How did the Ancient Egyptians find this volume without Algebra?

In 1892/1893 the Russian archeologist Vladimir Golenishchev traveled to Thebes where he purchased a papyrus that would later be known as the “second most important document” to our understanding of ancient Egyptian mathematics or the Moscow Papyrus for short [1]. It contains 25 ancient Egyptian practice problems with their solutions and provides an invaluable demonstration of how ancient Egyptian mathematicians approached and solved problems, including problems of calculating payments, finding ratios for beer making, geometry, and (very) basic algebra. The most remarkable of all of these is problem number 14 concerning the volume of a truncated pyramid. From the papyrus it is clear that the ancient Egyptians knew the same formula we do for this volume, but one thing is unclear… how?
In modern mathematics we would find the volume of such a pyramid quite easily using algebra, but from every other source we have found it very much appears that the ancient Egyptians had no substantial knowledge of algebra. This seeming contradiction was first noticed by B.A Turaev in his 1917 paper [2] and expanded on in a 1929 paper by Gunn and Peet who provided a potential way the ancient Egyptians could have discovered this formula [4]. However, while Gunn and Peet found an elegant way to find this formula without the use of algebra (in the modern sense), their solution would require the ancient Egyptians to have known a version of what is called “Greek algebra” which is also not demonstrated in any source as pointed out by Vetter in his 1933 roast on the same topic [4]. There have been many more voices in this discussion over the years, including Kurt Vogel in 1930 and and Siegmund-Schultze in 2022 (who suggested the Egyptians used the same method as the Chinese mathematician Liu Hui), however none have provided a potential solution convincing enough to gain a consensus in the academic community [5, 6]. So the question still remains… how did they do it??
In this video we will explore this very question and even take a look at my own potential solution which I believe to be the most practical and least objectionable I have seen yet! But the question I have is, what do you think? Do you think my solution was doable by the ancient Egyptians? Do you think that is how they actually did it? Or do you have your own ideas?
You may learn, you may laugh, and if I’ve done my job you may even not cry. But no matter how you react, if our video makes your day better please remember to like and subscribe and tell your friends. Have a great day!
Sources
1. M. Clagett, Ancient Egyptian Science. 1989.
2. B. A. Turaev, “The Volume of the Truncated Pyramid in Egyptian Mathematics,”in Ancient Egypt (1917), 100-102.
3. Gunn, B., & Peet, T. E. (1929). Four Geometrical Problems from the Moscow Mathematical Papyrus. The Journal of Egyptian Archaeology, 15(3/4), 167-185.
4. Vetter, Q. (1933). Problem 14 of the Moscow Mathematical Papyrus. The Journal of Egyptian Archaeology, 19(1/2), 16-18.
5. Vogel, K. (1930). The Truncated Pyramid in Egyptian Mathematics. The Journal of Egyptian Archaeology, 16(3/4), 242-249. doi.org/10.2307/3854215
6. Siegmund-Schultze, Reinhard (2022). Another look at the two Egyptian pyramid volume ‘formulas’ of 1850 BCE, British Journal for the History of Mathematics. British Journal for the History of Mathematics. ISSN: 2637-5451. 37s 171 - 178. doi:10.1080/26375451.2022.2106061.
Intro Music
"Cambodian Odyssey" Kevin MacLeod (incompetech.com)
Licensed under Creative Commons: By Attribution 4.0 License
creativecommons.org/licenses/b...

[@alaingamache3908]

Every now and often I teach a unit on Egyptian Math to high school students. This demonstration is the best I have encountered as a POSSIBLE way to could have reasoned. It is also worth saying that numerous arguments by the Greeks were credited to be coming from the Egyptians, and that argument seems to fall exactly in line. Thank you!

[@imaginaryangle]

I spilled my drink at 0:06 🤣 This is awesome, I loved it! You picked a topic that is just naturally interesting and then topped it with really well produced visuals and non-stop entertainment. Good luck! 🕺

[@chillaxiommath]

Haha thanks! Sorry about your drink.

