Last week I wrote about confusing coffee pricing: Wawa, a Philadelphia-area convenience store chain, charges $1.25 for 32 ounces of coffee and $2.99 for 64 ounces. $2.99 is more than twice $1.25. Various commenters pointed out other counterintuitive pricing (train or airline fares that don't obey the triangle inequality, for example). Paul Soldera pointed out in a comment that the reason for this may just be that there aren't that many mathematicians out there, and $3 for 64 ounces of coffee sounds like a bargain to most people.

Paul Soldera may be right -- but I discovered another candidate explanation for this pricing today. Namely, I took a closer look at a sign (at a different Wawa from the one I normally go to), and it said that the 64-ounce "includes supplies". In other words, they're not selling this as a giant cup of coffee for one person to drink, but as something from which you can pour multiple cups for multiple people. Thus, they provide the cups, and perhaps other coffee paraphernalia as well.

## 30 April 2008

## 29 April 2008

### "New York geometry"

From Kitchen Table Math, I found out that there is a textbook entitled "New York Geometry". It is, unsurprisingly, a textbook for high school geometry to be used in the state of New York. The state of New York has something called the Regents Examinations which are, roughly speaking, standardized final exams in certain subjects. (Not being too familiar with the system, I don't want to say more.)

But is geometry really so different in New York than in other states that it needs its own special book? If it is, I can't tell from the table of contents of the text; it sounds like standard high school geometry.

But is geometry really so different in New York than in other states that it needs its own special book? If it is, I can't tell from the table of contents of the text; it sounds like standard high school geometry.

### Everybody (okay, not *everybody*) knows a racist

From The Dilbert Blog (by Scott Adams), on the fact that most people are not racists but most people say they

Still, it's a good point; we are not our friends, and our friends can believe different things than we do, and that's not a problem. (Incidentally, Barack Obama is not Jeremiah Wright.)

*know*racists, and how it affects the upcoming presidential election:The other inference is something I call math. If there are ten friends, and only one is a racist, then it is true that 90 percent are not racists while everyone knows someone who is. It’s that one guy.This isn't quite true -- if the average person has ten friends, and ten percent of people are racist, then the average person has one racist friend. But even if friends are randomly distributed, the probability that I have no racist friends is (0.9)

^{10}or about 35 percent. And friends*aren't*randomly distributed. Most people tend to have people like themselves as friends. So the probability of having no racist friends is higher.Still, it's a good point; we are not our friends, and our friends can believe different things than we do, and that's not a problem. (Incidentally, Barack Obama is not Jeremiah Wright.)

## 27 April 2008

### The English language is not equipped for metric spaces

From Lydia Millet's novel Oh Pure And Radiant Heart, in which Robert Oppenheimer, Enrico Fermi and Leo Szilard, three of the chief minds behind the Manhattan Project, find themselves in contemporary America (p. 290):

If the scientist is saying that the distance in some abstract civilizedness-space (here caricatured by the unit interval) from Westerners to Marshall Islanders is less than the distance from Westerners to mice, then yes. The last clause could be rephrased as "it is nevertheless true that they are more like us than mice

Of course, in this particular case, I would claim the scientist intended the second interpretation; regardless of what one thinks about how civilized various groups of humans are, it is obvious that all such groups are more civilized than mice. (I apologize to fans of the

Also, this shouldn't need saying, but the value 1/3 above is

To my ear, something about this last sentence is ambiguous -- and I suspect that a mathematician is more likely to spot this ambiguity than an average person. Let's assume that degrees of civilizedness fall on a scale from 0 to 1, with mice at 0 and Westerners at 1. Say that we have a magical civilizedness-measuring meter, and the Marshall Islanders fall at 1/3. Is the scientist's statement true?

Scientists at the Atomic Energy Commission took advantage of the testing in the Marshall Islands to study the effects of radiation on people.

In 1956, at an AEC meeting, one official admitted that Rongelap was the most contaminated place on earth. He said of the Marshall Islanders, reportedly without irony, "While it is true that these people do not live, I would say, the way Westerners do —civilized people— it is nevertheless true that they are more like us than mice.

If the scientist is saying that the distance in some abstract civilizedness-space (here caricatured by the unit interval) from Westerners to Marshall Islanders is less than the distance from Westerners to mice, then yes. The last clause could be rephrased as "it is nevertheless true that they are more like us than mice

**are like us**." But if the scientist is saying that the distance from Westerners to Marshall Islanders is less than the distance from Marshall Islanders to mice, then it's not true if the islanders fall at 1/3; to force this interpretation, the original sentence could be rephrased as "it is nevertheless true that they are more like us than**they are like**mice." (I make no claim that these are the most elegant possible rephrasings, just that they clear up the ambiguity.)Of course, in this particular case, I would claim the scientist intended the second interpretation; regardless of what one thinks about how civilized various groups of humans are, it is obvious that all such groups are more civilized than mice. (I apologize to fans of the

*Hitchhiker's Guide*series.) So there is no need to even make the statement under the first interpretation! There is not much point in telling people something they already know.Also, this shouldn't need saying, but the value 1/3 above is

*entirely*hypothetical, and I do not mean to make any statements about the civilizedness of actual groups of life forms.## 25 April 2008

### Fractions are not about pizza

Study Suggests Math Teachers Scrap Balls and Slices, from today's New York Times.

The

The

The

*Times*article is about a study reported on in today's issue of*Science*(Jennifer A. Kaminski, Vladimir M. Sloutsky, Andrew F. Heckler1. The Advantage of Abstract Examples in Learning Math. Science 25 April 2008: Vol. 320. no. 5875, pp. 454 - 455). Researchers taught the idea of the group**Z**_{3}to some students who weren't familiar with it; some learned it "abstractly" (the elements of the group were represented as funny-looking symbols) and some learned it "concretely" (by considering the slices in a pizza with three slices, or thirds of a measuring cup, or tennis balls in a three-ball can). It seems that the ones who learned the "abstract" version more easily picked up the rules of yet another "concrete" version (a children's game) than those who learned the original "concrete" version.The

*Science*authors claim that this is because "Compared with concrete instantiations, generic instantiations present minimal extraneous information and hence represent mathematical concepts in a manner close to the abstract rules themselves." This seems like the whole point of mathematics -- a lot of what we do as mathematicians is to strip away extraneous details of a problem while retaining those that are actually significant. If you learn about fractions by thinking about slices of pizza, perhaps you will always think that fractions are about pizza. And then whenever you hear about them, you'll think "where's lunch"?
Labels:
education,
New York Times,
Science

## 22 April 2008

### Delegate math in the Pennsylvania primary

In tonight's Pennsylvania primary, the structure of the delegate allocation heavily favors to Obama, according to cnn.com video coverage.

