Mathematicians on roofs

I am a mathematician, and I would like to stand on your roof (video, 17 minutes). Ron Eglash talks about African fractals. It seems that fractals of one sort or another show up naturally in designs used both decoratively and functionally in various modern African cultures. (As far as I can tell from the video, Eglash is not claiming this is something unique about Africa. I'm curious if he addresses this question in his book, African Fractals: Modern Computing and Indigenous Design, which I have not yet read because I didn't know it existed until this morning.) The title of this post comes about because the best way to see how a village is designed is to stand on the roof of the tallest building.

McMullen on the geometry of 3-manifolds

The Geometry of 3-Manifolds, a lecture by Curt McMullen. This is one of the Science Center Research Lectures; in which Harvard professors talk about their research to the interested public; the series doesn't appear to have a web page, but here's a list of videos available online in that series; these include Benedict Gross and William Stein on solving cubic equations. There are non-mathematical things too, but this is at least nominally a math blog, so I won't go there.

McCullen apparently also gave this lecture at the 2008 AAAS meeting in Boston, and has a couple other video lectures available online.

And now I want a do(ugh)hnut-shaped globe with just North and South America on it. This is a fanciful example of what an "armchair Magellan" might suspect the world looked like if humans had reached the North and South poles starting from somewhere in the Americas but had never crossed the Atlantic or Pacific; they might suspect that the cold area to the north and the cold area to the south are actually the same. McMullen uses to illustrate that tori and spheres are not the same, since loops on the sphere are contractible but loops on the torus are not. The lecture, which leads up to telling the story of the Poincaré conjecture, begins by using this as an example of how topology can distinguish between surfaces.

Finally, here's an interesting story, which may be well-known to some people but wasn't to me: Wolfgang Haken, one of the provers of the Four-Color Theorem, may have intended that (famous, computer-assisted) proof to be a "trial balloon" for a brute-force proof of the Poincare conjecture.

The Theorem

Via Keith Devlin: The Theorem. (To give more of a description would spoil the punchline; I'd recommend not reading Devlin's article until after you've seen the video.) This should be familiar to anybody who's seriously thought about mathematics, although my life doesn't come with background music like the video does.

Yellow books

I'm currently watching Science Saturday at bloggingheads.tv, which this week features Peter Woit (Not Even Wrong) and Sabine Hossenfelder (Backreaction).

When the video started, I thought "hmm, Woit has an awful lot of yellow books behind him for a physicist".

Woit, it turns out, works in the mathematics department at Columbia, as I was reminded when he started talking about the different job situations in physics and mathematics about fifteen minutes in. Basically, Woit says that jobs in physics are scarcer compared to PhDs in physics than the analogous situation in mathematics, so physicists feel more pressure to "do everything right" -- in his view this means they feel unnatural pressure to work in string theory, which Woit sees as a bad thing. After all, what if string theory's wrong? Physics as a discipline should diversify.

Three beautiful quicksorts

Jon Bentley gives a lecture called Three Beautiful Quicksorts, as three possible answers to the question "what's the most beautiful code you've ever written?" (An hour long, but hey, I've got time to kill.)

Watch the middle third, in which some standard code for quicksort is gradually transformed into code for performing an analysis of the number of comparisons needed in quicksort, and vanishes in a puff of mathematical smoke.

Although I must admit, I'm kind of annoyed that he slips into the idea that an average-case analysis is the most important thing somewhere in there. The first moment of a distribution is not everything you need to know about it! Although I admit that at times I subscribe to the school of thought that says "the first two moments are everything", but that's only because most distributions of normal.

(Note to those who don't get sarcasm: I don't actually believe that most distributions are normal.)

Video abstracts?

From John Baez at the n-Category Cafe: the Journal of Number Theory is now inviting authors to post their abstracts on YouTube.

Something about putting them on YouTube -- as opposed to on the journal's website -- strikes me as saying that they're not "really" part of the paper.