[@michaeledwards2251]

@@chillaxiommath By simply looking at a pyramid as a stack of slices, and shifting the slices to fit into a corner, it can be demonstrated 3 pyramids fill a cube. (This seems especially likely given the building of stacked pyramids ) Given the ancients were known to have determined the values of all the parts of a cubic, not needing imaginary numbers, all the dissections you demonstrated, would have been known. The assumption they didn't know algebra seems unlikely : today algebra is taught to primary school children. Likely it would have been known to an elite minority who kept their work hidden. (The way they used numbers and not the type of measurement, height, base width, top width, suggests they were aware of the general concepts of algebra. ) (Only a tiny amount of the total amount of excavatable material has been dug, and much of the easy to access material has been sold off to collectors who would have simply thrown nearly all of it away. (It was fashionable to collect material from Egypt in the 19th century. ) ) The scroll with the practice problems was an accidental find : the odds it would have been sold to some one who would recognise its significant is small. This suggests it was a standard exercise text for the children of scribes.

[@alphalunamare]

It went straight over my head because I hadn't yet tuned in to the diction. KZbin subtitles aren't too brilliant but I got most of it. I think this video is a real treasure :-)

[@gametimewitharyan6665]

Oh hello there imaginaryangle, I watch your videos, you too are an excellent maths communicator like him

[@imaginaryangle]

@@gametimewitharyan6665 Thank you! 😊

[@adityakhanna113]

Holy shit! Your aesthetic is so nice and I love your presentation style. Absolute KZbinr material waiting to explode

[@adityakhanna113]

I can't believe you went through the trouble of *carving* a pyramid!

[@chillaxiommath]

Thanks so much!

[@chillaxiommath]

@@adityakhanna113 And you have no idea how hard carving playdoh is! Not sponsored

[@hallfiry]

Should they have known that the volume doesn't change when you shift the "lid", then there's a shorter, very elegant solution: 1) Shift the lid (a²) diagonally, so one of its corners is above one of the corners of the bottom (b²) and its shadow is entirely outside b². 2) Above b² and below a² there are now obvious pyramids with height h and bases b² and a². 3) The two remaining bits are diagonal sixths of a cuboid with side length a, b and h. Since there are two of them, they are a third in total. 4) Profit.

[@alphalunamare]

Not being fussy but could you break your explanation down a little bit more? What is the lid for example?

[@LetsGetIntoItMedia]

Absolutely awesome video! I love the takeaway about big perspective shifts in math, and how that's what's taught in schools. Super fun and entertaining, insightful, and memorable! Your videos are exactly the format and delivery I'm going for too, so I really feel and appreciate the work that went in and the polish you achieved

[@chillaxiommath]

Thank you so much! From years of math classes and late nights on Wikipedia I’ve learned so many math facts that I view as epic math stories, and so our goal is to try and communicate these ideas just like that. As the epic stories they are. I wish you luck on your stories as well!

[@aiden_3c]

Came here from Grant's video and just finished watching your example, that was an absolutely amazing solution It seriously did just pop out

[@chillaxiommath]

Right?? I really wish I had time to explore the explanations in the literature, because most of them really do not pop out in the same way. They all do something like assume the Egyptians knew something called Greek algebra or that they used multiple truncated pyramids being dissected together. It’s a whole thing! This solution took a lot of trial and error to get to, but once I found it it just made so much sense!

[@MrSilversMathSheets]

Congratulations on getting an honorable mention. I remember reviewing this video and I liked it better than the other one. I didn’t think it would get a mention though.

