Pythagoras on Pizza
October 1, 2017 9:30 AM Subscribe
Surprising Uses of the Pythagorean Theorem. A nice intuitive exploration of the old mathematical chestnut. With just a little bit of simple logic, we find that the Theorem doesn't just apply to triangles. From this we learn for instance, that the energy used to accelerate one bullet to 500 mph can accelerate two others to 400 and 300 mph. And whether that large pizza is better than two mediums.
A circle or radius 5 has the same area as a circle of radius 3 and radius 4 added together.
Includes handy dandy calculator for deciding the pizza questioni.
A circle or radius 5 has the same area as a circle of radius 3 and radius 4 added together.
Includes handy dandy calculator for deciding the pizza questioni.
They ought to revoke your diploma, if you never did the pizza calculation before graduation (although as stated above, it seems to have little to do with the Pythagorean theorem, except it's got squares in it)
posted by thelonius at 10:16 AM on October 1, 2017
posted by thelonius at 10:16 AM on October 1, 2017
Several years ago I picked up a copy of Euclid's elements, with parallel English and ancient Greek. I was surprised to learn that the classic theorem is not that "the square on the hypotenuse..." but rather that "a figure on the hypotenuse equals that figure on the other two sides," a far more general statement: any figure whatsoever. They do not teach that in school. (Not my school anyway.)
posted by sjswitzer at 1:21 PM on October 1, 2017 [1 favorite]
posted by sjswitzer at 1:21 PM on October 1, 2017 [1 favorite]
That extra generality (not just squares!) is charmingly illustrated in Numberphile's Mathematical Fable (you might want to jump to 3:00, if you can't afford to spend three minutes on the simple version).
posted by Wolfdog at 2:22 PM on October 1, 2017 [8 favorites]
posted by Wolfdog at 2:22 PM on October 1, 2017 [8 favorites]
Several years ago I picked up a copy of Euclid's elements, with parallel English and ancient Greek. I was surprised to learn that the classic theorem is not that "the square on the hypotenuse..." but rather that "a figure on the hypotenuse equals that figure on the other two sides," a far more general statement: any figure whatsoever.
This follows immediately from the theorem for squares.
If the figures are similar, then their areas are ka^2, kb^2 and kc^2, for some non-zero constant k that depends on the shape. Multiplying both sides of a^2 + b^2 = c^2 by this k gives ka^2 + kb^2 = kc^2, which implies the claim about a general figure.
posted by James Scott-Brown at 4:04 PM on October 1, 2017 [1 favorite]
This follows immediately from the theorem for squares.
If the figures are similar, then their areas are ka^2, kb^2 and kc^2, for some non-zero constant k that depends on the shape. Multiplying both sides of a^2 + b^2 = c^2 by this k gives ka^2 + kb^2 = kc^2, which implies the claim about a general figure.
posted by James Scott-Brown at 4:04 PM on October 1, 2017 [1 favorite]
Im sure I mentioned this before, but years ago I was living in Melaka and would regularly order pizza from Papa John's (in the big Jusco mall if you are familiar with the area). Anyway, one day the pizza came and it had like four inches of naked crust surrounding a tiny circle of sauce and cheese.
Irritating, to be sure, but more funny to me than anything. I mean who would send a pizza out like that and think it acceptable? Anyway, I took a couple pictures and sent an email to Papa John's headquarters. Rather than leave it at that, I did the math explaining how much pizza was actually missing. I thought it was funny, and figured they would send me a coupon or something.
The next day I get a phone call from a strange number. It was the Papa John's manager. He pleads to make things right. I say ok and he offers to deliver me a new pizza right then.
No one would refuse that, so I say ok. He asks what toppings I like and I say I like everything. Twenty minutes later the delivery guy pulls up, except it isn't the delivery guy. It is the manager himself. He begs my forgiveness, takes a huge amount of food from the delivery box and opens each container for my approval before handing it to me. Then, when I have all the food, he prostrates himself in my driveway and begs my forgiveness again. I am so taken aback by this that I decide I will never complain about my pizza again.
I'm not sure if that was his angle, but it was one of the most awkward moments of my life. But, hey, tons of free pizza I guess.
posted by Literaryhero at 4:37 PM on October 1, 2017 [16 favorites]
Irritating, to be sure, but more funny to me than anything. I mean who would send a pizza out like that and think it acceptable? Anyway, I took a couple pictures and sent an email to Papa John's headquarters. Rather than leave it at that, I did the math explaining how much pizza was actually missing. I thought it was funny, and figured they would send me a coupon or something.
