Needles, Noodles, Doodles, and the Length of a Potato
June 28, 2022 9:08 AM Subscribe
In 1777, the Comte de Buffon studied the odds of landing a round coin on a single square of a tile floor. (Those early probabilists, always thinking about gambling!) Buffon then proposed a famed variation with needles. Though there are no overt circles in this version of the problem, π still shows up in the answer. For a beautiful explanation, trade needles for noodles (YT, 9m 59s; text alternative). You can simulate tossing noodles to empirically calculate π, or count grid crossings to estimate the length of a mountain trail. But that's not the end of the story…
Imagine measuring an object in every direction with a pair of calipers. The average measurement is known as mean width. In the noodle argument, the grid acts like calipers, implying that for convex objects, mean width is just another way to measure perimeter -- they're the same up to a factor of π. Similar arguments link the surface area of a three-dimensional object to the average size of its shadows (YT/3Blue1Brown, 40m).
Ravi Vakil's inviting yet deep The Mathematics of Doodling connects these ideas with a diverse constellation of problems, from proving you can't cheat the Russian railroads (puzzle 2 in Seven Puzzles You Think You Must Not Have Heard Correctly) to the work of Maryam Mirzakhani.
If volume and surface area measure a three-dimensional object in three and two dimensions, how should we measure it in one or zero? What Is the Length of a Potato?
Imagine measuring an object in every direction with a pair of calipers. The average measurement is known as mean width. In the noodle argument, the grid acts like calipers, implying that for convex objects, mean width is just another way to measure perimeter -- they're the same up to a factor of π. Similar arguments link the surface area of a three-dimensional object to the average size of its shadows (YT/3Blue1Brown, 40m).
Ravi Vakil's inviting yet deep The Mathematics of Doodling connects these ideas with a diverse constellation of problems, from proving you can't cheat the Russian railroads (puzzle 2 in Seven Puzzles You Think You Must Not Have Heard Correctly) to the work of Maryam Mirzakhani.
If volume and surface area measure a three-dimensional object in three and two dimensions, how should we measure it in one or zero? What Is the Length of a Potato?
I don't like the way the character pi looks in Metafilter's font; I only realized it was pi, not "n" because I recognized the needle example.
posted by xris at 9:36 AM on June 28, 2022 [3 favorites]
posted by xris at 9:36 AM on June 28, 2022 [3 favorites]
Great stuff. From Vakil's paper: "The average length of the shadow of a convex region of the plane, multiplied by π, is the perimeter!" That reminds me of the meandering river theorem, inspired by Einstein but attributed to Stolum, that the average sinuosity of all (idealized) rivers tends to pi. Sinuosity of river = (actual length) / (length crow flies).
So, (perimeter)/(average shadow length) = pi = (average length)/(average length crow flies).
I'm not sure how equivalent these theorems are without some more work, but at least they are equally believable when the convex region is a disk, and the river must flow across that disk from one point on the boundary to its antipode.
posted by TreeRooster at 11:47 AM on June 28, 2022
So, (perimeter)/(average shadow length) = pi = (average length)/(average length crow flies).
I'm not sure how equivalent these theorems are without some more work, but at least they are equally believable when the convex region is a disk, and the river must flow across that disk from one point on the boundary to its antipode.
posted by TreeRooster at 11:47 AM on June 28, 2022
Ooh, do I get to link to Chris Staecker's Steinhaus Longimeter Review video now?
(NB: if anyone - especially outside the USA - is finding .edu sites hard to reach, that's because of Shields Up)
posted by scruss at 3:19 PM on June 28, 2022 [1 favorite]
(NB: if anyone - especially outside the USA - is finding .edu sites hard to reach, that's because of Shields Up)
posted by scruss at 3:19 PM on June 28, 2022 [1 favorite]
Buffon's needle makes an appearance in one of Samuel R. Delaney's Nevèrÿon stories, I believe in the book-length Neveryóna, or: The Tale of Signs and Cities; I remember reading it in pre-Wikipedia high school days and laboriously trying to work out for myself if the math seemed right.
posted by snarkout at 7:34 PM on June 28, 2022
posted by snarkout at 7:34 PM on June 28, 2022
Oooh, the Steinhaus longimeter is new to me, and Staecker's YouTube channel is looking like a mighty dangerous rabbit hole. Thanks, scruss (I think)!
I like that Staecker compared the longimeter with the map measurer in practice and got a discrepancy of almost 20% -- that tracks with my understanding of how slowly the estimate of π in the Buffon needle process actually converges (you can experience this for yourself in the simulators). In that light, it's not too surprising that the longimeter never caught on. But I love the concept.
