

** UPDATE 8/20/2013: It's unlikely that the actual pyramid had a huge external ramp. A new theory, which has plenty of credibility as it makes testable predictions, is that the pyramid contains an internal spiral ramp. This is well-thought out, and explains a number of features of the great pyramid that are otherwise puzzling (to say the least). In addition to requiring very little additional material (a ramp would only be used for the bottom third of the pyramid), it explains the "grand gallery" as a channel for a counter-weighted sledge used to hoist the granite slabs used for the roof of the King's Chamber. Also, this technique agrees with my assumption that safety is a major concern, and describes how the limestone cladding would have been placed first, back-filled with the sandstone blocks. As you can read below, I have always maintained that the cladding would have been placed as the construction progressed rather than separately, but Houdin's theory not only ups the safety, but makes construction much easier. Click on the cover of Archaeology to read more about it. The major point here is that neither magic, divine intervention, nor super-science is in any way ever required for construction of this impressive structure.

Casing the Joint 
Some people might consider cladding the pyramid with casing stones after the main structure is built. I prefer to plan the cladding as each level is done. Again, this is a safety issue. The casing stones are smooth. How would you possibly plan to put the stones there after the fact? You really don't want to wind up sliding on an unstoppable path to the plateau floor to be crushed by a falling casing stone. Cladding as we go keeps it safe, and allows us to better track our angle as we move up the structure. It also allows us to build and dismantle our ramp only once.

Capping it off 
The capstone ("pyramidium") is the last thing we'll add. Prior to putting it in place the architect and each team leader might sign or mark the base. After all, this is something to be truly proud of. The Pharoah has a surprise for us... he's provided specialists to apply gold leaf to the capstone. The reflection of the sun can be seen for miles!

Cleaning up 
Goodbye ramp. It's dismantled and hauled away by the millions of bucketloads.. The white limestone casing is washed and polished on the way down. The polishing is done by scrubbing the casing with large flat stones, using the ever-present sand as an abrasive.

What Did We Just Build? Math and Wonder in Hindsight. 
Isn't this an amazing structure? And though the construction took sweat and determination, the design itself was dead simple, requiring only integer math, a wheel, and some plumb bobs and string. But it was only simple because we looked at the problem through the eyes of a Bronze Age engineer. Imagine we show the finished product to a mathematician who had no experience with the engineering challenges that faced us. What does he see?
  • The ratio of the base to the height is pi/2. 
Comment: Sure it is! We used half the number of turns of the odometer wheel as the number of cubits in height we'd planned. Since each full turn is of length pi, then the ratio of the base to height is one-half pi, or 1.570796326795. Our design is theoretically exact. Using the values of height and base from Wikipedia ( b=440, h=280), the calculated ratio for the actual Great Pyramid is approximately 1.571428571429. The difference isn't worth stressing over.
  • The perimeter of the base equals the circumference of a circle whose radius is equal to the height of the pyramid. 
Comment: Naturally! The circumference of a circle is 2 * pi * the radius. We'll just call the height "one unit", so it simplifies to 2*pi. Since each side is to the ratio pi/2 (above), and we have four sides, then the perimeter is 2*pi. It couldn't possibly be otherwise. Of course, that's true for ours, but the actual Great Pyramid is an approximation. 
  • The stones at the base are placed with high accuracy. The variance of the length of each side is on the order of scant inches. 
Comment: This is a result of using an odometer to do the measuring. On the actual Great Pyramid, the length of the sides vary by as much as 8 inches, which is fantastically good for a structure of this size, but hardly supernatural.
  • The angles are 51°45'27" 
Comment: Well, of course. This is because we took great care to build our angle surveying equipment (the angles and plumb bobs we used to sight with) from the original blueprints, which were created with a scale of our circular odometer. Using the measurments, 200 cubits of height and 100 turns of the cubit wheel at the bottom, we can calculate the angle as follows: 
base angle = ASIN((1/2 * base)/ height)
base angle = ASIN ((1/2 * 140 cubits * pi) / 280 cubits) = 51.75751851602 degrees = 51°45'27"
This is purely determined by the ratio of the height to the base. If it weren't this, then the height/base ratio we noticed above wouldn't be accurate. Again, our design is theoretically exact. Interestingly, the numbers given for the real Great Pyramid vary according to the source. Wikipedia gives it as 51°50'40". However, if we took the reported height (280 cubits) and width (440 cubits) values as accurate, then the calculated value should be 51°47'12.4" . Various sources give the value as any of these, plus a dozen more. Some are the result of approximation, poor observation, or are simply calculated using whatever values for height and width strike the fancy of the observer. 
Remember that the top of the Great Pyramid is missing and has to be calculated. so either the base/height ratio is off or the angle is. So much for supernatural accuracy.
  • The geographical location is the "center of balance" of the landmass of the Earth. 
Comment: We had some requirements for the location, none of which involved surveying the entire Earth. Besides, we live where we live. It's not like we migrated to Egypt just to build this one pyramid. Go back to and look at the location again. How many other locations in Egypt match our requirements as well as this one? None. That's why so much of this megalithic work is centered at Giza. And there is no bloody "center of balance" for our globe, unless you count the center of gravity in the core. What's described by this observation is an artifact of the map projection being used. 
  • The coup de grace: The sides bow inward ever so slightly. In fact, the arc of the bow is equal to the curvature of the Earth! Surely this couldn't possibly be a coincidence! 
Comment: If it were noticeable, this would be such an amazing effect that it's tempting to claim you actually were cleverly encoding your "astounding knowledge" of the exact curvature of the Earth. The reality is much more mundane. Our technique for leveling the plateau involved the flow of water under the influence of gravity. It ensured that all points on the plateau were equidistant from the center of the Earth. On the other hand, our method of measuring the sides (measuring out from a centerline), is valid only for plane geometry. On a curved surface like the Earth, though, the best definition of a "line" is a Great Circle route. Try this experiment yourself. Cut a perfectly square piece of paper. Then place it on a globe and pin down the corners. What happens to the sides? They appear to bow inward, to a degree that mirrors the curvature of the surface of the globe! So this "encoding" has nothing whatsoever to do with actual knowledge of the curvature of the Earth... rather, it's evidence that the engineer didn't even take it into account (and probably is totally ignorant of it!) This would be noticeable only in a monument of stupendous size. 
Update: OK, I've been asked to explain this one further, so I'm including a graphic to help folks visualize this: Start at the center and measure a line North/South, and another East/West. Now lay out a grid, starting from those center lines and working outward in both directions, making sure that all of the N/S lines are parallel to the original N/S line. Likewise, all the E/W lines are parallel to the original one (like lines of latitude on both sides of the Equator. When you look at the result you'll see that only the center lines trace Great Circle routes. The lines on the edges are bowed compared to a Great Circle route when viewed from above, just like lines of latitude. An azimuthal stereographic map projection illustrates this perfectly (think the Pan Am logo).
Compare this to Great Circle routes, which is what you'd get if you measured your straight lines with a line or surveying equipment. Great Circle routes are straight as you get on a globe, but appear to bow outward, like lines of longitude converging at the poles and spreading at the equator. Now, measuring with a wheel from a centerline might not be how the Pyramids were actually laid out, but it explains so much so easily that I double-dog-dare you to find a more elegant approach that matches the actual geometry of the Great Pyramid.