The next day I get a phone call from a strange number. It was the Papa John's manager. He pleads to make things right. I say ok and he offers to deliver me a new pizza right then.
No one would refuse that, so I say ok. He asks what toppings I like and I say I like everything. Twenty minutes later the delivery guy pulls up, except it isn't the delivery guy. It is the manager himself. He begs my forgiveness, takes a huge amount of food from the delivery box and opens each container for my approval before handing it to me. Then, when I have all the food, he prostrates himself in my driveway and begs my forgiveness again. I am so taken aback by this that I decide I will never complain about my pizza again.
I'm not sure if that was his angle, but it was one of the most awkward moments of my life. But, hey, tons of free pizza I guess.
posted by Literaryhero at 4:37 PM on October 1, 2017 [16 favorites]
Oh but I didn't use the Pythagorean theorem, I used Google.
posted by Literaryhero at 4:38 PM on October 1, 2017 [1 favorite]
posted by Literaryhero at 4:38 PM on October 1, 2017 [1 favorite]
This follows immediately from the theorem for squares.
True, but Euclid's proof did not proceed that way. It was derived in terms of an unspecified figure and the square case just falls out.
posted by sjswitzer at 4:44 PM on October 1, 2017 [1 favorite]
True, but Euclid's proof did not proceed that way. It was derived in terms of an unspecified figure and the square case just falls out.
posted by sjswitzer at 4:44 PM on October 1, 2017 [1 favorite]
Literaryhero, are you sure it was Papa Johns you ordered from, and not Uncle Enzo's?
posted by Pope Guilty at 5:09 PM on October 1, 2017 [14 favorites]
posted by Pope Guilty at 5:09 PM on October 1, 2017 [14 favorites]
I once used my amazing math skills to determine the carbohydrates in an eight inch flour tortilla, vs a six inch flour tortilla. The two inches in diameter make a huge differnce. Huh huh huh.
posted by Oyéah at 7:46 PM on October 1, 2017
posted by Oyéah at 7:46 PM on October 1, 2017
Twenty minutes later the delivery guy pulls up, except it isn't the delivery guy. It is the manager himself. He begs my forgiveness...
Was his name Uncle Enzo?
posted by exogenous at 8:29 PM on October 1, 2017 [2 favorites]
Was his name Uncle Enzo?
posted by exogenous at 8:29 PM on October 1, 2017 [2 favorites]
This reminds me of another blog post from a long time ago that pointed out that the Pythagorean theorem is useful for calculating distance between two points. And by that they meant any two points. One of the examples they used was finding the distance between two colors.
I am almost positive I saw it linked on here, but I couldn't find it.
Blew my mind at the time. I didn't even realize that was a problem that needed a solution.
posted by ArgentCorvid at 8:47 PM on October 1, 2017
I am almost positive I saw it linked on here, but I couldn't find it.
Blew my mind at the time. I didn't even realize that was a problem that needed a solution.
posted by ArgentCorvid at 8:47 PM on October 1, 2017
AgentCorvid, you might like to read about metrics, which are just functions between two objects in a set that satisfy certain nice properties. And it turns out that our usual distance function in 2D (d^2 = (x1-x2)^2 + (y1-y2)^2) is useful for many different sets such as colors.
posted by dilaudid at 8:55 PM on October 1, 2017
posted by dilaudid at 8:55 PM on October 1, 2017
The L2 or Euclidean distance between colors is only useful if the colors are represented in a color space that aims to be perceptually uniform, like CIELAB. In particular, in an RGB color space the distance sqrt((r2-r1)^2 + (g2-g1)^2 + (b2-b1)^2) is a poor measure of how far apart two colors are perceptually. Don't even consider it in HSL.
posted by Pyry at 10:20 PM on October 1, 2017 [1 favorite]
posted by Pyry at 10:20 PM on October 1, 2017 [1 favorite]
like I said, it was a long time ago, but I think something about the colorspace thing did come up. It sounds familiar anyway.
posted by ArgentCorvid at 6:51 AM on October 2, 2017
posted by ArgentCorvid at 6:51 AM on October 2, 2017
They ought to revoke your diploma, if you never did the pizza calculation before graduation (although as stated above, it seems to have little to do with the Pythagorean theorem, except it's got squares in it)
Not letting this go. If I want to know if two 12-inch pizzas have more pizza than one 16-inch pizza, why would I go through solving the equation, when it is easy to see that 2*36 > 64?