More interestingly, though, the few "video abstracts" there are just people reading their abstracts. I think it's a good idea, but mathematical speech is just not mathematical writing read out loud. (This is, of course, the converse of the fact that mathematical writing is not just mathematical speech transcribed.) Perhaps there's potential here, though. The first few minutes of a good talk would work as a good "video abstract", I think.

But you've got to start somewhere.

I Will Derive

"I Will Derive", a parody of "I Will Survive".



(via reddit.)

Now, the original song is clearly about female empowerment, and there is somebody who could be described as a "jerk" in the song. (Here are the lyrics, in case you've been living in a cave, or you're not from the US -- I'm not sure how well-known this song is in other countries.) I think this parody would be improved if it mentioned jerk, the third derivative of position.

Large Rubik's cubes and asymptotic thoughts

A video of the solution of a Rubik's cube of side 100, from YouTube.

I don't know the details of the algorithm, but it has the interesting property that at first it looks like nothing's happening, and then larger and larger blocks of each color seem to form -- I'm not sure if this is because of the low resolution or if this really is some sort of "phase transition" in the algorithm. This is the sort of thing one just doesn't see in the normal-sized cube.

I came to this from a solution of the size-20 cube, which works in a much different way; each of the faces of the cube gets "filled in" in turn. So in the second case the algorithm appears to be making steady progress; in the first case it doesn't. There are essentially two different families of algorithm.

It would be interesting to know how the solution time (measured in number of moves) of these families of algorithms -- and I'm calling them "families" because I suspect that the actual side length of the cube isn't all that important -- scales with time. Also, how do these algorithms compare with the asymptotically fastest one? One could compute a lower bound on the number of moves needed to solve a Rubik's cube of arbitrary size by just computing the number of possible positions (analogously to the calculation of that number for the normal cube I outlined here) and comparing that to the number of possible moves one can make from a given position. An absolutely useless question -- is that lower bound (I worked it out once, but I forget what it is, and it was the sort of the back-of-the-envelope work that might have been wrong anyway) asymptotically tight? That is, does there exist a solution procedure for the Rubik's cube for which the number of moves needed to solve the size-n cube is within a constant factor of this obvious lower bound?

As for what that lower bound is, I may write that up as another post; I'd either need to find the notes I made on it a few months ago or (more likely) rederive them.

Carl Sagan on Flatland

Carl Sagan on Flatland (from Blake Stacey at Science after Sunclipse).

I've got other things I want to talk about, but tomorrow morning's class won't prepare itself.

Cosmology on bloggingheads.tv

I'm watching a fascinating video on bloggingheads.tv in which Sean Carroll of Cosmic Variance and John Horgan talk about modern comsology. An interesting point from my point of view is that the basic reason modern cosmology exists is because the idea of maximum likelihood doesn't seem to apply to the universe -- probabilistically the universe "shouldn't be" like it is. But it is! From an epistemological point of view that's why explaining the way the universe is is interesting -- you don't expect, for example, large-scale homogeneity and isotropy, but it's there!

(Also, bloggingheads has a feature by which you can speed up videos by a factor of 1.4, without the expected raise in pitch of the sound by a tritone. This is nice, because when listening to recorded speech I often find it to be too slow. That's fine -- I don't insist that people should speak faster -- but I can take in information faster than an extemporaneous speaker can produce it.)

Nothing particularly new if you know something about these things, but a good way to spend 47 (sped-up) or 66 (normal-speed) minutes on a Saturday afternoon.

Arthur Benjamin's mental arithmetic

Arthur Benjamin is a research mathematician (I've actually mentioned him before, although I didn't realize that until I looked at his web page and saw that the title of one his papers looked familiar...) and also a "mathemagician" -- he has a stage show in which he does mental calculations. See this 15-minute video of his show at ted.com.