The argument is the following: there are 55 at-large delegates which are assigned proportionally to the popular vote in the entire state. There are also 103 delegates divided up among the 19 Congressional districts, with more heavily Democratic districts receiving more votes. (For example, the 2nd district -- mine -- gets nine delegates, which I think is the most of any district nationwide. That's basically the western half of Philadelphia.) The 9th district gets the fewest, with three; numbers for other districts are here.

Now, the formula that assigns the delegates (I can't find it right now) basically says that the number of delegates that a district gets is proportional to the number of Democratic votes in the last few elections.

So assuming turnout is stable, the outcome really isn't any different than it would be if all the delegates were assigned "at large" -- up to rounding errors from the fact that delegates are quantized, but I don't believe the rounding errors break consistently one way or the other. (Roundinf error often do.)

For a small example, consider a hypothetical state with two districts. The first district historically has a turnout of 45,000 Democrats and gets three delegates; the second district historically has a turnout of 105,000 Democrats and gets seven delegates. In addition there are five at-large delegates.

Now say the turnouts in the election are the same; and 65% of Democrats in the first district vote for Clinton, and 65% of Democrats second district vote for Obama. So the first district breaks 29,250 to 15,750 for Clinton, and the second 68,250 to 36,750 for Obama. The state as a whole goes 84,000 to 66,000 for Obama -- 56% to 44%.

Then Obama gets 4.55 delegates in the second district, 1.05 in the first, and 2.8 in the state as a whole -- guess what! He gets 8.4 delegates, 56% of the total of fifteen. (Rounding those, Obama gets nine.)

Something similar is true for the state as a whole.

If anything, the district-based allocation helps

The argument is the following: there are 55 at-large delegates which are assigned proportionally to the popular vote in the entire state. There are also 103 delegates divided up among the 19 Congressional districts, with more heavily Democratic districts receiving more votes. (For example, the 2nd district -- mine -- gets nine delegates, which I think is the most of any district nationwide. That's basically the western half of Philadelphia.) The 9th district gets the fewest, with three; numbers for other districts are here.

Now, the formula that assigns the delegates (I can't find it right now) basically says that the number of delegates that a district gets is proportional to the number of Democratic votes in the last few elections.

So assuming turnout is stable, the outcome really isn't any different than it would be if all the delegates were assigned "at large" -- up to rounding errors from the fact that delegates are quantized, but I don't believe the rounding errors break consistently one way or the other. (Roundinf error often do.)

For a small example, consider a hypothetical state with two districts. The first district historically has a turnout of 45,000 Democrats and gets three delegates; the second district historically has a turnout of 105,000 Democrats and gets seven delegates. In addition there are five at-large delegates.

Now say the turnouts in the election are the same; and 65% of Democrats in the first district vote for Clinton, and 65% of Democrats second district vote for Obama. So the first district breaks 29,250 to 15,750 for Clinton, and the second 68,250 to 36,750 for Obama. The state as a whole goes 84,000 to 66,000 for Obama -- 56% to 44%.

Then Obama gets 4.55 delegates in the second district, 1.05 in the first, and 2.8 in the state as a whole -- guess what! He gets 8.4 delegates, 56% of the total of fifteen. (Rounding those, Obama gets nine.)

Something similar is true for the state as a whole.

If anything, the district-based allocation helps

*Clinton*relative to allocation based purely on the popular vote, because new voters tend to break for Obama (or at least they have in previous contents), and districts with a lot of new voters will be slightly*undercounted*.### Confusing coffee pricing

Here in the Philadelphia area, we have an oddly-named chain of convenience stores named Wawa.

At Wawa, you can buy coffee for the following prices: $1.09, $1.19, $1.29, $1.39 for 12, 16, 20, 24 ounces respectively. This makes sense -- basically you pay $.79 for wandering around in their store taking up space and such, and then 10 cents for each four ounces of coffee.

However, things get weird if you bring your own cup (I'm talking about the "travel mug" sort here, not a paper cup). Then 12, 16, 20, 24 ounces cost $0.85, $0.95, $1.05, or $1.15 -- so far, so good. You save twenty-four cents by bringing your own cup.

32 ounces, in your own cup, is $1.25. So now they're really starting to reward you for buying in bulk -- another ten cents gets you

But then guess what happens? 64 ounces costs $2.99. That,s right -- I can fill two 32-ounce cups for $2.50, but filling one 64-ounce cup will cost $2.99. If you extrapolate the linear trend from 12, 16, 20, and 24 ounces, 64 ounces should cost $2.15. If I had a sixty-four-ounce travel mug, I'd go in there, fill it up, and try to get it filled for $2.50 just to see how the cashiers explained it.

Perhaps they're trying to say that you really just shouldn't be drinking that much coffee. I'd have to agree -- and I'm a mathematician.

Another argument is that perhaps they are attempting to discourage people from taking that much coffee because then there's less coffee for the people after them, and people won't be happy if the store runs out of coffee. This may be true -- it seems a bit doubtful, though, since a typical Wawa store might have a dozen or so pots of coffee at once, each holding 64 ounces or so.

At Wawa, you can buy coffee for the following prices: $1.09, $1.19, $1.29, $1.39 for 12, 16, 20, 24 ounces respectively. This makes sense -- basically you pay $.79 for wandering around in their store taking up space and such, and then 10 cents for each four ounces of coffee.

However, things get weird if you bring your own cup (I'm talking about the "travel mug" sort here, not a paper cup). Then 12, 16, 20, 24 ounces cost $0.85, $0.95, $1.05, or $1.15 -- so far, so good. You save twenty-four cents by bringing your own cup.

32 ounces, in your own cup, is $1.25. So now they're really starting to reward you for buying in bulk -- another ten cents gets you

*eight*more ounces.But then guess what happens? 64 ounces costs $2.99. That,s right -- I can fill two 32-ounce cups for $2.50, but filling one 64-ounce cup will cost $2.99. If you extrapolate the linear trend from 12, 16, 20, and 24 ounces, 64 ounces should cost $2.15. If I had a sixty-four-ounce travel mug, I'd go in there, fill it up, and try to get it filled for $2.50 just to see how the cashiers explained it.