posted by thelonius at 7:38 AM on October 2, 2017
Not letting this go. If I want to know if two 12-inch pizzas have more pizza than one 16-inch pizza, why would I go through solving the equation, when it is easy to see that 2*36 > 64?
posted by thelonius at 7:38 AM on October 2, 2017
ArgentCorvid, is this what you're looking for?
posted by _Mona_ at 12:32 PM on October 2, 2017 [1 favorite]
posted by _Mona_ at 12:32 PM on October 2, 2017 [1 favorite]
I think that might actually be it!
posted by ArgentCorvid at 2:59 PM on October 2, 2017
posted by ArgentCorvid at 2:59 PM on October 2, 2017
I didn't even realize that the post I was talking about was a part of a series including the original post until just now.
That would explain why I thought of it though.
posted by ArgentCorvid at 6:59 AM on October 3, 2017
That would explain why I thought of it though.
posted by ArgentCorvid at 6:59 AM on October 3, 2017
Literaryhero, I got the same pizza once! I also took pictures and tried to complain but unfortunately mine fizzled as I sent the complaint to the American dominoes, not the Canadian one.
posted by Naib at 8:59 AM on October 3, 2017
posted by Naib at 8:59 AM on October 3, 2017
Not letting this go. If I want to know if two 12-inch pizzas have more pizza than one 16-inch pizza, why would I go through solving the equation, when it is easy to see that 2*36 > 64?
Well the existence of similar triangles, and thus similar shapes in general such as circles that allows you to make that inference is equivalent to the parallel postulate and thus the Pythagorean theorem. I'd say that makes it pretty related, all things considered.
posted by Proofs and Refutations at 5:02 AM on October 4, 2017
Well the existence of similar triangles, and thus similar shapes in general such as circles that allows you to make that inference is equivalent to the parallel postulate and thus the Pythagorean theorem. I'd say that makes it pretty related, all things considered.
posted by Proofs and Refutations at 5:02 AM on October 4, 2017
But all you need to know is that the area of a circle is pi * r^2
posted by thelonius at 6:43 AM on October 4, 2017
posted by thelonius at 6:43 AM on October 4, 2017
Well the existence of similar triangles, and thus similar shapes in general such as circles that allows you to make that inference is equivalent to the parallel postulate and thus the Pythagorean theorem.
gotta admit, I'm not following this....where do I use the parallel postulate to calculate the pizza size?
posted by thelonius at 6:45 AM on October 4, 2017
gotta admit, I'm not following this....where do I use the parallel postulate to calculate the pizza size?
posted by thelonius at 6:45 AM on October 4, 2017
But all you need to know is that the area of a circle is pi * r^2
This formula can't be derived without assuming the Pythagorean theorem or some equivalent statement (like the parallel postulate) though. If the Pythagorean theorem were not true, and we know of many geometries where it is not*, then the familiar pi * r^2 formula could not be used to find the area of a circle. Hyperbolic geometry, for example, has circles that have an area proportional to e^r rather than r^2.
You're quite right though that none of that is necessary to perform the calculation, of course, as long as you're willing to believe the formula holds*.
*Which it doesn't quite in the "real world", as the space we live in isn't quite flat in the presence of gravity, as Einstein famously showed.
posted by Proofs and Refutations at 12:22 AM on October 5, 2017 [1 favorite]
This formula can't be derived without assuming the Pythagorean theorem or some equivalent statement (like the parallel postulate) though. If the Pythagorean theorem were not true, and we know of many geometries where it is not*, then the familiar pi * r^2 formula could not be used to find the area of a circle. Hyperbolic geometry, for example, has circles that have an area proportional to e^r rather than r^2.
You're quite right though that none of that is necessary to perform the calculation, of course, as long as you're willing to believe the formula holds*.
*Which it doesn't quite in the "real world", as the space we live in isn't quite flat in the presence of gravity, as Einstein famously showed.
posted by Proofs and Refutations at 12:22 AM on October 5, 2017 [1 favorite]
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The Pythagorean theorem is the specific statement that the lengths of a triangle's sides satisfy the Pythagorean equation if, and only if, it contains a right-angle: the theorem really does "just apply to triangles".
posted by James Scott-Brown at 9:38 AM on October 1, 2017 [13 favorites]