All set

How long is the coastline of Britain?

Answer: as long as you want it to be.

This is not some kind of abstruse joke, and if it sounds like it, blame the mathematicians.  This is what's known as the coastline paradox, which is not so much a paradox as it is the property of anything that is a fractal.  Fractals are patterns that never "smooth out" when you zoom in on them; no matter how small a piece you magnify, it still has the same amount of bends and turns as the larger bit did.

And coastlines are like that.  Consider measuring the coastline of Britain by placing dots on the coast one hundred kilometers apart -- in other words, using a straight ruler one hundred kilometers long.  If you do this, you find that the coastline is around 2,800 kilometers long.

[Image licensed under the Creative Commons Britain-fractal-coastline-100km , CC BY-SA 3.0]

But if your ruler is only fifty kilometers long, you get about 3,400 kilometers -- not an insignificant difference.

[Image licensed under the Creative Commons Britain-fractal-coastline-50km, CC BY-SA 3.0]

The smaller your ruler, the longer your measurement of the coastline.  At some point, you're measuring the twists and turns around every tiny irregularity along the coast, but do you even stop there?  Should you curve around every individual pebble and grain of sand?

At some point, the practical aspects get a little ridiculous.  The movement of the ocean makes the exact position of the coastline vague anyhow.  But with a true fractal, we get into one of the weirdest notions there is: infinity.  True fractals, such as the ones investigated by Benoit B. Mandelbrot, have an infinite length, because no matter how deeply you plunge into them, they have still finer structure.

Oh, by the way: do you know what the B. in "Benoit B. Mandelbrot" stands for?  It stands for "Benoit B. Mandelbrot."

Thanks, you're a great audience.  I'll be here all week.

The idea of infinity has been a thorn in the side of mathematicians for as long as anyone's considered the question, to the point that a lot of them threw their hands in the air and said, "the infinite is the realm of God," and left it at that.  Just trying to wrap your head around what it means is daunting:

Teacher: Is there a largest number?
Student: Yes. It's 10,732,210.
Teacher: What about 10, 732,211?
Student: Well, I was close.

It wasn't until German mathematician Georg Cantor took a crack at refining what infinity means -- and along the way, created set theory -- that we began to see how peculiar it really is.  (Despite Cantor's genius, and the careful way he went about his proofs, a lot of mathematicians of his time dismissed his work as ridiculous.  Leopold Kronecker called Cantor not only "a scientific charlatan" and a "renegade," but "a corrupter of youth"!)

Cantor started by defining what we mean by cardinality -- the number of members of a set.  This is easy enough to figure out when it's a finite set, but what about an infinite one?  Cantor said two sets have the same cardinality if you can find a way to put their members into a one-to-one correspondence in a well-ordered fashion without leaving any out, and that this works for infinite sets as well as finite ones.  For example, Cantor showed that the number of natural numbers and the number of even numbers is the same (even though it seems like there should be twice as many natural numbers!) because you can put them into a one-to-one correspondence:

1 <-> 2
2 <-> 4
3 <-> 6
4 <-> 8
etc.

Weird as it sounds, the number of fractions (rational numbers) has exactly the same cardinality as well -- there are the same number of possible fractions as there are natural numbers.  Cantor proved this as well, using an argument called Cantor's snake:

Because you can match each of them to the natural numbers, starting in the upper left and proceeding along the blue lines, and none will be left out along the way, the two sets have exactly the same cardinality.

It was when Cantor got to the real numbers that the problems started.  The real numbers are the set of all possible decimals (including ones like π and e that never repeat and never terminate).  Let's say you thought you had a list (infinitely long, of course) of all the possible decimals, and since you believe it's a complete list, you claimed that you could match it one-to-one with the natural numbers.  Here's the beginning of your list:

7.0000000000...
0.1010101010....
3.1415926535...
1.4142135623...
2.7182818284...

Cantor used what is called the "diagonal argument" to show that the list will always be missing members -- and therefore the set of real numbers is not countable.  His proof is clever and subtle.  Take the first digit of the first number in the list, and add one.  Do the same for the second digit of the second number, the third digit of the third number, and so on.  (The first five digits of the new number from the list above would be 8.2553...)  The number you've created can't be anywhere on the list, because it differs from every single number on the list by at least one digit.

So there are at least two kinds of infinity; countable infinities like the number of natural numbers and number of rational numbers, and uncountable infinities like the number of real numbers.  Cantor used the symbol aleph null -- ℵ₀ -- to represent a countable infinity, and the symbol c (for continuum) to represent an uncountable infinity.

Then there's the question of whether there are any types of infinity larger than ℵ₀ but smaller than c.  The claim that the answer is "no" is called the continuum hypothesis, and proving (or disproving) it is one of the biggest unsolved problems in mathematics.  In fact, it's thought by many to be an example of an unprovable but true statement, one of those hobgoblins predicted by Kurt Gödel's Incompleteness Theorem back in 1931, which rigorously showed that a consistent mathematical system could never be complete -- there will always be true mathematical statements that cannot be proven from within the system.

