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The Great Pyramid and Irrational Numbers

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The Great Pyramid

The Great Pyramid of Giza is the biggest of the pyramids of Egypt. Built in about 2570 BC as a tomb for the Pharaoh Khufu, it is one of the largest constructions in the world before the modern era1, and was considered one the Seven Wonders of the Ancient World. It is located about 12km southwest of Cairo, the capital of Egypt. There are many claims made about the Great Pyramid, ranging from the mundane (the Pharaoh's remains are still lying inside the pyramid in a secret chamber) to the insane (it was originally built the other way up, with the point downwards and the square base on top as a landing place for alien spaceships).

One of the reasonable-sounding theories is that the dimensions of the pyramid are based on the irrational number π, also known as Pi, or the irrational number φ, also known as Phi or the Golden Ratio. Both of these are often quoted as fact.

So as not to keep you in suspense, we'll say straight-away that these appear not to be true. Although the measurements of the pyramid can be manipulated to produce very close approximations to these numbers, it appears to be a coincidence.

This Entry outlines the arguments for and against these claims and presents a much more likely explanation for the design of the pyramid, based on the simple way the ancient Egyptians expressed the concept of slope.

The Irrational Numbers π and φ

The term irrational here does not mean 'crazy'. Rather, it means 'not a ratio'. Numbers such a half, three quarters etc can be written as a ratio or fraction - one whole number divided by another whole number. For example, the number 1.234234234234... with a repeating 234 can be written as 137/111. Some numbers, however, can not be written as a ratio of whole numbers. If we draw a square with the sides 1 unit in length, then the diagonal will be √2, the square root of 2 - a number approximately equal to 1.4142. This cannot be written as a ratio of whole numbers (a fact proven in about 500 BC in ancient Greece), so it is called 'irrational'.

Another well known irrational number is π (spelled Pi in English and pronounced 'pie'), the ratio of any circle's circumference to its diameter. Pi is defined as a ratio, but it is not a ratio of whole numbers. If the diameter is a whole number of centimetres, for example, then the circumference will not be a whole number of centimetres. So π is also irrational.

The number π was extensively studied by the ancient Greeks although they didn't refer to it by that name. Archimedes in about 250 BC proved that it was somewhere between 223/71 and 22/7. Although it is closer to the former figure, the latter is easier to remember, so 22/7 is often used as a handy way of remembering Pi. Modern mathematics makes frequent use of π, not only in the study of circles, cylinders and spheres, but also in odd places such as complex integration and the distribution of prime numbers.

The irrational number φ (spelled Phi in English and pronounced to rhyme with pie and bye) was also known to the ancient Greeks, and again, they didn't know it by that name. They divided a line into what they called 'extreme and mean ratio'. This means that they took a line of say 1 metre long, and divided it into two unequal lengths of 0.618m and 0.382m. We find that the length of the full line, 1, divided by the length of the long section, 0.618, is equal to the length of the long section divided by the length of the short section, 0.382. This works out as a ratio of (1+√5)/2 which has a value of approximately 1.618. This number has been credited in recent years with having a pleasing look to the eye - a rectangle with sides of 1 and 1.618 is claimed by some to be the most pleasing looking rectangle to the human eye, although others find it a bit short. The number has been given the name φ (phi) and the grandiose name of the 'Golden Ratio'. It is also claimed that ancient buildings were designed using the golden ratio because it gave a pleasing look, even if the designers were not aware of the number.

The Claims about the Great Pyramid

Theories linking the Great Pyramid to π date back to the 19th Century. The pyramid has a square base with four sides almost exactly the same length. Simply put, the claim is that twice the length of the base of any side divided by the height of the pyramid is π. Since there are no circles involved in the pyramid, if this is more than coincidence it must have been done for some mystical reason.

The claim linking the Great Pyramid to φ is more recent - from the late 20th Century at the earliest. The slope length of the pyramid, that is, the distance from the top of the pyramid to the middle of the base of any of the sides, divided by half the length of the base is equal to φ the golden ratio.

Before looking at these claims in detail, let's look at what the ancient Egyptians knew about maths.

Egyptian Mathematics

There's a lot of measurement involved in Egyptian mathematics so it is worth mentioning their units of measurement. The basic unit was the cubit - this was used all over the ancient world but not all cubits were the same size. The Egyptian one was about 524 millimetres (20.6 inches). It was divided into seven 'palms' so a palm was 74.9mm (2.95in) and this was further divided into four 'digits', a digit thus being about 18.7mm (0.74in). There were therefore 28 digits in a cubit.

Most of what we know about ancient Egyptian mathematics comes from a document called the Rhind Papyrus. This was bought by a Scottish antiquarian Alexander Henry Rhind in Luxor in 1858. Most of it is now in the British Museum in London. The document was written in about 1550 BC, but is a copy of an older document from a few hundred years earlier. Even so, the contents are from more than seven centuries after the Great Pyramid was built. It contains the equivalent of 30 A4 pages of mathematics, written in the hieratic script2.

