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Those of you who remember the formula for the volume of a pyramid and its rounder cousin the cone may have, like me, have been told to simply accept it, at least until we learned calculus – that is, if we learned calculus. When I was in school, this bugged me. Bugged me enough to doodle a lot of pyramids until I discovered the primordial pyramid. The primordial pyramid is the only pyramid I know of which makes its volume obvious without the use of calculus – heck, nearly without the use of math! It is, however, a pyramid with specific proportions, incapable of answering for all pyramids and cones. To make it do that takes magical cheese.
Imagine a perfect cube. It could be a perfect cube of cheese, but at this point it’s more helpful to picture the cube as transparent – made of jello, for example. Picture lines inside the cube connecting each corner of the cube to its most opposite corner. The surfaces connecting these lines divide the cube up into six identical pyramids, primordial pyramids. The height H of each primordial pyramid is one-half the height of the cube, so the volume of the cube is (2H)^3. Because six of these pyramids together form the cube, the volume of each pyramid is (2H)^3/6. The base of each pyramid has area A = (2H)^2. Writing the volume of the primordial pyramid in terms of base and height, the volume is 2HA/6 = 1/3 HA. Now suppose we build a primordial pyramid out of that American kitchen staple, cheese singles (very thin, identical squares of cheese food product):
We’d start with a stack of cheese singles half as tall as it was wide, evenly trimming the higher cheese singles into smaller squares till they reached a point. If we built two of these cheesy primordial pyramids, we could combine them into one pyramid of twice the height and twice the volume by taking turns stacking the trimmed cheese slices. If the cheese singles were magic cheese singles, capable of uniformly changing their thickness at our command, we could use them to make a square pyramid of any proportions without having to disassemble and reassemble primordial pyramid slices. Each slice in the pyramid is a box, or very nearly so, so if its thickness t becomes rt, its volume v becomes rv during the magic change. Again taking H as the height of the primordial pyramid before its slices are bewitched into changing thickness, the height of the cheesy pyramid after its slices are bewitched is h = rH, and its volume becomes 1/3 hA. Now we have a picture for why any square pyramid has volume 1/3 hA, where h is its height and A is the area of its square base.
Since we’re already working with magic cheese, why don’t we magic up a magic cookie cutter, too? Whatever shape it cuts, this cookie cutter cuts it out of a square of dough (or, in this case, cheese) in proportion to the size of the square it’s cutting from. From a square of area A, the cutter cuts a shape of area a, and whatever square it cuts from, the ratio of the area it cuts to the area of the square it cuts from is always a/A. All we need to construct any pyramid we like (including cones) is a magic cookie cutter of the right shape and a square pyramid made of cheese slices. We just use the cookie cutter on each slice in the square pyramid. The volume of the pyramid made by the magic cookie cutter is a/A of the volume of the original square pyramid. (Can you picture this?) Since we know we can write the volume of the square pyramid as 1/3 hA, the volume of the pyramid cut out by the magic cookie cutter is
a/A * 1/3hA = 1/3ha, where h is the height and a is the area of the base.
This shows us that all we need to know to calculate the volume of any pyramid (including cones) is its height and a formula for the area of its base.
The magic cheese slices and the magic cookie cutter are of course “kiddie calculus” – informal ways to picture what calculus does formally. Doing even “kiddie calculus” justice takes a bit of work. You want to be sure you’ve really pictured how, if the volume of every slice of an object changes by ratio r, the volume of the whole object changes by ratio r, and so on. It is, however, picture work, which takes advantage of the brains we have. As Dominic Cummings (he of Brexit fame) notes,
[I]n the words of Stanislas Dehaene, the leading researcher of neurophysiology of mathematical thinking:
“We have to do mathematics using the brain which evolved 30,000 years ago for survival in the African savanna.”
In humans, the speed of totally controlled mental operations is at most 16 bits per second. Standard school maths education trains children to work at that speed.
The visual processing module in the brain crunches 10,000,000,000 bits per second.
I offer a simple thought experiment to the readers who have some knowledge of school level geometry.
Imagine that you are given a triangle; mentally rotate it about the longest side. What is the resulting solid of revolution? Describe it. And then try to reflect: where the answer came from?
The best kept secret of mathematics: it is done by subconsciousness.
Mathematics is a language for communication with subconsciousness.
Occasionally, our intuitive pictures of math can lead us astray, but what’s remarkable is how often they don’t (and how, even when they do, we can often train our intuition to change). That said, invoking both magic cheese and magic cookie cutters in order to get the general formula for the volume of a pyramid does bug me a bit. Is it needlessly elaborate? Is there a cleaner way to do it, short of using actual calculus, that doesn’t involve quite so much “kiddie calculus”?
Just because my little kitchen fantasy happens to be how I would naturally picture the derivation doesn’t mean it’s how other people should learn it. It does, however, illustrate the lack of sophistication that can go into mathematical thought. Mathematics scrupulously tests and verifies its intuitions, lending them a refinement that can seem otherworldly or even mystifying. But the intuition you started out with? Yes, sometimes it’s very cheesy.