Pyramids: Newsletter Challenge (08/21/07)

BTW, I admit it was a great challenge, even if the "official answer" was a bit of a disappointment.

Hi STL

If I said my version was simpler than yours, it would have been when I was under the misapprehension that you were attempting to solve the challenge in a mathematics-based way as opposed to a physics-based way, which I couldn't square with what I saw - so I assumed some complexity which I was missing.

Where I do claim my technique is relatively simple is by comparison to those who were trying to cut all sorts of angles into a rectangular block to produce three (or more) congruent pyramids (other than the six that fit into a cube), and temporarily convincing themselves they had succeeded. I'm not actually saying this is impossible - just that I haven't managed it, and the construction required could not be simple (I'd be happy to be proved wrong, because this would be a proper proof because it doesn't even depend on the scaling assumption).

So, I agree that my version would be marginally more complex than yours* - though I imagine the cuts would be easier to make if you were to try them in practise. But mine is making a less stringent assumption, so I don't find that small difference in complexity a problem.

*Equivalent descriptions of our basic constructions:
. Mine: start with a double-height pyramid. Make five cuts right through, one parallel to the base and half-way up, and four including the edges of the frustum and perpendicular to the base. Now sort the pieces and calculate based on the areas of those that have already been proved (the prisms), and the 2h pyramid having eight times the volume of a single-height pyramid. The assumption here is the scaling of volumes for similar shapes. Establish likelihood of the assumption by making multiple measurements of pairs of x1 and x2 scaled pyramids.

. Yours: based on a pyramid, w x l x h. Create a rectangular parallelepiped w x l x 2h. Make eight cuts through the edges of length w and l to the half-height mark to define two of the prisms that are mirror images of each other. That creates six pyramids. Assume that a law exists of form volume = λ.w.l.h where lambda is an unknown constant. Observe that the value for such a fixed lambda would need to be 1/3. Establish likelihood of assumption by making multiple measurements of volumes.

BTW, there is no way that my disections are beyond you - especially not the basic version (which was part of the reason that I made the challenge about square-based pyramids in the first instance). If it appears so, it is either because I have written badly**, or upset you (apologies in either case), or you are distracted for some other reason (coffee). So, put all that behind you, and see if you can improve on what I've done
**I deliberately avoided mathematical description. If you really want it, I'm sure someone will be happy to provide it - but it will not be I in this case, because it doesn't actually add anything.

Hope that defines our real differences, as opposed to any that were personality-driven, and puts the difficulty behind us.

Fyz

No personality issues. I just don't understand why you think that an algebraic solution to this problem is not valid and you keep saying that I am ASSUMING there is a universal constant. I did not ASSUME anything. I stated a hypothesis, which could be based on experience working with pyramids of various shapes. I just happened to have the benefit of hindsight in knowing the formula, but it could have been done without knowing that one existed, because I solved for the constant based on formulas created only from known geometry and the hyothetical formula and found one, and only one, unique rational number in the solution.

Your basic solution is not that much different from mine and only encompasses regular rectangular pyramids, both right and basic oblique, which have an apex over the base. That solution and proof is simple enough. Does your basic solution as given at the end of the challenge question even deal with oblique pyramids at all? Mine does.

You go on to say you can extend your solution beyond the base rectangle and encompass the extremely oblique pyramids, but then your solution becomes extremely complex, and that is what I am having trouble following, as apparently do many others. I am not saying you are wrong, just that this extension is neither "simple", nor "elegant", IMHO.

And since I was the first to describe a solution, that works, beyond the limited case proven by the cube method, I was only a bit miffed that I got no recognition, only criticism, that I still consider unwarranted.

I now step down off my soapbox and return to my corner to sulk and continue to munch my sour grapes ("I tell ya', I don't get no respect!")

Maybe this is just a language issue - you probably regard a well-justified physical model as being 'proven', whereas I just regard it as a working theory until displaced by something more precise.

You state a hypothesis, which generates a numerical result. I think you will agree that that says nothing about whether or not the the hypothesis is true. If you can now generate that same result by a different route (in this case measurement?), you will have started on the process of a physics- (or engineering-) type model of a physical process. That is not the same as a proof, which takes specific assumptions regarded as "givens" (generally called axioms) and demonstrates that these inevitably lead to the outcome you desire. Anything else is either a measurement-based model, an assumption, or elevating the hypothesis to the status of an axiom. I am perfectly happy to agree that what you have can become a very good measurement based model, or even a theory - just like Newton's laws or (later) Einstein's special theory of relativity. But it could in principle become subject to refinement - except that in this case it won't because it was proved elsewhere (based on axioms and the use of limits).

I admit to adding an unjustified hypothesis, and using it as an axiom - one which I can demonstrate with the same level of accuracy that you can demonstrate yours. The only merit of my unproved hypothesis is that is apparently less restrictive than yours, and also more intuitive if you haven't previously seen the result [show overbalancing pyramids and right pyramids (each with equal bases and heights) to people who haven't previously seen the formula or the layering technique, and you will see what I mean].

Actually my praise was deliberately over the top, just to contrast with my endless criticism of your own efforts. I'd even considered saying: "With Fyz handling the logic and math, and you handling the illustration, you'd make a great team." But I thought that would be a little too far over the top... your illustration isn't all that great!