[@elijahdschultz]

Very fun discussion. I will note that this formula generalizes for pyramids with top and bottom faces that are arbitrary 2D shapes. V = h*[A1 + (A1*A2)^(1/2) + A2]/3 where A1 and A2 are the upper and lower face areas, however computed. This was proven by Heron of Alexandria (an Egyptian by birth and possibly also by ethnicity). I use this version of the formula to estimate excavation volumes around buried structures in civil engineering, where the walls of the excavation must often slope to prevent collapse.

[@James639]

Great video! Loved the graphics!

[@joelsmith3473]

Criminally undersubscribed. I've subbed, look forward to more, and wish you success to get the view numbers you deserve.

[@MrTheblackopsdude]

With the quality of your videos, I thought you'd have several hundred thousand subscribers. The subtle humor is 💯

[@chrishelbling3879]

Gotta love any math video that insults my mom right off the bat. Your disection / reärrangement theory is brilliant.

[@Freytana]

Fantastic video, thank you for being enthusiastic about communicating math!

[@jeffreybernath6627]

Thanks for making this great video for SoME3, so I could learn about your channel! Most fun I've had with math since Matt Parker.

[@chillaxiommath]

Thank you so much for the kind words! You have no idea how much being named in the same sentence as Matt Parker means to me. This is better than my Dad saying he’s proud of me

[@AlexandreMacabies]

Hey, this is actually a very enticing and believable "proof", in my humble opinion. Replacing the tiny pyramid with the appropriate block is probably the most far-fetched step but not completely bonkers. Great video and actually moving the needle!

[@rongarza9488]

Yes, replacing the tiny pyramid with the appropriate block smacked of hocus pocus. I see the tiny pyramid as a scaled down version of the original problem. So then, its top half becomes a tinier pyramid. Wait, this sounds like Calculus!

[@iout]

​@@rongarza9488 It's not really a scaled down version of the original problem. As the video said, they already knew how to find the volume of a regular pyramid and that they could have believably figured out how to do so through trial and error. The real leap is the understanding that you could replace the pyramid with a prism of equivalent volume, but it's not wholly unbelievable either since the trial and error method already does this to begin with.

[@entcraft44]

Without doing some research, it can be very difficult to judge whether a concept would be far fetched or not to an ancient civilization. Knowledge expands, but thinking patterns shift.

[@zarblitz]

@@rongarza9488 It's the largest leap, to be sure, but I don't think it's unreasonable. In fact I believe that such a realization could be the natural consequence of approaching the problem just like this, with clay, as some ancient Egyptian mathematician or architect may have done. I could imagine them arriving at the step with all these rectangular prisms and this outlier pyramid, only to smash the clay pyramid out of frustration and realize they can simply alter its shape without changing the volume and that doing so would make the rest of the puzzle much simpler. Certainly seems plausible, at the very least.

[@marktube5732]

I’m so happy you’re back. I think you have the potential to become huge on KZbin

[@MrSilversMathSheets]

This is a nice video. It was funny and informative. I wonder if the dissection method was used to get the original full pyramid formula.

[@alphalunamare]

8:48 The Egyptians knew about rationals but not to the extent of their manipulation that we know today. They could conceive of 'parts' which when aggregated gave 'answers'. They'd say take a fifth then add a sixth and remove a fourth to arrive at a solution but they couldn't just say take seven 60th's because that level of manipulation was beyond them. That we can express their ways of doing things via algebra just makes us clever in figuring out whether they were correct or not, it has no baring on whether they were as bright as us or not or indeed how they arrived at their ways of calculation. All it really says is that our slick way of figuring things out via algebra means that we have forgotten the old ways of doing things. 12:00 Bravo! :-) I think you have explained exactly how they did it! I am well impressed :-)

[@BritishBeachcomber]

Ancient Egyptians did not know algebra (equations) but they were masters of geometry (measurement). They would realise that you can easily divide a truncated pyramid into easily manipulated rectangles and trapezoids by dissection.

[@neutronstarmerger]

Congrats on the (much deserved) honorable mention for SOME3!