He starts out by squaring some two-digit numbers... this didn't impress me much, because I could almost keep up with him. (And 37 squared is especially easy for me. One of my favorite coffeehouses in Cambridge was the 1369 Coffee House, and at some point I noticed that that was 37 squared. So I'll always remember that one.) Squaring two-digit numbers is just a feat of memory. Three- and four-digit numbers, though... that's a bit more impressive. And of course I'm harder to impress in this area than the average person.

One trick that might not be obvious how it works: he asks four people to each find a number 8649x (the 8649 was 93 squared, from the number-squaring part of the show) for some three-digit integer x, and give him six of the seven digits in any order; he says which digit is left out. How does this work? 8649 is divisible by 9. So the sum of the digits of 8649x must be divisible by 9. So, for example, say he gets handed 2, 2, 2, 7, 9, 3; these add up to 25, so the missing digit must be 2, to make 27? (How could he tell apart a missing zero and a missing nine? I suspect there's a workaround but I don't know what it is; the number 93 was given by someone in the audience, so I don't think it's just memory.)

He also asks people for the year, month, and day which they were born and gives the date; I found myself trying to play along but I can't do the Doomsday algorithm quite that fast... and I suspect he uses something similar. (I noticed that he asked three separate questions: first the year, then the month, then the day. This gives some extra time. I know this trick well; when a student asks a question I haven't previously thought about, I repeat it. I suspect I'm not the only one.)

The impressive part, for me, is not the ability to do mental arithmetic -- I suspect most mathematicians could, if they practiced -- but the ability to keep up an engaging stage show at the same time.

(The video is on ted.com, which shows talks from an annual conference entitled "Technology, Entertainment, and Design"; there look to be quite a few other interesting videos on there as well.

Stop-motion Tetris

Stop motion tetris is made out of people! Some people got together and played the part of the squares in the game of Tetris; a lecture hall with seats in a grid was the background.

What surprised me is how, at least in this particular game of Tetris, each individual piece got broken up pretty quickly, with only one or two pieces out of the original four remaining on the board. (The four people who made up each dropping piece were wearing the same color T-shirt, and there were enough different colors that for the most part one could assume that people wearing the same color and next to each other were part of the same piece.) This isn't obvious if you're used to, say, the NES version of Tetris (which is the one I played the most); the color scheme is different there. If you stop to think about it for a moment, though, it makes perfect sense that this should happen; it's very rare that four squares which fall at the same time all get eliminated at the same time.

Incidentally, the people writing that post say "Now, we're not mathologists". What is a mathologist? I would argue that mathologists are to mathematicians as musicologists (i. e. people who study music in a scholarly fashion, as opposed to those who produce it) are to musicians. However, there don't seem to be many mathologists in this sense; it's quite difficult to get the sort of deep appreciation for mathematics that one needs without actually doing mathematics, or so it seems to me.

By the way, Tetris is also a traditional Russian board game involving vodka. (Not really. And the amounts of vodka that this description invokes would kill even a Russian.)

What is power?

From Walter Lewin's 8.01 Lecture 14, about 14 minutes in:

"What is power? Power is work that is done in a certain amount of time. dw/dt, if w is the work, is the instantaneous power at time t. We also know power in terms of political power. That's very different. Political power, you can do no work at all in a lot of time and you have a lot of power. Here in physics, life is not that easy."

This reminds me of the classic claim that, say, bricklayers do more work than lawyers, because work is force times distance, and bricklayers are exerting much larger forces when they lift things, since bricks are heavier than pieces of paper. This is a sort of translingual pun, the two languages being English and scientific jargon.

Scaling arguments

There's a classic argument, supposedly due to Galileo, for why mice and elephants are shaped differently, and there are no organisms that have the general mammalian body plan but are a hundred feet tall.