Gadzooks, am I evil incarnate, or what? Your illustrations are great. And I don't really think you should be the illustrator, with Fyz being the brains behind the operation. You're clearly a bright and competent guy, capable of the logic, the math, the illustration -- all while chewing gum!

Thinking like an Egyptian seems harder than walking like an Egyptian.

I may be saying the same thing as some others, but here goes... Think like an Egyptian and an engineer -> make an assumption.

Assume that the volume of the pyramid is given by :

V = f w d h

where f is a geometric scale factor and w, d base dimensions, h = height

also assume that all rectangular base pyramids obey the same relationship. Now, the block volume will be equal to the core pyramid volume + 2 other pyramid volumes + 2 other pyramid volumes :

wdh = f wdh + 2 f d h (w/2) + 2 f w h (d/2)

solve : f = 1/3

Definition of "calculus" : the heart of calculus is the concept of the limit. So, if a limit is involved in a derivation, I would (roughly) say it is a calculus based derivation.

Just a side comment... the official answer is nice / correct, but I think there is an "assumption" there as well. The asumption is that if you double all the dimension properties of the pyramid, the volume of the larger pyramid will be 8 times the smaller (2^3). This is, of course, true and intuitive, but not necessarily known at the time (let alone proven at the time).

Agreed - as the party guilty of introducing this question and solution, I have stated that many times. I'm not claiming to have proved anything - just to have provided a sufficient basis for an Egyptian surveyor to say he has a viable formula. The curious thing is that the people who say "measure the volume of a scaled pyramid, because you can't prove a formula" are assuming that this is "true". Otherwise, it is only some of the proponents of the "fixed lambda" hypothesis who are claiming they have a proof.

The scaling appears to be a less restrictive assumption than fixed lambda, however, because it does not immediately imply the fixed-constant formula for different shapes. It is also better supported than the "fixed formula using orthogonal measurements", in that it is already proved for every shape for which we can calculate the volume. Obviously, its support (as a working model of the universe rather than a proof) by direct measurement of scaled pyramids is equally valid or contentious as that of the "fixed lambda".

Hey Goober,

IMHO that points out the one thing none of us know and all of us at least thought about: where did the idea of volume originate for the Egyptians, how did they think about it, and what shape was their starting point. My best guess would be some kind of granary, but what shape? If we knew that, and I don't think we do, it would help a lot in deciding what we could do in a proof. Oh well, a heck of a good challenge was had by all.

Tom

STL may not be willing or able to draw this up, but I am! It may take me a day or two, because I haven't done an assembly in Inventor for a couple of years. I need more experience with Inventor anyway.

Here's a rough drawing in TurboCad.

1.) Top and corner ready.

2.) Top and corner off.

3.) Complete cuts

Separate three prisms: the two triangular prisms are each exactly half the shape and volume of the rectangular (or parallelogram) based one.

I needed to draw it to see what Fyz meant. Great challenge!

I imagine that they had learned to stack fruit and veggies back in 1800 BC, and that the pyramid shape was the most stable way to stack. I also imagine that they had box-like containers or baskets, and noted volumes. They apparently knew something of weights and measures in order to trade. Why do modern people assume that the ancients were stupid just because they had no cell phones or iPods? Lee

It is simple during the great floods they measured how much water it displaced by counting the number of drowned slaves 22.1 miles down stream and subtracting 2 litres for every new born upstream. Of course this only works during the alignment of Ra with the moon goddess any other time they just used buckets and filled it up with water the headman using his trusty abacus.

Here's my illustration of how I visualized Fyz's concept:

To save time, I recycled parts from my first illustration. I did offset the apex 1 inch towards the front (blue arrow on the UCS symbol).

If Fyz requests, I will show the pieces as solids. Otherwise, I'll leave it as is unless I hear a public outcry. However, I can be bribed with chocolate.

The only difficulty is that in this (isometric?) perspective the front top and rear bottom edges of the parallelepiped are co-linear. If it's reasonably simple to do, could you build it around the parallelepiped and make the three dimensions of the parallelepiped unequal (I think that should separate the edges without requiring you to change the angles) ?

Thanks, Fyz

Maybe It's too simple, but if you take a cube you can cut it into 6 pyamids with each side forming the base. The volume has to be 1/6th that of a cube.

Wow, that is only the gazillionth time that one has been mentioned! Like all the others before you, that solution only works for right pyramids with square bases and which have a height equal to half of the length of one side of the base. Try changing any one of those (off center apex, rectangular base, random height) and you will not be able to fit your 6 identical pyramids together.

How does 8.Volume can be explained???

Ohhh I get it 2^3!

And how is it proven?

As I said earlier (when I gave the hint) - it's an assumption, possibly it could even have been an axiom (no worse than angles between parallel lines, for example). But I've used it as the minimal assumption I can make, and it's validity is based on being true for any shape whose volume we know how to calculate (parallelepipeds and prisms). We can also demonstrate that any sum of a group of pyramids that adds together to give parallelepiped obeys this, using STL's construction, but that is about all. The idea was not to present a proof (this wasn't intended to be one), but that the ancient Egyptian surveyors could reasonably have come up with a geometrically demonstrable (not proven) formula without the use of infinitesimals.

In that respect, STL answered the challenge, though (in my opinion if not his) with a more sweeping assumption than mine.

Fyz