TreeRooster, the river sinuosity result popped up while I was collecting material for this post! But as I dug in, I started to frown. Stølum's paper doesn't give any clear reason for the average to be π; he says it's the "sinuosity of a circle", but if you divide the semicircular arc between antipodes by the diameter, you get π/2, not π. Following Stølum's claim down into the footnotes, it looks like he cooked up a figure of π in (to put it delicately) a rather post hoc manner.
I'm not sure I believe that sinuosity is a dimensionless ratio, actually -- it seems like longer rivers should have higher sinuosity, since if you stick a bunch of river segments end to end, their curvy lengths add but the crow lengths sub-add (the crow doesn't have to visit all the splice points). However, Stølum does say to measure length in terms of units of river width, which is an interesting subtlety that might address my objection. (This seems not to have been taken into account by this guy who tried to test Stølum's result empirically.)
posted by aws17576 at 8:35 PM on June 28, 2022 [1 favorite]
I like that Staecker compared the longimeter with the map measurer in practice and got a discrepancy of almost 20% -- that tracks with my understanding of how slowly the estimate of π in the Buffon needle process actually converges (you can experience this for yourself in the simulators). In that light, it's not too surprising that the longimeter never caught on. But I love the concept.
TreeRooster, the river sinuosity result popped up while I was collecting material for this post! But as I dug in, I started to frown. Stølum's paper doesn't give any clear reason for the average to be π; he says it's the "sinuosity of a circle", but if you divide the semicircular arc between antipodes by the diameter, you get π/2, not π. Following Stølum's claim down into the footnotes, it looks like he cooked up a figure of π in (to put it delicately) a rather post hoc manner.
I'm not sure I believe that sinuosity is a dimensionless ratio, actually -- it seems like longer rivers should have higher sinuosity, since if you stick a bunch of river segments end to end, their curvy lengths add but the crow lengths sub-add (the crow doesn't have to visit all the splice points). However, Stølum does say to measure length in terms of units of river width, which is an interesting subtlety that might address my objection. (This seems not to have been taken into account by this guy who tried to test Stølum's result empirically.)
posted by aws17576 at 8:35 PM on June 28, 2022 [1 favorite]
(Sorry, I should have said I'm not sure I believe that sinuosity is scale-free. A length divided by a length is certainly dimensionless...)
posted by aws17576 at 8:41 PM on June 28, 2022
posted by aws17576 at 8:41 PM on June 28, 2022
aws17576 Following Stølum's claim down into the footnotes, it looks like he cooked up a figure of π in (to put it delicately) a rather post hoc manner.
So, any ratio between 1:2 and 1:4 rounds to 1:π if you really like π ?
In March I was correctly pulled up for some loose talk about φ and the human body.
posted by BobTheScientist at 12:19 AM on June 29, 2022
So, any ratio between 1:2 and 1:4 rounds to 1:π if you really like π ?
In March I was correctly pulled up for some loose talk about φ and the human body.
posted by BobTheScientist at 12:19 AM on June 29, 2022
Sorry if it has been linked in this post, Buffon’s name triggers joy in me and I have to let everyone know that ants implement Buffon’s Needle to estimate potential nest size. How they reach consensus on which nest is best is another great story.
PDF of the paper.
posted by Dr. Curare at 6:32 AM on June 29, 2022 [1 favorite]
PDF of the paper.
posted by Dr. Curare at 6:32 AM on June 29, 2022 [1 favorite]
Thanks for the link aws17576! On Stølum's result: It looks like the proof (just a sketch) is in his footnote 14. Not easy to parse! By "cuts" he means "arcs" of the circle. I think he is describing a fractal kind of like a Koch Snowflake, or more like a Koch curve, one side of the snowflake. It begins with the straight line, then that semicircle, so sinuosity pi/2 as you point out. Then there is a fractal generating process that makes meanders in the meanders...but how is it defined? It appears that he knows, but leaves it unclear, then just says "approximately" and skips a proof! This needs work.
posted by TreeRooster at 7:03 AM on June 29, 2022
posted by TreeRooster at 7:03 AM on June 29, 2022
As someone who uses math most of the day but very often doesn't really understand why math folks are excited about most of the things math folks are excited about, this is all really interesting and mostly new. (I've seen variations on the coins and needle ones. But the writeups are quite good.) The noodles are cool, and the potato paper is great fun. Thanks!
posted by eotvos at 7:22 AM on June 29, 2022
posted by eotvos at 7:22 AM on June 29, 2022
The average measurement is known as mean width.The word "average" is doing an unconscionable amount of work in all of those sentences and is about to go on strike.