Closing Thoughts 
Prior to writing this page I looked up the dimensions of the Great Pyramid, and but pretty much discarded everything except the height in designing my building approach. Happily, the approach then pretty closely matched what was actually built for the Pharoah Khufu. Though they didn't have our advantage of 5,000 years of accumulated technology, the Pharoah's builders were no less intelligent than modern engineers. And if I can figure this out on my own, they could, too.

I didn't do any particular archeological research before writing this How-To. It wasn't necessary. My goal here isn't to tell you how the Egyptians did build the Great Pyramid, but to show that it could have been done with the technology of the time. And it absolutely could. No advanced technology, no alien visitors, no divine intervention need apply. In fact, the argument that the Egyptians conceived and built it themselves is so conclusive that to refute it you'd have to actually produce a witness to the intervention. Good luck.

That said, the building of such a structure is an astounding feat. And the coincidental relationships of its measurements, driven as they were by necessity, truly interesting. I don't care if somebody reads whatever meaning they want into them, and uses them to illustrate whatever message they have to give. Understand, though, that this is an exercise in retcon -- retroactively assigning intent where there was none by the builders. There's no evidence that the builders intended to encode anything beyond what we know of their culture... at all.

Postscript (2007): 
I think I ought to explain that, since this page has been online for half a day and I'm getting multiple feedback from multiple directions.

I think that, coincidental or not, such numerical relationships can be used in an instructive way, much as St. Patrick is said to have used the shamrock to explain the Trinity or Richard Middleton used a deck of playing cards as a prayer aid. Our ability to use patterns in this way does not provide evidence that this is the purpose of the object we're using as illustration. For instance, we are a race of people that see bunny rabbits in the clouds, yet we don't imagine that clouds were designed to display images of bunny rabbits. That said, the fact that something is used to illustrate a message in no way diminishes the message itself. The illustration should help you to understand the message; it is not the message itself.

Of course, just because there's no evidence of something doesn't necessarily mean it's not so, either. But it's always good to be clear about what we can and can't prove. In the case of the Pyramids, we can reasonably expect them to designed with the symbology of their culture in mind (and the same would hold true for structures built in Mesoamerica, or China, or anywhere else). I don't think it's necessary to ascribe to the builders of these objects an understanding of concepts that are foreign to what we know about their culture, anymore than we can reasonably say they were incapable of work that can clearly be demonstrated as possible using (again) what we know about their culture and technology.
  • Must the Egyptians have known the exact value of pi? NO.  
  • Must they have had help from aliens? NO
  • Is it valid to use the Great Pyramid to illustrate the joining of the finite to the infinite? Definitely. Illustrate whatever you like. Just don't confuse your analogy with the builders' intent.
  • Is the technique I describe sufficient to disprove that space aliens built it? Unless you introduce me to a space alien, I'll take that as my working assumption: YES. Extraordinary claims require extraordinary evidence.

Postscript (2011)
On Hulu now is a miniseries entitled "The Pyramid Code". Although there is mixed in here some facts and a few valid conjectures, In my opinion this is, in aggregate, a huge honking sack of bovine excrement. In it is postulated all sorts of magical thinking under the very thinnest veneer of scientific language. This does a grave disservice to the ancients by taking their very real accomplishments, dismissing them as impossible, and replacing them with rainbow unicorn fantasies of resonant crystals and 'subtle energies'. Yes, it was written by a Ph.D, Carmen Boulter, and that's a very sad thing for her university.

Copyright 2007-2011 by David F. Leigh