Perhaps they're trying to say that you really just shouldn't be drinking that much coffee. I'd have to agree -- and I'm a mathematician.

Another argument is that perhaps they are attempting to discourage people from taking that much coffee because then there's less coffee for the people after them, and people won't be happy if the store runs out of coffee. This may be true -- it seems a bit doubtful, though, since a typical Wawa store might have a dozen or so pots of coffee at once, each holding 64 ounces or so.

Labels:
coffee,
economics,
Philadelphia

## 21 April 2008

### Mecca is the center of the earth?

Muslim call to adopt Mecca time, from BBC News. Muslim scientists are arguing that we should use Mecca time because they believe Mecca is the center of the earth.

Of course, the Earth is spherical; its center is not on the surface! The only points that we can really pick out are the north and south poles, and indeed time zones are based on meridians which pass through those two points -- and some third point.

I can't find the actual source. But it would not surprise me to learn that the argument's circular -- Mecca is the center of the earth because Muslims pray to it, and Muslims pray to it because it is the center of the earth. Can anyone confirm or refute this?

Of course, the Earth is spherical; its center is not on the surface! The only points that we can really pick out are the north and south poles, and indeed time zones are based on meridians which pass through those two points -- and some third point.

I can't find the actual source. But it would not surprise me to learn that the argument's circular -- Mecca is the center of the earth because Muslims pray to it, and Muslims pray to it because it is the center of the earth. Can anyone confirm or refute this?

## 20 April 2008

### Mathifying Gore-Lieberman 2000

Gore-Lieberman: A Hyphen Apart? Try Poles, from today's New York Times, by John M. Broder.

The article points out that since Al Gore and Joe Lieberman ran together in the 2000 presidential election, Gore has gone to the left, and Lieberman to the right; would that have happened had they been elected in 2000?

The article ends:

I don't remember having to do problems like that. This may be why I still like mathematics. (And why is it always "trains" in these problems? I suspect they may actually have been more common in a time when trains were common; if the probems were written now they would be about cars or airplanes.)

The article's not bad re: politics; this is just yet another mathification, i. e. gratuitous insertion of math where it doesn't really make sense.

The article points out that since Al Gore and Joe Lieberman ran together in the 2000 presidential election, Gore has gone to the left, and Lieberman to the right; would that have happened had they been elected in 2000?

The article ends:

Ron Klain, who was Mr. Gore’s vice presidential chief of staff, said the Gore-Lieberman schism reminded him of Algebra I.

"It’s like one of those old math problems,” he said. “Two trains leave Chicago at the same time traveling in opposite directions. How far apart are they in three hours? Very far apart."

I don't remember having to do problems like that. This may be why I still like mathematics. (And why is it always "trains" in these problems? I suspect they may actually have been more common in a time when trains were common; if the probems were written now they would be about cars or airplanes.)

The article's not bad re: politics; this is just yet another mathification, i. e. gratuitous insertion of math where it doesn't really make sense.

## 18 April 2008

### Visualizing the Sieve of Eratosthenes (in the May Notices of the AMS)

From the May

The basic idea of the article is that we can color the point (n, m) whenever m and n are positive integers and m divides n; then interesting patterns appear among lattice points in the first quadrant. Alternatively, the row y = n contains colored points at x = n, 2n, 3n, ... and in general every

One pattern that appears comes about when we mark (3,1), (6,2), (9,3), and so on, so all the lattice points on a diagonal through the origin of slope 1/3 are colored; something similar happens for every

Cox points out some mysterious-seeming

(In the interests of the sentence preceding this one being entirely correct, I will define "simple" as "things I thought of while writing this post".)

*Notices of the AMS*: Visualizing the Sieve of Eratosthenes, by David N. Cox.The basic idea of the article is that we can color the point (n, m) whenever m and n are positive integers and m divides n; then interesting patterns appear among lattice points in the first quadrant. Alternatively, the row y = n contains colored points at x = n, 2n, 3n, ... and in general every

*n*th column. The procedure generalizes the sieve of Erastosthenes; the number of marked points in the column x = n is just the number of divisors of n.One pattern that appears comes about when we mark (3,1), (6,2), (9,3), and so on, so all the lattice points on a diagonal through the origin of slope 1/3 are colored; something similar happens for every

*k*. But diagonals also appear "radiating" from points on the x-axis which are not the origin. For example, radiating out of the point (2520, 0) we have a diagonal of slope 1, which passes through the marked points (2521, 1), (2522, 2), ..., (2530, 10); this occurs because 2520 is divisible by each of 1 through 10. In general one seesCox points out some mysterious-seeming

*parabolic*patterns of the marked points; for example he mentions a left-opening parabola with vertex (17956, 134). Now, 134^{2}is 17956 -- and so the parabola constains the points (17956-x^{2}, 134 ± x) for each*x*. In fact, we have a parabola containing the points (k^{2}-x^{2}, k ± x) for each integer*k*. Cox says that no right-opening parabolas are observed; these would correspond to factorizations of the form (k^{2}+x^{2}, k ± ix) where i is the imaginary unit, but he's working over the integers! Of course, if you stare at the points you might see what you*think*are right-opening parabolas, but the appearance of those is probably just a coincidence. At this point I will wave my hands and intone the magic words "Ramsey theory". In any case, right-opening parabolas, should they exist, are certainly not given by such a simple rule.(In the interests of the sentence preceding this one being entirely correct, I will define "simple" as "things I thought of while writing this post".)

Labels:
number theory,
visualization

## 17 April 2008

### Answers to the Wall Street Journal quiz

Last week I wrote about a quiz on probability in the Wall Street Journal.

I promised I'd link to the answers when they came up; here are the answers.

For the record, I didn't send in answers.

I promised I'd link to the answers when they came up; here are the answers.

For the record, I didn't send in answers.

## 16 April 2008

### Edward Lorenz dies

Edward Lorenz, father of chaos theory and butterfly effect, dies at 90. (Link goes to MIT press release; I found out from Greg Laden's blog.)

He traditionally gave a lecture to the course on chaos, and he did when I took that class in 2003. I wish I remember what he said! I suspect it was something interesting. Steven Strogatz wrote, in his book

Strogatz wasn't at MIT when I was there -- Dan Rothman taught the class using Strogatz's text -- but Lorenz still gave an annual lecture to that class at least as late as fall of 2006. I'd like to think they did the same silly little dance.