The argument goes as follows: assume that all animals have roughly similar skeletons. Consider the femur (thigh bone), which supports most of the weight of the body. The length of the femur, l, is proportional to the animal's linear size (height or length), S. The animal's mass, m, is proportional to the cube of the size, S³. Now, consider a cross-section of the bone; say the bone has cross-sectional area d. The pressure in the bone is proportional to the mass divided by the cross-sectional area, or l³/d². Bones are probably not much thicker than they have to be (that's how evolution works), and have roughly the same composition across organisms. So l³/d² is a constant, and so d is proportional to l^3/2 or to s^3/2. SSimilar arguments apply to other bones. So the total volume of the skeleton, if bones are asusmed to be cylindrical, scales like S^7/2 -- and so the proportion of the animal taken up by the skeleton scales like S^1/2. What matters here is that if S is large enough then the entire animal must be made up of bone! And that just can't happen.

Unfortunately, the 3/2 scaling exponent isn't true, as I learned from Walter Lewin's 8.01 (Physics I) lecture available at MIT's OpenCourseWare. It's one of the canonical examples of dimensional analysis... but it turns out that it just doesn't hold up. I suspect, although I can't confirm, that this is because elephant bones are actually substantially different from mouse bones. It looks like for actual animals, d scales with l (or perhaps like l^α for some α slightly greater than 1), not with l^3/2. Lewin uses this as an example of dimensional analysis; he also predicts that the time that it takes an apple to fall from a height is proportional to the square root of that height, which is true, but that's such a familiar result that it seems boring.

(Watching the 8.01 lecture is interesting. The videotaped lectures available on OCW are from the fall of 1999, which is two years before I entered MIT; occasionally the camera operator pans to the crowd of students, and a few of them look vaguely familiar.)

P. S. By the way, this blog is six months old. Thanks for reading!

Edited, Wednesday, 9:14 AM: Mark Dominus points out that the argument is in fact due to Galileo, and can be found in his Discourses on Two new Sciences. This is available online in English translation.

The binomial pinball machine

The Binomial Pinball Machine, from Kim Moser's repository of videos of Commodore 64 programs, via the Statistics Consulting Blog.



It's too bad the video stops before you actually start seeing a decent approximation to the normal distribution.

By the way, I almost named this blog quincunx, which is a name for this machine, but I decided that people might have trouble spelling it.

A Mobius strip video

A video about the Mobius strip, with an interesting choice of background music.

This video illustrates certain classical facts about the Mobius strip:
- if you draw a line down the middle of the Mobius strip, it runs down "both sides" of it (by which I mean both sides of the original sheet of paper; the Mobius strip isn't orientable)
- if you cut along that line, you end up with a long twisted strip;
- if you cut along a line one-third of the way from one of the edges of the original strip of paper to the other, you get two linked strips, of equal thickness but different lengths;
and another fact I didn't know:
- if you cut along a line one-quarter of the way from one of the original edges to the other, you get three linked strips (I assume at least one of them is Mobius), but now one is twice as wide as the other two. (But I think there's something trickier going on with the cutting here; it's hard to get a good look.)

Also, breaking the underwater one-handed Rubik's-cube-solving record. I can't make this stuff up.

(These are both from is from sciencehack, which is a site for science videos that claims that "every science video on ScienceHack is screened by a scientist to verify its accuracy and quality". Apparently they don't host the videos; they just index other people's science-related videos.) Here is their index of mathematics videos.0

Moebius transformations revealed

Moebius transformations revealed, from YouTube, via MathTrek -- a visual demonstration of the folk theorem that Möbius transformations correspond to moving around the Riemann sphere in various ways. By Douglas Arnold and Jonathan Rogness.

The Menger sponge

Zooming in and out on the Menger sponge, from Microsiervos.



You always hear that fractal objects are "self-similar" (that's what makes them fractal) but it's hard to get your head around that if you're looking at an ordinary drawing, since intuitively you think that the "small" parts are somehow qualitatively different than the "big" parts. But that's not really true. And this video illustrates that quite vividly -- as you zoom into the Menger sponge it very clearly looks the same on smaller and smaller scales.