[@__Bruh]

Plot twist: the Egyptians used calculus to find the frustum’s volume 💀 But seriously, accurately finding volumes without algebra as we know it, that’s impressive

[@dluxdoggdlux]

Thanks for this interesting video! Just a quick remark on your "Before they discovered the wheel" remark - I know that was meant to be a joke or offhand comment but this is incorrect. They knew about wheels. Egyptians primarily used the Nile for transport (and thus are "sea faring" society); wheels in the desert make little sense. As many historians have observed, the issue isn't about "inventing the wheel" - the main technological challenge was to create a working axle.

[@dirkroosendaal2254]

this video is super interesting. I have never heard anything like this story!

[@learn_cpp]

I agree that the method you presented is very likely to be the way this formula was found, but thought it's worth mentioning if they had some form of Cavalieri's principle and very basic geometry (no algebra required), then it also naturally drops out as the sum of a pyramid with base a^2, an inverted pyramid of base b^2, a cuboid of base ab (and height h). minus a pyramid of base ab, and minus an inverted pyramid of base ab. Although at first glance this decomposition seems impossible to stumble across without algebra, it actually arises naturally from doing an exercise where we try constructing a square which matches the cross-section of a layer of the frustum at a specific given height using rudimentary geometry.

[@learn_cpp]

Also by the same method you present in the video, the Egyptians would have known a^3 - b^3 = (a - b)(a^2 + ab + b^2), you can physically remove a smaller cube from the corner of a larger cube then split the shape that remains into a cube of dimension (a-b) x (a-b) x (a-b), and 3 cuboids of dimension (a-b) x (a-b) x b, and 3 cuboids of dimension (a-b) x b x b, then rearrange them just as in your video (or better yet you can split the shape directly into the cuboids (a-b) x a x a, and (a-b) x a x b, and (a-b) x b x b). Also it is trivially implied by the frustum formula: Just consider two (similar) square based pyramids of base a x a and height a, and base b x b and height b. The difference in volumes the Egyptians would know was both a^3 / 3 - b^3 / 3 and by the formula in the papyrus (a-b) (a^2 + ab + b^2) / 3.

[@agnelomascarenhas8990]

Excellent presentation! The ⅓ of height was a worrying step with integer counting.

[@AxiomTutor]

So nice!!!!!!!!!!

[@anti-troll-software6151]

The egyptians built enough pyramids to come up with some heuristic with the amount of stone needed to reach a certain layer, so guessing was not impossible. Also I noticed the egyptians used the case where b = 2a as an example. This is easier because the problem simplifies to 1/3×2h×(2a)² - 1/3×h×a² which is 7/3×h×a². So if the formula is noted as you specified it they indeed knew that the values were interchangeable which is impressive. Algebra before algebra existed.

[@chillaxiommath]

Certainly not impossible you’re right! It would be a remarkable guess though. Regardless, I think it’s fun playing with the different clever ways they could have done it. And a juicy part we left out is exactly what you pointed out, that in their example b = 2a. Some people think that the Egyptians really were able to get the formula for this specific case (which would be simpler because all the a by b - a squares just turn into a by a squares) and then guess that it applies to all truncated pyramids. So that’s the frustrating part! We’ve only found this one example, so it’s hard to know what details matter and which are coincidence here

[@tristanridley1601]

When discussing volume, it's helpful to know that they worked with A LOT of stone blocks underwater. Underwater can mean displacement, and reveals volume in a completely straight forward way.

[@timperkin9]

Omg!! Valvo from film club! Congrats on the 3b1b mention!

[@chillaxiommath]

Haha what’s up Tim?? Thanks buddy! I’m so curious what your KZbin subscriptions are because I didn’t even know you were a math guy. You just have good taste all around it would seem!

