The height of the pyramid of Cheops

[Ad VanderVen]

On the site https://www.wisfaq.nl/show3archive.asp?id=2537&j=2002 it is said that the ratio of the height of the Cheops pyramid (##a##) to the length of one side of the base (##a + b##) is equal to $$ \frac{2 \, (a + b) / Pi}{ a + b}= 2 / Pi = 0.6366197722$$ But if you use the golden ratio as a criterion, you get $$\frac{a}{b} = \frac{a + b}{a},$$ where ##a > b## and it follows, that $$a = (\frac{1}{2} \sqrt{5} +\frac{1}{2} ) b$$ and from that it follows that
$$\frac{a}{ a + b} = \frac{1 + \sqrt{5}}{ 3+\sqrt {5}} = 0.6180339887.$$
Now the values ## 0.6366197722## and ##0.6180339887## are very similar. How can you be sure that the first calculation is correct?

 

[Borek]

Pure numerology. If you take any set of numbers and try hard enough, you will find ways to combine them into something resembling any number you can think of.

It doesn't mean anything, it is just a random coincidence.

 

[Rive]

How can you be sure that the first calculation is correct?

The issue starts with measurement. According to the referred page:

de hoogte van deze piramide 232,52 Hebreeuwse el is.

It translates as 'the height of this pyramid is 232.52 Hebrew ell. '

Regarding the unit of measure the only thing sure is that it's not accurate.
Regarding the accuracy of measurements of that age, it's even worse.
Two digit is just out of question.

So to have any calculation what can be referred as 'slightly correct' ( ) it would be a basic courtesy to add all the error calculations and ranges instead of three- or six digit values

When you have just the multiple digit values for such a topic, you can be sure it's just pure speculation.

Ps.: honestly, my humble assessment is, that within the relevant accuracy both 0.6366 and 0.6180 is as good as 'a bit over half'.

 

[gmax137]

I think this tells us how seriously the Egyptians took the dimensions and angles.

 

[Ad VanderVen]

Today we could measure exactly how large the ratio really is. Since it is a ratio, we do not need to know the exact height of the square base. Any horizontal section of the pyramid can be used as a base and the height must then be measured from that base. I wonder if such a measurement has been done before. If you have an exact replica of the pyramid, you can determine the ratio quite accurately. I wonder if there is such an exact three-dimensional replica available on the internet, something like is shown in the attached files.

 

[Vanadium 50]

I think this tells us how seriously the Egyptians took the dimensions and angles.

This is very interesting. Did you know that all future pyramids used the angles of the top part?

 

[gmax137]

all future pyramids used the angles of the top part

Is that true? Interesting. Maybe someone found an error and recalculated the golden ratio?

 

[russ_watters]

On the site https://www.wisfaq.nl/show3archive.asp?id=2537&j=2002 it is said that the ratio of the height of the Cheops pyramid (##a##) to the length of one side of the base (##a + b##) is equal to $$ \frac{2 \, (a + b) / Pi}{ a + b}= 2 / Pi = 0.6366197722$$ But if you use the golden ratio as a criterion, you get $$\frac{a}{b} = \frac{a + b}{a},$$ where ##a > b## and it follows, that $$a = (\frac{1}{2} \sqrt{5} +\frac{1}{2} ) b$$ and from that it follows that
$$\frac{a}{ a + b} = \frac{1 + \sqrt{5}}{ 3+\sqrt {5}} = 0.6180339887.$$
Now the values ## 0.6366197722## and ##0.6180339887## are very similar. How can you be sure that the first calculation is correct?

The first calculation is just 2/pi, so it is what it is. The a+b stuff is just a decoration that isn't really part of the calculation. You could just as easily throw in the distance to the moon divided by the distance to the moon.