He traditionally gave a lecture to the course on chaos, and he did when I took that class in 2003. I wish I remember what he said! I suspect it was something interesting. Steven Strogatz wrote, in his book

*Sync*,Every time I taught my chaos course, we'd go through the same ritual each year, and I'd come to look forward to it. I'd call up Professor Lorenz and invite him to give a guest lecture to the class. He'd say, with genuine puzzlement, as if it were an open question, "what should I talk about?" And I'd say, How about the Lorenz equations? "Oh, that little model?" And then, as predictable as the seasons, he'd show his face to my awestruck class, and tell us not about the Lorenz equations but about whatever he was working on then. It didn't matter. We were all there to catch a glimpse of the man who'd started the modern field of chaos theory.

Strogatz wasn't at MIT when I was there -- Dan Rothman taught the class using Strogatz's text -- but Lorenz still gave an annual lecture to that class at least as late as fall of 2006. I'd like to think they did the same silly little dance.

Labels:
chaos theory,
Lorenz,
Strogatz

### All Spanish youth have cell phones

Todos los jÃ³venes espaÃ±oles tienen mÃ³vil -- that is, all Spanish youth have cell phones. From tuexperto.com, via Microsiervos, both in Spanish.

As the authors of Microsiervos point out, this is almost certainly not true. Rather, some Spanish youth have no cell phone, and some have two or more. The average Spanish youth may have one cell phone, but the number of cell phones one has is

(And although the pictures show babies, "youth" here are people from 15 to 35.)

As the authors of Microsiervos point out, this is almost certainly not true. Rather, some Spanish youth have no cell phone, and some have two or more. The average Spanish youth may have one cell phone, but the number of cell phones one has is

*not*an indicator random variable. (That is, it doesn't take the value 0 if you have no cell phone and 1 if you have a cell phone.)(And although the pictures show babies, "youth" here are people from 15 to 35.)

## 15 April 2008

### The napkin ring problem

Keith Devlin writes about The Napkin Ring Problem in his column this month. The problem is the following: consider a sphere with a cylindrical hole drilled out from the center. What is the volume of the resulting solid? At first one thinks that this should depend on the radius of the sphere and the radius of the hole, but it turns out to depend only on the height of the resulting solid; Devlin gives the computation. As it turns out, the volume is 4π

This is a problem that is usually assigned in our calculus courses when we get to solids of revolution. The way that it's phrased in Stewart's calculus text (which I don't have at hand right now) is in two parts. The first part refers to two such napkin rings -- call them A and B -- and says that napkin ring A comes from a larger-radius sphere than B but also has a larger-radius hole; the student is asked to guess whether A or B has greater volume.

At this point the students invariably ask "what if my guess is wrong?" I laugh and tell them that it doesn't matter; the point is that they should try to think about it for a moment before they plunge into the calculations.

The second part, if I remember correctly (my officemate has the book, so I'll check this tomorrow) asks the students to express the volume as a function of the two radii and the height; often, even if they manage to get the correct answer (which is a bit tricky) they do not seem

(Incidentally, it is always amusing to look at reviews of textbooks at amazon.com; I often wonder how many of them are written by students who are disgruntled about their class and want somewhere to take it out.)

*h*^{3}/3, where*h*is the half-height of the resulting solid; this is the volume of a sphere of radius*h*. In fact, the corresponding cross-sectional areas of a "napkin ring" of half-height*h*and a sphere of radius*h*are the same, as one can see without calculus; by Cavalieri's principle the volumes are equal.This is a problem that is usually assigned in our calculus courses when we get to solids of revolution. The way that it's phrased in Stewart's calculus text (which I don't have at hand right now) is in two parts. The first part refers to two such napkin rings -- call them A and B -- and says that napkin ring A comes from a larger-radius sphere than B but also has a larger-radius hole; the student is asked to guess whether A or B has greater volume.

At this point the students invariably ask "what if my guess is wrong?" I laugh and tell them that it doesn't matter; the point is that they should try to think about it for a moment before they plunge into the calculations.

The second part, if I remember correctly (my officemate has the book, so I'll check this tomorrow) asks the students to express the volume as a function of the two radii and the height; often, even if they manage to get the correct answer (which is a bit tricky) they do not seem

*surprised*by this fact, which I was when I first saw it. Then again, it is often difficult to judge whether students are surprised by some statement made in class or on the homework; it's not the sort of thing your average college student is going to let show! (Penn students taking the calculus courses are mostly pre-professional, either in the engineering or business schools; I wonder if the future mathematicians or even the future "pure" scientists would react differently.)(Incidentally, it is always amusing to look at reviews of textbooks at amazon.com; I often wonder how many of them are written by students who are disgruntled about their class and want somewhere to take it out.)

### Fermat on FLT

...not Fermat's

A plaintext message

Anyway, the point of this post really wasn't to talk about RSA cryptography, but to point to a letter written from Fermat to Frenicle in 1640. One often hears that Fermat very rarely proved things but just made motions saying "I have a proof", but the only such remark one actually hears repeated is the famous statement of Fermat's Last Theorem:

which translates as something like:

(I don't know Latin, so I take no responsibility for the correctness of this translation.) It turns out that there's a less famous remark regarding the Little Theorem:

That is:

(The "progressions" refer to Fermat's phrasing of the theorem, in which he refers to the

*last*theorem, but Fermat's*little*theorem. When I took an introductory number theory course I was inordinately amused by this coincidence of abbreviations. Fermat's little theorem is more useful; I came across it most recently because we're explaining RSA cryptography to the Ideas in Mathematics class. For those of you who don't know it, the RSA algorithm works as follows. In order to enable people to encrypt messages to be sent to you, in such a way that only you can decrypt them, calculate three numbers*n*,*e*, and*d*. The number*n*is the product of two large primes*p*and*q*;*e*is some number relatively prime to φ(*n*) = (*p*- 1)(*q*-1);*d*is the multiplicative inverse of*e*modulo φ(*n*). Then publicize the numbers*n*and*e*; keep*d*secret. Calculating*d*from these in the obvious way requires factoring φ(*n*).A plaintext message

*T*is then encrypted as a ciphertext*C*, where C =*T*^{e}mod φ(*n*); a ciphertext message C is then decrypted by taking*C*^{d}mod φ(*n*). The reason that decryption is the inverse of encryption -- which is of course what one wants -- is Euler's theorem, which states that*a*^{φ(n)}mod*n*= 1 whenever*n*and*a*are relatively prime positive integers. Assuming*T*and φ(*n*) are relatively prime, then, applying the encryption map and then the decryption map to some plaintext message*T*gives*T*^{de}mod*n*; but*de*is one more than some multiple of φ(*n*), so that's just*T*. (This is only true when*T*and φ(*n*) are relatively prime.)Anyway, the point of this post really wasn't to talk about RSA cryptography, but to point to a letter written from Fermat to Frenicle in 1640. One often hears that Fermat very rarely proved things but just made motions saying "I have a proof", but the only such remark one actually hears repeated is the famous statement of Fermat's Last Theorem:

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

which translates as something like:

It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.

