Surveying for Your Own Egyptian Pyramid

Surveying for Your Own Egyptian Pyramid The Great Pyramid

| Site's Table of Contents | Site's Index |

PLEASE NOTE: The following scheme is only that of this one author. You may have even better ways to do this. This is only offered to plant ideas in your thinking.

Starting materials

Hundreds of feet of strong twine that doesn't stretch very much.
Several hundred feet of garden hose
An indelible marker of a bright fluorescent color.
A dozen tent stakes
A half dozen 2"x2" wooden stakes that are 3 or 4 feet long.
A square, flat, LEVEL field of at least 13 acres (approx. 750 ft on each side)
A 12 foot tall pole
A sunny day on March 21 or September 22
Four or five friends to be helpers.

INTRODUCTION

Of course, you are not going to build your own pyramid, but you are going to plan it, and that means you must deal with all the activities necessary to survey the building site so the thousands of imaginary workers can soon start placing the large stone blocks in the right places. There are several things you must do as an "experimental archeologist." Some are simple, and others are probably simple IF you can figure out how to do them without the aid of modern concepts such as the fine points of Euclidean geometry, trigonometry, and even, perhaps, the finer points of common arithmetic such as division and multiplication! Among other things to be re-invented after several thousand years will be engineering feats such as how to prepare the huge foundation so that it is level and not on a slight slope. With such a massive thing, even the slightest slope might eventually cause it to slip and form a giant rock slide. So let's enumerate what we need to do, and then have a few comments in purple:

Need to find a solid base upon which to build the pyramid. This is obvious: build it on bedrock, whicht the Eqyptians were smart enough to do.
Need to make that base level so that no point is higher than any other point. How the Eqyptians did this was figured out in the past 50 years.
For orientation purposes, we need to find the cardinal compass directions (N, E, S, W). Magnetic compasses are not allowed. The North Star, Polaris, is usually lost in the horizon haze in the Egyptian latitudes.
Need to make a huge perfect square such that the sides exactly face the cardinal directions. Once you figure out the cardinal directions, this is easy.
Need to determine the exact center of the square so that the apex (point) of the pyramid is somewhere directly overhead. This is very easy.
Need to determine the exact angles that the sides slope inwards. Remember, no trigonometry is allowed: it didn't exist back then.

LET'S BEGIN!

1 What you want You are not actually going to level a field, but rather you are going to hammer four stakes into the ground in just the right places (where the corners of your pyramid will be), and just far enough into the ground so that the tops of the stakes are all level with each other. If you were Pharoah, you would order your workers to scrape off enough of the bed rock here and there so that the whole surface is level. That would take a week or two. The picture here shows your ultimate goal. Following are steps needed to arrive at that goal.

2 Finding the compass directions Plant the tall pole upright near the center of your level field. On a sunny morning, you will see the pole's casting a shaddow. Set a rock or brick or hammer in a tent stake where the shadow of the top of the pole is cast.

Every couple of hours hammer in other stakes wherever the pole's top casts its shaddow. The stakes should form a straight line in one of the two cardinal directions, namely, E-W or N-S. But which one? And this will happen on only two days of the year - March 21 and September 22. Neat!

3 Placing the pyramid's first side. Now "draw" a line that is parallel to your solar shadow line. This new line will be one of the edges of your pyramid. (If you were really smart, you would have placed your pole so that its shadow points would form the pyramid's edge line! If you weren't that smart, you could either replant the pole for another sunny day, or be very careful to be sure that your new line is exactly parallel to the shaddow line. How to be sure: extend the shadow line a long distance in both directions so that it will be at least as long as the intended length of the pyramid's side. Using the string, cut two identical lengths by folding the string in half and cutting it. place one piece on the shadow line and the other where you think the pyramid's edge might be. The diagonals between the two strings should be the same (use more string to measure the diagonals. Once you have have adjusted the edge string so that both diagonals are in fact equal, hammer in a stake at each end of the edge string.