[@timperkin9]

@@chillaxiommath My subs are public under "channels" on my page if you're curious. I actually wound up minoring in math! Very cool to run into you. I remember you telling me about planning to make math videos

[@nekkowe]

Oh! The dissection method was what I ended up doing when you asked the audience to try and derive the formula on our own (or rather, when you provided the measurements and I paused the video because I got curious and wanted to figure out the volume for myself)

[@KUWAITGRIPSVEVO]

Egyptian math homework was literally “Chepsut has a ten foot pyramid and Nefer has one half that size. What is the ratio of their volumes?” You can’t make this up

[@JamesWanders]

I liked and subscribed. Gonna be expecting that 10/22/6023 video.

[@StatiCraft3712]

I think you’re forgetting the new mathematicians greatest discovery tool. Doing an equation the “wrong way” but ultimately finding a new way to do it. I think the long equation was known but somebody made a “mistake” and proved something new

[@efrandsen72]

Know the volume of a pyramid? Yeah, I think the Egyptians had that one down.

[@hasanalharaz7454]

Will you publish this? I mean it’s completely bew

[@ABaumstumpf]

Getting the volume of a pyramid is easier than this - so i would assume that for the side-parts they just used that.

[@Xayuap]

for needing algebra for that, you are assuming they are starting like you did, knowing already the pyramid volume. maybe the did the other way around, maybe they knew the truncated volume and after that, they found the pyramid one, with the special case with a=0

[@alphalunamare]

lol, I had to turn subtitles on for this 🙂

[@Jaylooker]

Clever

[@GreenMeansGOF]

Something tells me that the same method could be used to prove the difference of two cubes formula.

[@vigilantcosmicpenguin8721]

before they had _WHEELS?!?!_

[@ardlight172]

Really loved this. I am looking forward to more content like this one. It was fresh, interesting, made me take pen and paper and think about a problem. It even has yo-mama jokes. Niiiice

[@msthurnell]

I like the pyramid shaped measuring tool. It’s my guess that the Nubians actually had such Pyamid shaped measuring tools to measure exact fractions of the cube or a cubit sized box of grain or whatever substance.

[@crowonthepowerlines]

4:03 Was the wheel originally used for pottery in Egypt as well? I knew it was used in China for pottery long before it was applied to transportation, but it's interesting to hear the same is true among other cultures.

[@tristanridley1601]

Absolutely. Egypt and Mesopotamia exchanged pretty much every decent idea they had, and Mesopotamia (probably) had pottery wheels before China. It was also probably used for niches as a roller (like logs under stone blocks when they couldn't use a water channel).

[@alphalunamare]

Rameses, without wheels, must have had a heck of a problem riding his chariot at the Battle of Kadesh as shown in the Hieroglyphs.

[@Ozymandi_as]

Egyptians also had very little wood that could be used for construction of anything. Granite wheels would make for a rather limited charioting sector, I imagine. That and the lack of horses. Besides, who needs wheels when you've got an unlimited supply of slaves?

[@4tbf616]

wait a minute, this ISNT a channel with a 100k+ subscribers!?

[@ianweckhorst3200]

What if they just noticed that pattern, I used purely my intuition to find the triangle number formula, I originally came up with n^2-((n+n^2)/2) and then just used that because I didn’t know algebra yet either, but I could have easily found an easier formula and just went with that, I suggest that at some point someone forgot the formula, but saw the numbers and just saw a pattern

[@JohnDlugosz]

Basically, you can do algebra (manipulate forms) physically using clay.

[@jujuteuxOfficial]

did they actually need to know the volume? this was done over years, and filling the thing with stones that doesn't pack neatly is one more reason that it's not really needed they likely just filled the thing up until it reached high enough it's >35% rubble, not cut stones

[@philippecoulonges4439]

Talking about Ancient Egypt is talking of about 3 millenniums, without more precision, we could say Israel had atomic bombs at the time of Christ. I've done a little research (well, I just read Wikipedia), the Moscow papyrus is a late document, 20th to 22nd dynasty, which means around 1000BC, effectively well before the Greek mathematicians, but more than a millennium after the construction of the great pyramids, they had time to think about it.