(I don't know Latin, so I take no responsibility for the correctness of this translation.) It turns out that there's a less famous remark regarding the Little Theorem:

Et cette proposition est généralement vraie en toutes progressions et en tous nombres premiers; de quoi je vous envoierois la démonstration, si je n'appréhendois d'être trop long.

That is:

This proposition [Fermat's Little Theorem] is generally true for all progressions and for all prime numbers; I would send you the demonstration if I did not fear that it were too long.

(The "progressions" refer to Fermat's phrasing of the theorem, in which he refers to the

*p*-1 term of a geometric progression*a*,*a*^{2},*a*^{3}, ... and claims that it is one more than a multiple of*p*.) The rest of the letter includes other such statements without proofs, but illustrated with copious examples. (The link I provided is to the version of David Zhao and Amanda Bergeron, which includes both the French text of the letter and their English translation; the English translation of Fermat's French I gave above is my own.)
Labels:
cryptography,
Fermat,
history of mathematics,
number theory

## 14 April 2008

### Knuth on teaching calculus

In 1998, Donald Knuth wrote a letter on using the O-notation in teaching calculus, which Alexandre Borovik has reproduced.

## 12 April 2008

### Red Bull gives you wings

So one of my favorite mathematical quotes is one that Rick Durrett (in his text

I just came across this paper from MIT's Undergraduate Seminar in Discrete Mathematics (18.304, Spring 2006), for which an anonymous student wrote some notes on simple random walks. Here we learn that a drunk man will eventually get home, but a drunk man who has had drinks containing Red Bull will not; as you know from the commercials, "Red Bull gives you wings!"

Another random walk which is transient would be that performed by a (non-flying) drunk with a time machine -- although not as viewed

(Incidentally, I almost wrote "Michiko Kauktani" there; she's a literary critic for the New York Times. It turns out that Michiko is Shizuo's daughter, which I didn't know.)

*Probability: Theory and Examples*) credits to Shizuo Kakutani: "A drunk man will eventually find his way home but a drunk bird may get lost forever." More formally, random walks on the square lattice in two dimensions return to the origin infinitely often; random walks in two dimensions with more general steps allowed, but in which the expected position at any time is still zero, return arbitrarily close to the origin infinitely often. In three dimensions this is not true. (Random walks that return arbitrarily close to the origin infinitely often are called "recurrent", the others "transient".)I just came across this paper from MIT's Undergraduate Seminar in Discrete Mathematics (18.304, Spring 2006), for which an anonymous student wrote some notes on simple random walks. Here we learn that a drunk man will eventually get home, but a drunk man who has had drinks containing Red Bull will not; as you know from the commercials, "Red Bull gives you wings!"

Another random walk which is transient would be that performed by a (non-flying) drunk with a time machine -- although not as viewed

*by*the drunk, but as viewed by an observer who is*fixed*at one point in space-time. (The model I have of space-time in my head for this problem is something like that which one sees in certain movies, where if you're not careful you can run into a copy of yourself.)(Incidentally, I almost wrote "Michiko Kauktani" there; she's a literary critic for the New York Times. It turns out that Michiko is Shizuo's daughter, which I didn't know.)

## 11 April 2008

###
Two quotes from *Principia Mathematica* (Russell and Whitehead)

"From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." -- a comment to proposition 54.43.

"The above proposition is occasionally useful." -- a comment to proposition 110.643, which proves that 1+1=2.

Quite the understatement, don't you think?

"The above proposition is occasionally useful." -- a comment to proposition 110.643, which proves that 1+1=2.

Quite the understatement, don't you think?

## 10 April 2008

### Fractal cookies

Fractal cookies, from Evil Mad Scientist Laboratories

Take nine "square cylinders" (i. e. rectangular solids which are much longer in one direction than the other two) of dough, one of which has chocolate in it.

Arrange the nine sticks in a three-by-three grid with the chocolate one in the center; squish them together so that they are one big piece of dough.

Stretch the whole thing to eight times its current length; cut into eight pieces of equal length (the length of the original piece), each of which will have a chocolate center. (This can be done by stretching to twice the length, cutting in half, and repeating twice more.)

Add a piece of chocolate dough of the same size; again arrange in a three-by-three grid with the chocolate one in the center, stretch, and cut. Then do it again. Then cut the whole thing into slices and cook.

Of course, you get the Sierpinski carpet in cookie form.

However, at the level of iteration given here, (8/9)

See also the Sierpinski gaskets made from polymer clay, which are made by a similar process. These are inferior, because they cannot be eaten.

Take nine "square cylinders" (i. e. rectangular solids which are much longer in one direction than the other two) of dough, one of which has chocolate in it.

Arrange the nine sticks in a three-by-three grid with the chocolate one in the center; squish them together so that they are one big piece of dough.

Stretch the whole thing to eight times its current length; cut into eight pieces of equal length (the length of the original piece), each of which will have a chocolate center. (This can be done by stretching to twice the length, cutting in half, and repeating twice more.)

Add a piece of chocolate dough of the same size; again arrange in a three-by-three grid with the chocolate one in the center, stretch, and cut. Then do it again. Then cut the whole thing into slices and cook.

Of course, you get the Sierpinski carpet in cookie form.

However, at the level of iteration given here, (8/9)

^{3}, or about seventy percent, of the cookie will consist of non-chocolate dough! This is sad. I recommend interchanging the chocolate and non-chocolate doughs.See also the Sierpinski gaskets made from polymer clay, which are made by a similar process. These are inferior, because they cannot be eaten.