Rather than discussing abstract theory, it states a number of problems and then presents methods for solving them. It deals with a lot of basic arithmetic including fractions, methods for calculating areas and volumes, and techniques that would be helpful in measuring out areas of land - this was a major task in ancient Egypt as the annual flooding of the Nile washed away many of the boundaries between fields and they had to be laid out again after the floods.

The Rhind Papyrus makes it clear that the ancient Egyptians of the time knew about the number π. One problem includes a method for calculating the volume of a cylinder, phrased in terms of the volume of a cylindrical granary. The formula assumes a value of π = 256/81. This works out as about 3.1605, within 0.6% of the correct value. There was no mystical significance attributed to π in the Rhind Papyrus. It was just a number used to calculate lengths and areas of circles. The ancient Egyptians do not seem to have been aware of the more accurate value of 22/7 for Pi.

There's no mention of φ or any golden ratio in the Rhind Papyrus, but the ancient Egyptians did know about square roots so they could have calculated the value (1+√5)/2 if they had wanted to.

Crucially, the Rhind Papyrus has a section on how to express the slopes of pyramids. Rather than talking about angles, they measured a distance across and a distance up. We often talk about a steep road being '1 in 10'. This means that the road rises by 1 metre vertically for every 10 metres you go horizontally. The smaller the number after the 'in', the steeper the road. The Egyptians used a very similar system - they would say a pyramid had a seked (slope) of 30 digits. This meant that it rose by one cubit for a horizontal distance of 30 digits. Again, the smaller the number of digits, the steeper the slope. This method of expressing slope meant that the mason cutting the facing blocks of the pyramid didn't need any device for measuring angles - he just needed his cubit rod, a wooden stick one cubit in length with palms and digits marked on it.

Dimensions of the Pyramid

In the 19th Century, the English archaeologist William Flinders Petrie did a thorough study of the Great Pyramid. He used a surveyor's measuring tape that was accurate to within 0.1 of an inch. Because people were interested in the theory about π and the pyramid, he paid particular attention to the length of the base and the height.

This is not as easy as it sounds - the pyramid originally had a casing of white limestone that was smooth on the outside. When Cairo was destroyed by an earthquake in the 14th Century, the residents saw the pyramid as a handy source of cut stone for rebuilding the city. The outer limestone layer was almost completely stripped off the pyramid, leaving the present rough surface. This clearly made the pyramid a bit smaller. Quite a large amount was removed from the top, lowering the pyramid by about eight metres. By a careful examination of all the remaining casing stones, some of them below modern ground level, Petrie was able to make an estimate of the original base size, accurate to within 0.7 of an inch. From the slope of the pieces he found, he was also able to estimate the original height, accurate to within 7 inches.

Here are Petrie's estimates:

 Original Length
Angle of slope of sides51°52'

The lengths in inches are his values, taken from The Pyramids and Temples of Gizeh (1883). The lengths in metres are a modern conversion.

Calculating Pi and Phi

Taking Petrie's values for base and height, it is easy to double the base and divide by the height. This gives us the number 3.140166. This is close to the actual value of π (3.141592654) - in fact it is within 0.5 parts per thousand. That's close enough to make us at least consider whether it might be deliberate.

Of course we don't know what value the builders of the pyramids used for pi, but it seems likely that they would have used the value in the Rhind Papyrus: 256/81. Although the contents of the papyrus were written seven centuries after the pyramid was built, Egyptian culture was extremely stable and new ideas took a long time to take hold, so it is likely the pyramid builders used the same value. Comparing twice the base divided by the height with 'Egyptian Pi', we find it is still fairly accurate - within 6 parts in a thousand.

We can easily calculate the slope length using basic geometry. The slope length squared is equal to the height squared plus half the base length squared. Making the calculation, we find that the slope length equals 7,343.2 inches and this divided by half the base length gives us 1.619448. This is very close to the value of Phi, 1.6180339 - in fact it is within 0.9 parts per thousand.

So the pyramid does contain very accurate approximations to Pi and Phi in its dimensions, if we do a bit of juggling of the figures.

Is This Realistic?

Just because we can find these figures there, does it mean they were intended?

  • The Egyptians knew about Pi although they thought of it as a fraction. Its purpose was to calculate circle lengths and areas. Why build it into the dimensions of the pyramid in this peculiar way?

  • There's no evidence that the Egyptians had ever heard of the number Phi. The studies showing that Phi produces pleasingly shaped rectangles are very questionable - there was only one small study which is quoted again and again on the internet. Claims that other famous buildings such as the Parthenon in Athens are based on Phi are not just questionable, they're completely untrue. So why should the Great Pyramid use this number that the designers don't seem to have been aware of?