4 Planting the REFERENCE stake. Step back a bit from your edge line and scan the scene. You can now envision where the four corners of your pyramid will be. (If you are planning a full-scale model, those corners are going to be about 800 feet apart - probably enclosing an area larger than you have available!) Because the "lay of the land" is not likely to be flat or "level", you must choose which corner is on the highest ground. (As Pharoah, you would have your workers scoop it out so that the whole field is level with the lowest place!) Go to that approximate highest corner and drive in a wooden stake so that it is very firmly set in place, and then saw a notch in it about 6 inches above the ground. (The notch is a weather-resistant mark! All other points will be made level to this notch. It is mandatory that this notch's position not be lost. It is your standard, your "control," or "reference" stake.)

5 Establishing the first corner. Go to your edge line and hammer in a stake firmly where you want one of your pyramid's corners to be. This will be, as you will see, your "standard corner."

6 Levelling your first or "standard" corner. You want to cut a notch in the corner stake that is exactly level with that of the reference stake. What the ancient Egyptians did was dig a trench between the two stakes and then fill it with water. The water would "seek its own level" and they would measure up from the water level. You can imagine your digging such a ditch with shovels (no bulldozers then!), and that would appear to be sweaty work. But look how easy you would have it: the ancients dug their ditch in bedrock! (No dynamite then either!) So don't start digging because in your supplies you have a long length of garden hose. Fill it with water, and hold both ends over your heads as you and your partner separate. Your partner goes to stand by the reference stake and you stand by the standard corner stake. Slowly you lower your two ends of the hose (hand signals are in order for the long distances involved!) When the partner says that that end is even with the notch, you raise your end until the partner signals that water is flowing out that end. You look to see where the water is in your end of the hose. Shining a bright flashlight through the hose might help. Because the levels at both ends will be identical, draw a line on your corner stake. Then weatherproof it by notching it. Yes, you will have a few mishaps in this until you get the hang of it. So have a few buckets of water and a funnel handy to reload the hose. (Having a stopper for each end is also a good idea so you can lay the hose down without losing the water.)

7 Roughing in the perimeter. You want to get a piece of strong string that is equal to the perimeter of your pyramid. (If you are going for a full-scale model, then the string should be about 1 km long (about 2/3 mile)!) You want to divide this into four exactly equal lengths. (Because string or rope is a bit stretchy, Pharoah would order that you use chain!) So once you have all this string or cord or rope (or chain!) stretched out and untangled. Tie the two ends together.

Stretch the coubled cord out in a straight line. It might help to loop the knotted end over your corner stake, and then send your partner on a long hike to stretch it out.

Now have your partner double back with the far end and loop that over the corner stake along with the knotted end. Once back, you take a wooden stake and place it in the far loops and pull it tight as you set the stake on the edge line and hammer it in firmly. At this time use marker to color the cord strands at both stakes. This will divide the cord into four equal lengths. Don't be stingy with your marker.

Remove one loop from each stake and, using two additional helpers, stretch it out into what will approximately be the baselines of the pyramid. See how the red marks indicate the future corners? But you don't have it perfectly square yet so don't hammer in the other two corners yet.

8 Leveling Corner #2 Using the hose again, place a level notch on the second corner stake. Before cutting the notch, cross check with both the reference stake and the standard corner stake (corner #1). When all results seem within tolerance (perhaps a half inch), cut the notch.
9 Making it square. This will take three helpers and yourself. While you and your senior helper hold the two marked sections of the cord and stretch out a rough square, your other two helpers will stretch a diagonal cord. They tightly hold their measurements as they switch to see if the other diagonal is the same. Unless you are highly lucky, the second measurement will be different. You and your senior partner, keeping the perimeter cord tight, move to a position such that half the error on the diagonal is where you take your new position. The junior helper holds tightly now to the new measurement, and then the two junior partners measure the other diagonal. They will run back and forth doing this adjusting until the two diagonals are equal. (And modern surveyor rodmen think they run around a lot!) Hammer in your #3 and #4 corner stakes. "Level" them with the hose, and do lots of cross-checking to make sure all it level.