[@condellmaurice8597]

pyramid has 6 or 8 sides, then there is the bent one. Nice work though makes maths more fun

[@KeozFPV]

Or we just using the easy way and saying that Aliens have shown the Egyptians the formula. Joke aside, your guess is really convincing.

[@munimahmed7877]

nobody : absolutely nobody : conspiracy theories : " YOOOO we found the 69420 th (false) proof that aliens contacted with the ancient egyptians and made the pyramids. "

[@tristanridley1601]

It's funny, because the only people underestimating the Egyptians more than the old 'authorities' are the conspiracy theorists. "It couldn't have been that they were really smart and maximized use of the knowledge they had. It had to be ALIENS!"

[@XmarkedSpot]

How the F did you make an element of my GUI light up at the moment mentioning said element? I mean, it's fancy but rather creepy nonetheless. I can't help knowing from now on even though I comfortable chicken admittedly would prefer not to.

[@DavyCDiamondback]

Almost always, pre-algebra, some hipster used geometry

[@amandadierenfeldt7052]

Cause Egyptians didn’t build i! They stumbled on it later. Like after the comet, flood , ice age. The ancients built it using a simple string and holding it up to the stars. It’s called a plumberline!

[@whiterottenrabbit]

How is it even possible to speak that unintelligibly? Thank goodness for CC **facepalm**

[@LightPink]

Not enough likes and subs :/

[@piwi2005]

You don't need algebra to know that a square of a sum is the sum of squares plus two times the product, and you do not need algebra to know that the difference of cubes is the difference times the sum of squares and product. Fermat's proof on Fermat theorem for n=4 is in full latin without a single variable. This looks very much overfitted and completely over interpreted, just because of one sheet of papyrus. You do not even know if egyptians had complete valid proofs or were satisfied with heuristic equalities that happened to always work.

[@vesk4000]

6:45 "They were forced to make a bold conclusion -- the Ancient Egyptians clearly knew this formula, but they got it from a completely different path than we had ever considered" Hmm... 🤔🤔Well... have you ever considered... aliens👽 😂

[@vixguy]

Dam I've never seen a physical proof. Very cool

[@C0MPLEXITY]

bro is ryan gosling

[@Xeroxorex]

The Egyptians could have made KZbin videos, it's a bold claim to make, when there is no evidence they DIDN'T make KZbin videos.

[@TheZenytram]

aliens of course

[@TomTom-rh5gk]

Another internet hoax.

[@bigpopakap]

Don't forget the possibility that they got the formula from aliens!

[@debrucey]

please speak slower and enunciate

[@user-dn5bi4si5w]

Explains absolutely nothing. Will look for a better tutor

[@Xayuap]

you would win if you'd made your english more palatable. you sound so country specific with high speed commit

[@landback1491]

But they did have algebra and even tables to calculate the quadratic equation (aka: the diagonal rule). Even before Egypt in Sumeria, they had algebra. These methods are just much easier to calculate this way and western style algebra isn't needed.

[@Ozymandi_as]

Having solutions to algebraic problems is not the same as having algebra. Egyptians knew the right angle property of 3, 4, 5, triangles, but that doesn't man they knew the about the sum of the squares, or that they could be used in a generalised description of right triangles that we call Pythagoras' Theorem, or how to prove it, or whether 'proof' was a meaningful concept for them, or that they were able to generate Pythagorean triples at will. Likewise, we have no evidence that they were able to generalise the representation of numbers as symbolic variables and manipulate them to solve equations. We do know that at some point they had a method for calculating the volume of a truncated pyramid, we just don't know how they got to it, in a pre-algebraic world. Algebra is a more modern 'technology', and the fact that they managed mathematical tasks that we would perform with algebra does not mean that they in fact had that technology more than 3,000 years ago.