The rest I can't make heads or tales of in part because the website appears to be in Dutch and won't translate, but the numbers/equations don't seem to be what you are describing. Maybe I'm missing something, but they seem to just contain random combinations of numbers pulled out of the air. I'd call that worse than numerology; it's not even a coincidence, it's just randomly generated gibberish.

[edit] Oh; that's the "golden ratio". That explains my confusion: I don't see much value in the golden ratio. It seems itself to be numerology that has been picked-up as if it had significance, and then on which other numerology is based. Looks meaningless to me.

 

[Helios]

You can look up the Angle of Repose for
Granite 35–40°
Gravel (crushed stone) 45°
that is about the pyramid slope.

 

[Vanadium 50]

Maybe someone found an error and recalculated the golden ratio?

I think they found the error when the new stone started sliding down the slopes. Hope nobody was beheaded over it.

I looked into visiting it, but it is not easy. For a long time it was part of a military base's excluded region, then it was under repair, and now there is Covid.

 

[gmax137]

I should have included "/sarcasm" at the end of my post about recalculating Φ.

 

[Vanadium 50]

recalculating Φ.

It wasn't so easy back in the day. Lessee...LXXX + XX is C, and carry the X...

 

[Klystron]

Which golden ratio shall we use?

I made a series of space / seascapes on platforms cut to an eye pleasing 'golden' ratio of the length to the width as line segments. Continued this theme well into the undercoating color selections. Used reciprocals where appropriate and space filling.



For the last word on pyramidology consult Umberto Eco's "Foucault's Pendulum". There should be a Dutch version; original in Italian. Cheops pyramid specifically mentioned during Dr. Aglie's lecture on measurement.

 

[Ad VanderVen]

For the sake of clarity, I hereby repeat the two statements.

We have a regular square pyramid with a 4-sided base. If the height of the pyramid is related to the length of each base side according to the Golden Section then the ratio of the height of the pyramid to the length of a base side is equal to ##\frac{\sqrt{5} +1}{ \sqrt{5} +3}##.

We have a regular square pyramid with a 4-sided base. If the height of the pyramid is related to the length of each base side, according to the rule, that the ratio of the height of the pyramid to the length of one base side is equal to the ratio of the radius of the circle, which has a circumference which is equal to the sum of the lengths of the four base sides, to the length of a base side, then that ratio is equal to ##\frac{2}{\pi}##.

 

[russ_watters]

For the sake of clarity, I hereby repeat the two statements.

We have a regular square pyramid with a 4-sided base. If the height of the pyramid is related to the length of each base side according to the Golden Section then the ratio of the height of the pyramid to the length of a base side is equal to ##\frac{\sqrt{5} +1}{ \sqrt{5} +3}##.

We have a regular square pyramid with a 4-sided base. If the height of the pyramid is related to the length of each base side, according to the rule, that the ratio of the height of the pyramid to the length of one base side is equal to the ratio of the radius of the circle, which has a circumference which is equal to the sum of the lengths of the four base sides, to the length of a base side, then that ratio is equal to ##\frac{2}{\pi}##.

Ok. Your OP asked "How can you be sure that the first calculation[now the second calculation] is correct?"

Answer: I typed 2/##pi## into a calculator and got 0.6366197... which matches what you wrote in the OP. It's correct. Though strictly speaking, the answer to "how?" is "by verifying with a calculator."

Now what? Is that really what you wanted to ask?

 

[Ad VanderVen]

russ_watters

You wrote: "Now what? Is that really what you wanted to ask?" Of course not. My goal is to find out which of the two statements apply to the Pyramid of Cheops and maybe even another statement.

 

[Rive]

which of the two statements apply to the Pyramid of Cheops

They had some plans. They tried to build it according to that plan, but they had only limited capability. Then there was some thousand years and several rounds of recycling of raw materials, so at the end we can't even tell what was the original size.
What's your aim?
- guess the original plans/intentions: then the only way to prove anything would be looking for the build instructions themselves, since the building itself is not accurate enough to be a proof.
- check on the end result: we know that within a certain accuracy it's not deterministic anymore, so any numerology based on this is just pointless.