Amazingly, the edges (and thus the sides) of your pyramid will each face a cardinal direction.

10 Finding the center This is, of course, where the two diagonals cross.
11 Finding slopes of the sides and of the edges. While this is incorporated in Euclidean geometry, it is highly likely that Prof. Euclid's predecessors knew that little pyramids had the same angles as big ones. Afterall, it is downright conspicuous! So get 12 pieces of 2"x2" that are all of identical length, and make four triangles out of them. The lumberyard people have metal brackets that you can hammer directly onto the corners to hold them in place. Next stand your triangle up so the you have a pyramid with a square base, and the apex is up there somewhere. Fasten it all together.
12 Defining your big pyramid Move your wooden pyramid to the field and set it at each of the corners. Then hammer in a pipe in just the right angle that is parallel to the edge of the wooden pyramid. Make sure that the pipe crosses the corner stake at the notch.
13 Step back and imagine the lines defined by those pipes going up and up and finally meeting at the top of the pyramid. Amazing! One of the Seven Wonders of the Ancient World!

| Site's Table of Contents | Site's Index |

1	What you want	You are not actually going to level a field, but rather you are going to hammer four stakes into the ground in just the right places (where the corners of your pyramid will be), and just far enough into the ground so that the tops of the stakes are all level with each other. If you were Pharoah, you would order your workers to scrape off enough of the bed rock here and there so that the whole surface is level. That would take a week or two. The picture here shows your ultimate goal. Following are steps needed to arrive at that goal.
2	Finding the compass directions	Plant the tall pole upright near the center of your level field. On a sunny morning, you will see the pole's casting a shaddow. Set a rock or brick or hammer in a tent stake where the shadow of the top of the pole is cast.
2	Finding the compass directions	Every couple of hours hammer in other stakes wherever the pole's top casts its shaddow. The stakes should form a straight line in one of the two cardinal directions, namely, E-W or N-S. But which one? And this will happen on only two days of the year - March 21 and September 22. Neat!
3	Placing the pyramid's first side.	Now "draw" a line that is parallel to your solar shadow line. This new line will be one of the edges of your pyramid. (If you were really smart, you would have placed your pole so that its shadow points would form the pyramid's edge line! If you weren't that smart, you could either replant the pole for another sunny day, or be very careful to be sure that your new line is exactly parallel to the shaddow line. How to be sure: extend the shadow line a long distance in both directions so that it will be at least as long as the intended length of the pyramid's side. Using the string, cut two identical lengths by folding the string in half and cutting it. place one piece on the shadow line and the other where you think the pyramid's edge might be. The diagonals between the two strings should be the same (use more string to measure the diagonals. Once you have have adjusted the edge string so that both diagonals are in fact equal, hammer in a stake at each end of the edge string.
4	Planting the REFERENCE stake.	Step back a bit from your edge line and scan the scene. You can now envision where the four corners of your pyramid will be. (If you are planning a full-scale model, those corners are going to be about 800 feet apart - probably enclosing an area larger than you have available!) Because the "lay of the land" is not likely to be flat or "level", you must choose which corner is on the highest ground. (As Pharoah, you would have your workers scoop it out so that the whole field is level with the lowest place!) Go to that approximate highest corner and drive in a wooden stake so that it is very firmly set in place, and then saw a notch in it about 6 inches above the ground. (The notch is a weather-resistant mark! All other points will be made level to this notch. It is mandatory that this notch's position not be lost. It is your standard, your "control," or "reference" stake.)
5	Establishing the first corner.	Go to your edge line and hammer in a stake firmly where you want one of your pyramid's corners to be. This will be, as you will see, your "standard corner."