### e day

I have been informed that tomorrow is "e day", February 71st. (In American date notation, that's 2/71; compare "π day" which is March 14th.)

In non-leap years, "e day" is still February 71st, but that's equivalent to April 12th.

It's too bad Euler wasn't born a few days earlier; he was born on April 15, 1707.

In non-leap years, "e day" is still February 71st, but that's equivalent to April 12th.

It's too bad Euler wasn't born a few days earlier; he was born on April 15, 1707.

### Dimensionally inconsistent(?) spam

Received at my Penn e-mail address today: spam with the subject line "Increase girth and inches in one easy step!"

Presumably "inches" means "inches of length". (I'll leave it to you to figure out what length they're trying to increase.)

But I couldn't help but notice that they really should have said "girth" and "length", naming both the quantities. (Or "inches" and "inches", naming both the dimensions, but that would be silly and redundant.)

Presumably "inches" means "inches of length". (I'll leave it to you to figure out what length they're trying to increase.)

But I couldn't help but notice that they really should have said "girth" and "length", naming both the quantities. (Or "inches" and "inches", naming both the dimensions, but that would be silly and redundant.)

## 09 April 2008

### You mean there are people who *don't* write everywhere?

Harnessing Biology, and Avoiding Oil, for Chemical Goods, today's New York Times. I studied a fair bit of chemistry as an undergrad, so this is of interest to me academically. Basically, a lot of synthetic goods are made out of compounds with lots of carbon, which can eventually be traced back to petroleum; as you may have noticed, petroleum and its derivatives have gotten more expensive recently. So even if you were never a chemist you should still care.

The photo at the top of the article, though, is what got my attention. It's captioned "The scientists use the glass shield as a board on which to write chemical formulas", and I feel like there's the implication that they're doing this to conserve scarce resources (coming from the captions on the other photos). No! It's just that scientists of any sort write things everywhere -- every chemistry lab I was ever in had this property. I wonder what they'd think of mathematics departments. (One professor that I know often has about four different calculations going on simultaneously on the whiteboard of his office; they overlap each other, but they're in different colors, so he can tell them apart. I can't do that.)

In the hall of the dormitory floor I lived on as an undergrad we had several blackboards. They were often filled with mathematics of one sort of another. Of course, they were also often filled with transcriptions of the silly or obscene things some of us had said. I kind of wish I'd written them down... but let's face it, they were probably pretty embarrassing and are best left where posterity

It might be interesting to see pictures of well-known mathematicians' blackboards...

The photo at the top of the article, though, is what got my attention. It's captioned "The scientists use the glass shield as a board on which to write chemical formulas", and I feel like there's the implication that they're doing this to conserve scarce resources (coming from the captions on the other photos). No! It's just that scientists of any sort write things everywhere -- every chemistry lab I was ever in had this property. I wonder what they'd think of mathematics departments. (One professor that I know often has about four different calculations going on simultaneously on the whiteboard of his office; they overlap each other, but they're in different colors, so he can tell them apart. I can't do that.)

In the hall of the dormitory floor I lived on as an undergrad we had several blackboards. They were often filled with mathematics of one sort of another. Of course, they were also often filled with transcriptions of the silly or obscene things some of us had said. I kind of wish I'd written them down... but let's face it, they were probably pretty embarrassing and are best left where posterity

*can't*see them.It might be interesting to see pictures of well-known mathematicians' blackboards...

Labels:
chemistry,
mathematical culture,
New York Times

## 08 April 2008

### A couple of links from the Monkey Writes

(The headline is in honor of something I found out from the Internet Anagram Server: "Monkey Writes" is an anagram of "New York Times". I think this is unfortunate, because the New York Times is actually a decent paper.

By the way, monkey typing is illegal in the United States, or so Jonathan Block tells us in his notes on finance. (You really don't want to know how long I've been waiting to work that into a post.)

I had to go with that headline because of today's New York Times article Cognitive dissonance in monkeys (John Tierney), in which we learn that an incorrect solution to the Monty Hall problem is apparently used in the analysis of things that happen when monkeys are given choices between certain colors of M&Ms.

Also worth checking out in today's NYT is A Disease That Allowed Torrents of Creativity (Sandra Blakeslee).

*Metro*, on the other hand, is a free paper that feels like it was written by monkeys; "Swinkey Metro" is also an anagram of "Monkey Writes". I hereby submit that the Qankees, the baseball team this blog loves to hate, play in the city of Swinkey.)By the way, monkey typing is illegal in the United States, or so Jonathan Block tells us in his notes on finance. (You really don't want to know how long I've been waiting to work that into a post.)

I had to go with that headline because of today's New York Times article Cognitive dissonance in monkeys (John Tierney), in which we learn that an incorrect solution to the Monty Hall problem is apparently used in the analysis of things that happen when monkeys are given choices between certain colors of M&Ms.

Also worth checking out in today's NYT is A Disease That Allowed Torrents of Creativity (Sandra Blakeslee).

## 07 April 2008

### A quiz on probability from the Wall Street Journal

A Numbers Guy Quiz on Probability, from (you guessed it!) The Numbers Guy at the

The hardest of the eight questions, I think, is this one:

I won't give an answer, and I ask that you don't either -- but think about it; the answer is surprising. (Bialik says he'll be giving the answers next week.)

*Wall Street Journal*. The post gives an eight-question quiz derived from problems in Leonard Mlodinow's new book, The Drunkard's Walk: How Randomness Rules Our Lives. I claim that the people who designed the cover of that book got the idea for the cover from the title of my blog; it has dice on it, and since a lot of people's conception of God is invisible, I'll claim it has God on it as well. (But I really can't complain, since my title's cotaken from Einstein anyway. Yes,*co*taken, as in I took the complement of what Einstein said; there are no arrows getting reversed involved here.) Carl Bialik, who writes that blog, will be interviewing Mlodinow next week. Mlodinow apparently has a PhD (from the title of his thesis,*The Large N Expansion in Quantum Mechanics*, I'd have to guess it's in theoretical physics) and is also a screenwriter, which probably makes for a good story in itself; if I remember I'll link to the interview when it comes out.The hardest of the eight questions, I think, is this one:

You know that a certain family has two children, and you remember that at least one is a girl with a very unusual name (that, say, one in a million females share), but you can’t recall whether both children are girls. What is the probability that the family has two girls — to the nearest percentage point?