@russ_watters is right. You seems not know what's it you are after - and how to get that wrapped up.

 

[Vanadium 50]

A) You have not shown that either number (2/π or 1/Φ) has anything whatsoever to do with the Great Pyramid. Maybe it's buried somewhere in the Dutch webpage, but random Dutch webpages are hardly authoritative sources.

B) When you sweep away the mathematical brush - usually a good sign that someone is doing numerology - what you discover is [itex]\pi \approx \sqrt{5} + 1[/itex]. This is a substantially worse approximation than even 22/7.

 

[russ_watters]

You wrote: "Now what? Is that really what you wanted to ask?" Of course not. My goal is to find out which of the two statements apply to the Pyramid of Cheops...

The Wikipedia article on the Golden Ratio addresses this. It's close to both, but closer to the pi-based approximation (steeper than both). But as @Rive says, this doesn't get us into the mind of the designers - and there are other theories on the design that are considered more plausible. Ultimately it is impossible to know for sure without the original architectural drawings.

...and maybe even another statement.

Well that's provocative. So I repeat: now what? You might as well just stop treading water and tell us what you are after here.

 

[russ_watters]

B) When you sweep away the mathematical brush - usually a good sign that someone is doing numerology - what you discover is [itex]\pi \approx \sqrt{5} + 1[/itex]. This is a substantially worse approximation than even 22/7.

I'm trying hard to avoid starting a debate over whether this is numerology or historical conspiracy theory...

 

[Ad VanderVen]

A third statement would be:

In a regular square pyramid, each base edge and apex form a triangle called a lateral plane. If the lateral plane is an equilateral triangle, the height of the pyramid relative to the length of a baseline is equal to ##\frac{1}{2} \sqrt{2}##.

I don't understand all the reactions that well. It seems simple to me. The Pyramid of Cheops really does exist and you can measure everything directly and then see which of the three given rules the measurements meet best. That has nothing to do with numerology. That is a matter of measuring.

 

[Rive]

The Wikipedia article on the Golden Ratio...

Quite a good one, especially with all those references regarding both math and history.

 

[russ_watters]

It seems simple to me. The Pyramid of Cheops really does exist and you can measure everything directly and then see which of the three given rules the measurements meet best. That has nothing to do with numerology. That is a matter of measuring.

I agree it's not really numerology. But it's not just a matter of measuring either. What you really seem to be after (but for some strange reason don't want to say) is trying to figure out what the intent of the designers was...which may include an element of historical conspiracy theory.

I don't understand all the reactions that well.

The reactions are because you are being coy.

 

[Rive]

It seems simple to me. The Pyramid of Cheops really does exist and you can measure everything directly and then see which of the three given rules the measurements meet best.

Not that simple. The measurements and supposed original dimensions will give a range only, and for any range there will be plenty of 'rules' possible. Whichever is 'closest' means no proof for any of them being correct.
To have any reasonable pick you need a pretty complex analysis, mostly about historical sources: with the dimensions of the actual (ruined) state of that 'pile of rock' having only a limited value as supporting fact.

If you are after the math of ruined piles of rocks, then picking some hills or slopes in a national park would be a far more meaningful choice.

 

[Vanadium 50]

What you really seem to be after (but for some strange reason don't want to say) is trying to figure out what the intent of the designers was...which may include an element of historical conspiracy theory.

I agree that it does not bode well that the OP is not telling us what he's getting at. But...

The height of the Great Pyramid depends on two things: a) scale-independent quantities, and b) scale. I hope there's no argument there. When discussing (a), there is exactly one degree of freedom, after which all the others follow. We can take that as the slope. So the question "why is the pyramid as tall as it is relative to its base" is the exact same question as "why is the slope what it is?"