... the average size of its shadows
... average sinuosity of all (idealized) rivers
posted by Aardvark Cheeselog at 7:25 AM on June 29, 2022 [2 favorites]
Right, Aardvark! I think for the shadows, you'd have to integrate, as in the average size of a function over an interval: the integral of the function divided by the length of the interval. For the rivers, in the paper, "These opposing, processes self-organize the sinuosity into a steady state around a mean value of s = 3.14." (Then the footnote supposedly proves it.) Assuming it is true, it would mean that the ideal rivers are all on nearly flat planes of uniform soft soil and have existed there for long enough to approach that mean.
posted by TreeRooster at 7:47 AM on June 29, 2022
posted by TreeRooster at 7:47 AM on June 29, 2022
I’m not sure where shadows come in to it, but if you think about a convex shape’s average width, suitably defined, you’re describing a circle of that average’s diameter. But any shape with that average width would have to have a perimeter larger than the circle! So I don’t think this checks out.
posted by sjswitzer at 12:32 PM on June 29, 2022
posted by sjswitzer at 12:32 PM on June 29, 2022
On second thought, I’d have to think harder on the definition of “mean width” and its possible interpretations. It could be that deviations from a circle could increase the mean under suitable definitions. There’s a 3blue1brown that goes down the rabbit hole of random intersections. It might even already be linked here.
posted by sjswitzer at 12:44 PM on June 29, 2022
posted by sjswitzer at 12:44 PM on June 29, 2022
Some more about the rivers: this person thinks the real ratio should be pi/2, and the pictures are convincing. Real rivers aren't fractal curves, usually. This person talks a bit about the width of the river and size of meanders, to deal with the scale-free-ness. It has citations.
On the potatoes: I immediately though of the work of Tom Leinster, on general conceptions of magnitude which can be used to measure almost anything, from biological diversity to measuring systems (metric spaces) themselves. Then I noticed the potato paper is stored on Tom's website! Here's his main site, scroll down a bit to see his papers on Size. A paper that references potatoes is a good place to start.
posted by TreeRooster at 2:16 PM on June 29, 2022
On the potatoes: I immediately though of the work of Tom Leinster, on general conceptions of magnitude which can be used to measure almost anything, from biological diversity to measuring systems (metric spaces) themselves. Then I noticed the potato paper is stored on Tom's website! Here's his main site, scroll down a bit to see his papers on Size. A paper that references potatoes is a good place to start.
posted by TreeRooster at 2:16 PM on June 29, 2022
Ahhh, that ant paper is so cool, especially the bit with the "magic carpet" at the end. Thanks for linking, Dr. Curare.
TreeRooster is right -- mean width is an integral. For a plane figure, the directions you can measure in are indexed by one angle θ, so you measure width as a function of θ, integrate from 0 to 2π, and divide by 2π to normalize. In higher dimensions, you have to integrate over a spherical space of directions. Here's Wikipedia on mean width (I couldn't find a less dry and technical write-up, unfortunately).
sjswitzer, I'm not sure I understood your comment, but do you know about Reuleaux triangles? Those are examples of curves that have the same width no matter what direction you measure in, but they aren't circles. If you compare the Reuleaux triangle of width w to the circle of diameter w, they have equal perimeter! (The circle has more area, though.)
posted by aws17576 at 2:20 PM on June 29, 2022 [2 favorites]
TreeRooster is right -- mean width is an integral. For a plane figure, the directions you can measure in are indexed by one angle θ, so you measure width as a function of θ, integrate from 0 to 2π, and divide by 2π to normalize. In higher dimensions, you have to integrate over a spherical space of directions. Here's Wikipedia on mean width (I couldn't find a less dry and technical write-up, unfortunately).
sjswitzer, I'm not sure I understood your comment, but do you know about Reuleaux triangles? Those are examples of curves that have the same width no matter what direction you measure in, but they aren't circles. If you compare the Reuleaux triangle of width w to the circle of diameter w, they have equal perimeter! (The circle has more area, though.)
posted by aws17576 at 2:20 PM on June 29, 2022 [2 favorites]
→
Yeah, Chris Staecker's channel is a huge old time-suck. His one on the polar planimeter was the gateway drug for me.
But the real potato-quality area measurement device is the hatchet planimeter. Just accurate enough to give you an estimate of area that'll do, but so simple it almost defies belief.
posted by scruss at 6:53 PM on June 29, 2022 [1 favorite]
Thanks, scruss (I think)!
Yeah, Chris Staecker's channel is a huge old time-suck. His one on the polar planimeter was the gateway drug for me.
But the real potato-quality area measurement device is the hatchet planimeter. Just accurate enough to give you an estimate of area that'll do, but so simple it almost defies belief.
posted by scruss at 6:53 PM on June 29, 2022 [1 favorite]
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This is sort of a self-link, but I once used a Buffon's noodle argument on AskMeFi to estimate the number of bridges in Michigan.
posted by aws17576 at 9:09 AM on June 28, 2022 [3 favorites]