I won't give an answer, and I ask that you don't either -- but think about it; the answer is surprising. (Bialik says he'll be giving the answers next week.)

### Can a biologist fix a radio?

Can a biologist fix a radio?, by Yuri Labeznik, via Anarchaia. This article asks a question: say biologists decided to research radios in the same way that they research things like how cells work. Then they would buy a lot of radios, classify and dissect them, eventually conclude that there was some sort of evolutionary explanation for why the antenna is really long, and so on. Much work would be duplicated, because the biologists do not have a particularly good language for communicating to each other how complex systems work. (The author compares the language used by biologists to that of stock market analysts.) Engineers, the author claims, have this problem less, because they have found standardized ways to describe such systems, simulate their workings in computers, and so on. I found the following quote interesting:

Yes, but if the biologists figure this out, and they make their students take calculus, how do I feel about that? (I actually think I feel good about it; if I'm not mistaken the biology undergrads

In biology, we use several arguments to convince ourselves that problems that require calculus can be solved with arithmetic if one tries hard enough and does

another series of experiments.

Yes, but if the biologists figure this out, and they make their students take calculus, how do I feel about that? (I actually think I feel good about it; if I'm not mistaken the biology undergrads

*already*take calculus, but they think it's unnecessary for them.)## 05 April 2008

### Dear prospective graduate student.

Dear prospective graduate student:

1. I know you're reading my blog, and you found it from my UPenn web page, because my logs tell me that your ISP is the hotel that the prospectives are staying at. (Yes, I'm surprised too that that automatically appeared at the free site I use for such things.) Welcome.

2. Your mathematical interests will change during the first year in graduate school, because a lot of subjects "feel" different at the undergraduate level than at the graduate level, and there are some things you just don't see as an undergraduate at all. (This statement about "feeling" is incredibly difficult to make precise, but two examples are probability and number theory. Probability is usually taught in a "naive" way to undergrads and in a measure-theoretic way to grad students; number theory as taught to undergrads pretty much exclusively concerns itself with reasoning that takes place in the integers, whereas at higher levels it uses Big Fancy Algebraic Machinery. In addition, it may turn out that you think you are interested in X but in reality you had a particularly good teacher of X as an undergrad which colored your perception of that field.)

3. No matter where you go, the first year of graduate school will be painful. Maybe not so much physically painful -- but you will constantly wonder "am I the one person they admitted by mistake?" It gets better. (But bear in mind that those of us who are telling you this survived, or are about to be done surviving, the first year. The people who are currently first-years and are thinking they're going to leave the program are at this point avoiding coming to campus, so they're not here talking to you.)

4. Come to Penn! Our department is not so small that you will find no professors or other students interested in what you're interested in (with a few exceptions here and there), but not so large that you will feel like you are lost. Also, we pay well enough that you won't have to live on ramen. But you have to take a lot of classes. Maybe you like that, maybe you don't, but think about it.

(To give some context: many of our prospective graduate students are in town this weekend; they're encouraged to visit now although some have visited at other times. I said #2 through #4, in varying levels of detail, many times yesterday. I hope the advice of #2 and #3 can be useful to any of my readers who are currently attempting to choose a graduate school.)

1. I know you're reading my blog, and you found it from my UPenn web page, because my logs tell me that your ISP is the hotel that the prospectives are staying at. (Yes, I'm surprised too that that automatically appeared at the free site I use for such things.) Welcome.

2. Your mathematical interests will change during the first year in graduate school, because a lot of subjects "feel" different at the undergraduate level than at the graduate level, and there are some things you just don't see as an undergraduate at all. (This statement about "feeling" is incredibly difficult to make precise, but two examples are probability and number theory. Probability is usually taught in a "naive" way to undergrads and in a measure-theoretic way to grad students; number theory as taught to undergrads pretty much exclusively concerns itself with reasoning that takes place in the integers, whereas at higher levels it uses Big Fancy Algebraic Machinery. In addition, it may turn out that you think you are interested in X but in reality you had a particularly good teacher of X as an undergrad which colored your perception of that field.)

3. No matter where you go, the first year of graduate school will be painful. Maybe not so much physically painful -- but you will constantly wonder "am I the one person they admitted by mistake?" It gets better. (But bear in mind that those of us who are telling you this survived, or are about to be done surviving, the first year. The people who are currently first-years and are thinking they're going to leave the program are at this point avoiding coming to campus, so they're not here talking to you.)

4. Come to Penn! Our department is not so small that you will find no professors or other students interested in what you're interested in (with a few exceptions here and there), but not so large that you will feel like you are lost. Also, we pay well enough that you won't have to live on ramen. But you have to take a lot of classes. Maybe you like that, maybe you don't, but think about it.

(To give some context: many of our prospective graduate students are in town this weekend; they're encouraged to visit now although some have visited at other times. I said #2 through #4, in varying levels of detail, many times yesterday. I hope the advice of #2 and #3 can be useful to any of my readers who are currently attempting to choose a graduate school.)

**edit, April 8, 2:29 pm:**this post is also being discussed at Secret Blogging Seminar and Jordan Ellenberg's Quomodocumque. SBS in particular has some lively commenting going on.### Cookie Monster speaks

Cookie Monster says: "They don't call the vampire with math fetish monster, and me pretty sure he undead and drinks blood."

### Smale's problems

A lot of people refer to the Clay Mathematics Institute's seven "Millennium Prize Problems" as an analogue of Hilbert's problems for the 21st-century.

In 1998, Stephen Smale produced a list (of 18 problems) as well. (There is significant overlap with the Clay problems: both lists contain Riemann, P =? NP, Poincaré, and Navier-Stokes.) Two of them are Hilbert problems (the Riemann hypothesis and Hilbert's 16th problem). The list seems a bit biased, though, in that Smale made contributions to many of the problems mentioned; Smale acknowledges this as one of the criteria for forming his list, and the document isn't meant to stand alone; the essay was written in response to a query of Vladimir Arnold, and Smale was not the only person Arnold asked. One has to wonder if it would be possible for anyone to really be able to survey

In Smale's discussion of the Poincaré conjecture, after pointing out that a big part of the importance of the Poincaré conjecture is that it helped make manifolds respectable objects to study in their own right,he states:

This introduces the discussion of P =? NP, although the reason people study algorithms is not to answer that question; but one often hears statements like Smale's statement on Poincaré's conjecture, or statements that Fermat's Last Theorem is more important for the development in number theory that it spurred than for the result itself.