On the infinitude of primes

From Microsiervos: two videos: "Cómo Arquímedes calculó el volumen de una esfera" (5:52) and "Sobre la suma de los recíprocos de los números primos" 11:46 (That is, "how Archimedes calculated the volume of a sphere" and "On the sum of the reciprocals of prime numbers")

There's a bit of an apparent error. Translating the slide that's on the screen at 3:00 in the second video:
Proof
If P [the set of primes] is finite, P := {p_1, p_2, ..., p_n} in increasing order.
Let us form the number p_1 p_2 ... p_n + 1.
This number is greater than p_n, therefore it is not prime.
It is not divisible by p_1, nor by p_2, ..., nor by p_n, therefore it is prime.
Contradiction.

(There's nothing in what the speaker says that contradicts this; it's just easier for me to translate from the written text than the speaker.)

Indeed, the original post at Microsiervos points this out:

La demostración de Euclides de que hay infinitos primos que se presenta es incompleta: el 30031 sería un contraejemplo (se construye como 2 × 3 × 5 × 7 × 11 × 13 + 1 pero aun no siendo divisible entre 2, 3, 5, 7, 11, 13 no es primo: 59 × 509). Así que faltaría añadir «… o bien [el número creado] está compuesto por la multiplicación de otros números primos mayores que los de la lista original».

That is,

Euclid's demonstration that there are infinitely many primes, as presented here, is incomplete; 30031 would be a counterexample (it can be constructed as 2 × 3 × 5 × 7 × 11 × 13 + 1, which despite not being divisible by 2, 3, 5, 7, 11, and 13 is not prime: 59 × 509. One must add: "or else [the created number] is composite and made of oter prime numbers larger than those from the original list.

I'd claim that this isn't really an error; we've assumed there are no other primes, so the number p₁...p_n+1 can't be divisible by some other primes. It's hard to keep things straight because we all know there are infinitely many primes.

This also includes the classic demonstration that the sum 1 + 1/2 + 1/3 + 1/4 + ... is infinite, which leads to Euler's analytic proof of the infinitude of primes (roughly, the number of primes "is ζ(1)") before eventually getting to the proof that the sum of the reciprocals of primes diverges. (Wikipedia says that means that the primes are a large set, which seems like a good name, but I don't know if people actually use this term.) Also see this English Wikipedia article for a lot of the content of the second video.

Oh, and ζ in (Castilian) Spanish apparently sounds the same as θ in (American) English. How do Castilians pronounce θ?

Some mathematical videos

Some videos:

Meep's Math Matters is a new series of videos that look at some simple mathematics. (You don't see Meep; you hear her voice and see her writing on an imaginary "board".) The first two installments are available on YouTube, and embedded below. The first one looks at the standard trick for summing an arithmetic series, and includes one of about a zillion retellings of the Gauss 1 + 2 + ... + 100 = 5050 anecdote; if you want to read more such retelings go here.)



The second installment shows how to sum consecutive even and odd numbers:



From Science after Sunclipse: all of first-year physics (a ridiculously fast-pased explanation of derivatives and integrals, the basic conservation laws, and so on) in nine minutes! This is from Caltech's The Mechanical Universe, a series of computer animations by Jim Blinn.



and The story of π, from Caltech's Project Mathematics around the same time; the animation is supposedly due to Blinn, and the video is credited to Tom Apostol (the number theorist). From this, I learned that "one of the most practical uses of supercomputers is calculating a billion digits of π." Some other videos I found at Google Video include The Law of Large Numbers (which includes a wonderful animation illustrating the law of large numbers in terms of random walks, set to some sort of futuristic music...) and The Optiverse, by John M. Sullivan, George Francis and Stuart Levy, showing an optimal (in some energetic sense) way to turn a sphere inside out! More information is available here. Mostly it is good for the pretty pictures.