  • Most importantly, if either of these constants was important to the pyramid builders, why did they include them in only one of the hundreds of pyramids that were built over the centuries? You have to remember that the Great Pyramid was one of many pyramids, all slightly different. Each one improved on the previous one up as far as the Great Pyramid and its successor, the Pyramid of Khafre. It appears that the standards dropped after that, with smaller pyramids and less skilled masonry. But if something as simple as including Pi in the dimensions was only used once, it suggests that it was not part of the design and just came about by coincidence.

The Slopes of Pyramids

So why did the pyramid builders pick on 51°52' as the slope of the Great Pyramid? We don't know for sure but it appears to have been arrived at by experiment by building a few different pyramids.

  • Egyptian pharaohs were originally buried under a type of tomb called a mastaba. This was a rectangular block which was low compared with its length and breadth. Some of these had rooms inside them, others were solid.

  • The first Egyptian pyramid was created when some clever architect3 decided to stack smaller mastabas on top of larger ones, producing the Step Pyramid of the Pharaoh Djoser in about 2650 BC. This has seven levels, each one rectangular in plan and getting smaller towards the top. It is thought that the stepped shape suggested a 'stairway to heaven' for the soul of the departed pharaoh.

  • A few pharaohs after Djoser also built step pyramids, but these have not survived. The pharaoh Sneferu (also known as Snefru, Snofru, Seneferu or the Greek name Soris) built three major pyramids, trying to perfect the design. Sneferu's first pyramid is located at Meidum, about 75km south of Cairo. It started out as a step pyramid, but later he decided to convert it to the shape we now call a 'true pyramid', a square base with four flat, triangular sides meeting at a sharp point at the top. This may have been intended to represent the rays of the sun shining down on the pharaoh's grave, since the Egyptian religion venerated the sun-god Re. (Many of the pyramids were given names by the ancient Egyptians that involve light, such as 'shining', 'star' etc.) This first 'true' pyramid appears to have had a slope of about 52° which in the Egyptian style was a seked of 21.9. Unfortunately, at some point the outer parts collapsed leaving some of the original step pyramid core, so that it now looks like a giant tower surrounded by rubble. This may have even happened before the construction was complete.

  • Whether the collapse happened at that point or centuries later, Sneferu did not use this pyramid for burials. He started work on a second pyramid, now known as the 'Bent Pyramid', situated in Dahshur, about 28km south of Cairo. This one was designed as a 'true pyramid' from the start - that is, one with four triangular sides - rather than as a step pyramid.

    The steeper the slope, the taller and more impressive the pyramid. Sneferu's second pyramid started out with a very steep slope - 54°31'13" which in Egyptian style is a seked of 19.96 digits, clearly intended to be 20. Unfortunately, a large, impressive pyramid also weighs a lot - when the pyramid construction had reached a height of 47m, the design was changed to a much shallower slope of 43°21', a seked of 29.7 digits, presumably to reduce the weight on the base. This gives the pyramid its distinctive 'bent' shape. This pyramid was also considered not suitable for the burial of the king.

  • Sneferu's final pyramid, and probably his final resting place, was very large - the third largest pyramid ever built in Egypt. It is today known as the Red Pyramid because the stone is red limestone. Originally it would have had an outer casing of smooth white limestone. The slope of this pyramid is 43°21', a seked of 29.7, the same as the top of the Bent Pyramid. This was a conservative and safe option and it paid off - the pyramid is still standing.

  • The next pyramid to be built was the Great Pyramid, built by Sneferu's son Khufu. The steep slope of a seked of 20 had been tried and found too steep for the foundations. Khufu's pyramid has a seked of 21.996, obviously intended to be 22. This was steeper than the stable Red Pyramid but not too steep to stand, like the Bent Pyramid. So the slope of 22 may have been entirely chosen to make the pyramid as large as possible without it collapsing under its own weight.

  • The next major pyramid to be built was the pyramid of Khafre, Khufu's second son. The designers dared to push the slope a bit steeper, as far as a seked of 21. They got away with it and achieved a pyramid almost as tall as the Great Pyramid but narrower, making it look more elegant.

The clear indication is that the pyramid builders tried to pick a slope based on a seked of a whole number of digits. The value they chose for the Great Pyramid was 22, not too steep but steep enough to make an impact.

A seked of 22 means that the height is 28/22 times half the base length. Twice the base divided by the height thus gives us 22×4/28 which is 22/7. As mentioned above, this fraction is a very good approximation to π known to the ancient Greeks but not to the even-more-ancient Egyptians. The presence of Pi in this incredible monument is thus a coincidence.

And the Golden Ratio? The 28/22 mentioned in the previous paragraph when squared give us 1.619.., a reasonable approximation to Phi, so this one turns out to be a coincidence too.

The pyramids are mystical objects, enshrining concepts such as resurrection and the light of the gods shining on humanity, but the mathematics used to build them is entirely practical.

1It was the tallest building in the world until 1300 AD - that's nearly 4,000 years.2A flowing script based on the hieroglyphic writing system.3In modern times this is often credited to Djoser's Chancellor and High Priest, Imhotep, but this claim is not found in any ancient document.

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