In 1998, Stephen Smale produced a list (of 18 problems) as well. (There is significant overlap with the Clay problems: both lists contain Riemann, P =? NP, Poincaré, and Navier-Stokes.) Two of them are Hilbert problems (the Riemann hypothesis and Hilbert's 16th problem). The list seems a bit biased, though, in that Smale made contributions to many of the problems mentioned; Smale acknowledges this as one of the criteria for forming his list, and the document isn't meant to stand alone; the essay was written in response to a query of Vladimir Arnold, and Smale was not the only person Arnold asked. One has to wonder if it would be possible for anyone to really be able to survey

*all*of mathematics intelligently in the way I'm told Hilbert did. (I've read Hilbert's address but I don't know enough of the history to be able to assess whether it really covers all of mathematics at the time.)In Smale's discussion of the Poincaré conjecture, after pointing out that a big part of the importance of the Poincaré conjecture is that it helped make manifolds respectable objects to study in their own right,he states:

I hold the conviction that there is a comparable phenomenon today in the notion of a "polynomial time algorithm". Algorithms are becoming worthy of analysis in their own right, not merely as a means to solve other problems. Thus I am suggesting that as the study of the set of solutions of an equation (e.g. a manifold) played such an important role in 20th century mathematics, the study of finding the solutions (e.g. an algorithm) may play an equally important role in the next century.

This introduces the discussion of P =? NP, although the reason people study algorithms is not to answer that question; but one often hears statements like Smale's statement on Poincaré's conjecture, or statements that Fermat's Last Theorem is more important for the development in number theory that it spurred than for the result itself.

Labels:
mathematical culture,
Smale

## 03 April 2008

### The uniform distribution as a sum?

Yesterday, I was asked the following question: the sum of two uniformly distributed random variables with the same support has a triangular distribution. Is there a random variable

I don't know the answer, but I started thinking as follows. First, it's enough to show that there aren't independent identically distributed

At this point I think one wants to use Bochner's theorem, which says that the functions which are characteristic functions of measures on the real line are exactly the positive definite functions -- but how does one show that this function is positive definite?

The other thing to do is to look at the discrete analogue; consider the probability generating function of a random variable which is uniformly distributed on the set {0, 1, ...,

But what about the continuous case?

*X*such that*X + Y*has a distribution which is uniform, where*X*and*Y*are independent and identically distributed?I don't know the answer, but I started thinking as follows. First, it's enough to show that there aren't independent identically distributed

*X, Y*, such that*X*+*Y*has a distribution uniform on [-1, 1]; linearly scaling gets the general result. Now, the characteristic function of a uniform distribution on [-1, 1] is*φ*(*t*) = (sin*t*)/*t*. The characteristic function of*X*+*Y*is the product of the characteristic functions of*X*and*Y*. (If you're more familiar with analysis than probability, note that characteristic functions are basically Fourier transforms, and the probability density function of*X*+*Y*is the convolution of the probability density functions of*X*and*Y*.) Thus, if*X*exists it has characteristic function*ψ*(*t*) = [(sin*t*)/*t*]^{1/2}-- this is already a bit problematic, because we want ψ to be continuous, but even with that restriction we still have to specify which square root is being taken on each of the intervals ... [-3π, -2π], [-2π, -π], [-π, π], [π, 2π], [2π, 3π] ... (Informally, we have to make a new choice every time (sin*t*)/*t*goes through 0.At this point I think one wants to use Bochner's theorem, which says that the functions which are characteristic functions of measures on the real line are exactly the positive definite functions -- but how does one show that this function is positive definite?

The other thing to do is to look at the discrete analogue; consider the probability generating function of a random variable which is uniformly distributed on the set {0, 1, ...,

*n*-1}. This is χ(*x*) = (1+x+x^{2}+...+x^{n-1})/n. Now, if this random variable were the sum of two independent identically distributed random variables, its p.g.f. would be the square of a polynomial with positive real coefficients. It's not.But what about the continuous case?

## 02 April 2008

### The unexamined life?

From Bill James answers all your baseball questions, a long interview posted at the

*Freakonomics*blog:Q: Has looking at the numbers prevented you from actually just enjoying a summer day at the ballpark? Have we all forgotten the randomness of human ballplayers? By reducing players to just their numbers can we lose sight of the intangibles such as teamwork, friendships, and desire.This is yet another example of the "people who think about things are strictly better off than people who don't" meme -- roughly speaking, the usual justification for this is that we can turn off the thinking when we want to. But can we? I know I can't just turn off the part of my brain that is constantly counting things or figuring odds of things, and there are moments when that

A: Does looking at pretty women prevent one from experiencing love? Life is complicated. Your efforts to compartmentalize it are lame and useless.

*does*hurt my quality of life. I think in the end I come out ahead -- and most mathematicians probably would agree with me, otherwise they wouldn't be mathematicians -- but it is not so simple.## 01 April 2008

### Mathematical April Fool's hoaxes

The Museum of Hoaxes has a list of the top 100 April Fool's hoaxes of all time.

Of mathematical interest:

By the way, The Mandelbrot monk was a hoax, which I knew when I posted it; John Armstrong has debunked it. (A commenter pointed out that someone from now perhaps could have

Of mathematical interest:

- #7:Alabama changes the value of π (to exactly 3, which is supposedly the "Biblical value" -- but in interpreting the relevant verse of the Bible (2 Chronicles 4:2) one has to think about measurement error.
- #8: The left-handed Whopper, which had its condiments rotated 180 degrees for the benefit of the left-handed customers. But Whoppers are rotationally symmetric anyway! If you want a left-handed Whopper and rotate it 180 degrees. You could have a mirror-image whopper, but you wouldn't be able to digest it because the vast majority of the molecules in it would be enantiomers of what your body is set up to digest.
- #30: Operation Parallax, in which it's claimed that somehow Britain ended up two days ahead because of all the time changes.

By the way, The Mandelbrot monk was a hoax, which I knew when I posted it; John Armstrong has debunked it. (A commenter pointed out that someone from now perhaps could have

*explained*to a medieval monk how to compute the Mandelbrot set, which may be true, in the same sense that we can program computers to do something. But that wouldn't make the article any more true, unless one wants to posit time machines.)
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