We know the answer to that question. The slope is as large as they could make it without the pyramid falling down. We know that because they used to be steeper, but the Bent Pyramid of Sneferu demonstrated that that doesn't work at this scale. i..e. the height is as tall as they could make it. End of story.

 

[Rive]

...the height is as tall as they could make it.

Well, I think it's a bit less than that: they had settled with a working ratio and that's all. I don't think they had the ability or care to get experimental proof that if a bit steeper would also work or not.

I think the way they got the numbers has historical value as history of engineering/math and quite possibly: as history of beliefs. But then it should not be about (our) math but (mostly) about historical sources: otherwise its worth would be the same for any pile of rock.

 

[Vanadium 50]

I don't think they had the ability or care to get experimental proof that if a bit steeper would also work or not.

They tried steeper. It did not end well.

 

[russ_watters]

So the question "why is the pyramid as tall as it is relative to its base" is the exact same question as "why is the slope what it is?"

We know the answer to that question. The slope is as large as they could make it without the pyramid falling down. We know that because they used to be steeper, but the Bent Pyramid of Sneferu demonstrated that that doesn't work at this scale. i..e. the height is as tall as they could make it. End of story.

That's some good historical insight, but I suspect it doesn't end the story for the OP. It doesn't really explain where the angle came from. When they decided that it needed to get shallower, someone had to stick their neck out to pick the new angle.

Did you know that all future pyramids used the angles of the top part?

Do you mean after the Great pyramid, Because the wiki articles say:

• Bent Pyramid:
  • Bottom: 54 deg, 27 min
  • Top: 43 deg, 22 min
• Great Pyramid: 51 deg, 52min

It kinda seems like for the Bent Pyramid, they took a swag and decided they needed to greatly decrease the slope for the top part.

 

[Rive]

They tried steeper. It did not end well.

As far as I know they had only one which collapsed, and made them fell back to 43 degree from 54 degree immediately. Later they started to go for steeper again, but I'm not sure if those can be considered as 'experiments'.

 

[Vanadium 50]

The Giza pyramids have slopes of 42, 43.3 and 41.6 degrees. (Slope taken from the corners to the top).

The difference in height between 42 and 43.3 degrees is only about 3% or 7 meters. A guess as to construction precision can be obtained by noticing the base of Menkaure is not quite square: the long leg is 2.3% longer than the short leg. (And the base is the easy direction)

 

[Vanadium 50]

omeone had to stick their neck out to pick the new angle.

Literally.

 

[russ_watters]

The Giza pyramids have slopes of 42, 43.3 and 41.6 degrees. (Slope taken from the corners to the top).

You clearly have looked into this more than I have; all I have is what the wiki article says:

Slope	51°52'±2'

https://en.wikipedia.org/wiki/Great_Pyramid_of_Giza
And:

One Egyptian pyramid that is close to a "golden pyramid" is the Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu). Its slope of 51° 52' is close to the "golden" pyramid inclination of 51° 50' – and even closer to the π-based pyramid inclination of 51° 51'. However, several other mathematical theories of the shape of the great pyramid, based on rational slopes, have been found to be both more accurate and more plausible explanations for the 51° 52' slope.

https://en.wikipedia.org/wiki/Golden_ratio

Can you explain the discrepancy?

Literally.

You did what I saw there.

 

[Vanadium 50]

Slope is "rise over run", right? The rise is well defined, but the run is not. I took them from the corners. Wikipedia takes it from the midpoint of the sides, presumably.

 

[Klystron]

From the wiki article on "golden ratios" I cited in an earlier post also referred to by other posters:

Eric Temple Bell, mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem, nor any way to reason about irrationals such as π or φ.[99]

Bell, a noted mathematician and historian of mathematics, debunked many popular misconceptions surviving into the 20^th Century and, apparently, well into the 21^st.

 

[russ_watters]

Slope is "rise over run", right? The rise is well defined, but the run is not. I took them from the corners.

Oh...that's an interesting choice.