6	Levelling your first or "standard" corner.	You want to cut a notch in the corner stake that is exactly level with that of the reference stake. What the ancient Egyptians did was dig a trench between the two stakes and then fill it with water. The water would "seek its own level" and they would measure up from the water level. You can imagine your digging such a ditch with shovels (no bulldozers then!), and that would appear to be sweaty work. But look how easy you would have it: the ancients dug their ditch in bedrock! (No dynamite then either!) So don't start digging because in your supplies you have a long length of garden hose. Fill it with water, and hold both ends over your heads as you and your partner separate. Your partner goes to stand by the reference stake and you stand by the standard corner stake. Slowly you lower your two ends of the hose (hand signals are in order for the long distances involved!) When the partner says that that end is even with the notch, you raise your end until the partner signals that water is flowing out that end. You look to see where the water is in your end of the hose. Shining a bright flashlight through the hose might help. Because the levels at both ends will be identical, draw a line on your corner stake. Then weatherproof it by notching it. Yes, you will have a few mishaps in this until you get the hang of it. So have a few buckets of water and a funnel handy to reload the hose. (Having a stopper for each end is also a good idea so you can lay the hose down without losing the water.)
7	Roughing in the perimeter.	You want to get a piece of strong string that is equal to the perimeter of your pyramid. (If you are going for a full-scale model, then the string should be about 1 km long (about 2/3 mile)!) You want to divide this into four exactly equal lengths. (Because string or rope is a bit stretchy, Pharoah would order that you use chain!) So once you have all this string or cord or rope (or chain!) stretched out and untangled. Tie the two ends together.
		Stretch the coubled cord out in a straight line. It might help to loop the knotted end over your corner stake, and then send your partner on a long hike to stretch it out.
		Now have your partner double back with the far end and loop that over the corner stake along with the knotted end. Once back, you take a wooden stake and place it in the far loops and pull it tight as you set the stake on the edge line and hammer it in firmly. At this time use marker to color the cord strands at both stakes. This will divide the cord into four equal lengths. Don't be stingy with your marker.
		Remove one loop from each stake and, using two additional helpers, stretch it out into what will approximately be the baselines of the pyramid. See how the red marks indicate the future corners? But you don't have it perfectly square yet so don't hammer in the other two corners yet.
8	Leveling Corner #2	Using the hose again, place a level notch on the second corner stake. Before cutting the notch, cross check with both the reference stake and the standard corner stake (corner #1). When all results seem within tolerance (perhaps a half inch), cut the notch.
9	Making it square.	This will take three helpers and yourself. While you and your senior helper hold the two marked sections of the cord and stretch out a rough square, your other two helpers will stretch a diagonal cord. They tightly hold their measurements as they switch to see if the other diagonal is the same. Unless you are highly lucky, the second measurement will be different. You and your senior partner, keeping the perimeter cord tight, move to a position such that half the error on the diagonal is where you take your new position. The junior helper holds tightly now to the new measurement, and then the two junior partners measure the other diagonal. They will run back and forth doing this adjusting until the two diagonals are equal. (And modern surveyor rodmen think they run around a lot!) Hammer in your #3 and #4 corner stakes. "Level" them with the hose, and do lots of cross-checking to make sure all it level.
9	Making it square.	Amazingly, the edges (and thus the sides) of your pyramid will each face a cardinal direction.
10	Finding the center	This is, of course, where the two diagonals cross.
11	Finding slopes of the sides and of the edges.	While this is incorporated in Euclidean geometry, it is highly likely that Prof. Euclid's predecessors knew that little pyramids had the same angles as big ones. Afterall, it is downright conspicuous! So get 12 pieces of 2"x2" that are all of identical length, and make four triangles out of them. The lumberyard people have metal brackets that you can hammer directly onto the corners to hold them in place. Next stand your triangle up so the you have a pyramid with a square base, and the apex is up there somewhere. Fasten it all together.
12	Defining your big pyramid	Move your wooden pyramid to the field and set it at each of the corners. Then hammer in a pipe in just the right angle that is parallel to the edge of the wooden pyramid. Make sure that the pipe crosses the corner stake at the notch.
13	Step back and imagine the lines defined by those pipes going up and up and finally meeting at the top of the pyramid. Amazing! One of the Seven Wonders of the Ancient World!