So that's probably enough mind-blowing mathematics for one day.  I find it all fascinating, even though I don't have anywhere near the IQ necessary to understand it at any depth.  My brain kind of crapped out somewhere around Calculus 3, thus dooming my prospects of a career as a physicist.  But it's fun to dabble my toes in it.

Preferably somewhere along the coastline of Cornwall.  However long it actually turns out to be.

****************************************

Weird math

When I was in Calculus II, my professor, Dr. Harvey Pousson, blew all our minds.

You wouldn't think there'd be anything in a calculus class that would have that effect on a bunch of restless college sophomores at eight in the morning.  But this did, especially in the deft hands of Dr. Pousson, who remains amongst the top three best teachers I've ever had.  He explained this with his usual insight, skill, and subtle wit, watching us with an impish grin as he saw the implications sink in.

The problem had to do with volumes and surface areas.  Without getting too technical, Dr. Pousson asked us the following question. If you take the graph of y = 1/x:

And rotate it around the y-axis (the vertical bold line), you get a pair of funnel-shapes.  Not too hard to visualize.  The question is: what are the volume and surface area of the funnels?

Well, calculating volumes and surface areas is pretty much the point of integral calculus, so it's not such a hard problem.  One issue, though, is that the tapered end of the funnel goes on forever; the red curves never strike either the x or y-axis (something mathematicians call "asymptotic").  But calc students never let a little thing like infinity stand in the way, and in any case, the formulas involved can handle that with no problem, so we started crunching through the math to find the answer.

And one by one, each of us stopped, frowning and staring at our papers, thinking, "Wait..."

Because the shapes end up having an infinite surface area (not so surprising given that the tapered end gets narrower and narrower, but goes on forever) -- but they have a finite volume.

I blurted out, "So you could fill it with paint but you couldn't paint its surface?"

Dr. Pousson grinned and said, "That's right."

We forthwith nicknamed the thing "Pousson's Paint Can."  I only found out much later that the bizarre paradox of this shape was noted hundreds of years ago, and it was christened "Gabriel's Horn" by seventeenth-century Italian physicist and mathematician Evangelista Torricelli, who figured it was a good shape for the horn blown by the Archangel Gabriel on Judgment Day.

There are a lot of math-phobes out there, which is a shame, because you find out some weird and wonderful stuff studying mathematics.  I largely blame the educational system for this -- I was lucky enough to have a string of fantastic, gifted elementary and middle school math teachers who encouraged us to play with numbers and figure out how it all worked, and I came out loving math and appreciating the cool and unexpected bits of the subject.  It's a pity, though, that a lot of people have the opposite experience.  Which, unfortunately, is what happened with me in my elementary and middle school social studies and English classes -- with predictable results.

So math has its cool bits, even if you weren't lucky enough to learn about 'em in school.  Here are some short versions of other odd mathematical twists that your math teachers may not have told you about.  Even you math-phobes -- try these on for size.

1. Fractals

A fractal is a shape that is "self-similar;" if you take a small piece of it, and magnify it, it looks just like the original shape did.  One of the first fractals I ran into was the Koch Snowflake, invented by Swedish mathematician Helge von Koch, which came from playing around with triangles.  You take an equilateral triangle, divide each of its sides into three equal pieces, then take the middle one and convert it into a (smaller) equilateral triangle. Repeat. Here's a diagram with the first four levels:

And with Koch's Snowflake -- similar to Pousson's Paint Can, but for different reasons -- we end up with a shape that has an infinite perimeter but a finite area.

Fractals also result in some really unexpected patterns coming out of perfectly ordinary processes.  If you have eight minutes and want your mind completely blown, check out how what seems like a completely random dice-throwing protocol generates a bizarre fractal shape called the Sierpinski Triangle.  (And no, I don't know why this works, so don't ask.  Or, more usefully, ask an actual mathematician, who won't just give you what I would, which is a silly grin and a shrug of the shoulders.)

2. The Four-Color-Map Theorem

In 1852, a man named Francis Guthrie was coloring in a map of the counties of England, and noticed that he could do the entire map, leaving no two adjacent counties the same color, using only four different colors. Guthrie wondered if that was true of all maps.

Turns out it is -- something that wasn't proven for sure until 1976.

Oh, but if you're talking about a map printed onto a Möbius Strip, it takes six colors.  A map printed on a torus (donut) would take seven.

Once again, I don't have the first clue why.  Probably explaining how it took almost a hundred years to prove. But it's still pretty freakin' cool.

3. Brouwer's Fixed-Point Theorem

In the 1950s, Dutch mathematician Luitzen Brouwer came up with an idea that -- as bizarre as it is -- has been proven true.  Take two identical maps of Scotland.  Deform one any way you want to -- shrink it, expand it, rotate it, crumple it, whatever -- and then drop it on top of the other one.

Brouwer said that there will be one point on the deformed copy of the map that is exactly on top of the corresponding point on the other map.

[Nota bene: it works with any map, not just maps of Scotland.  I just happen to like Scotland.]

It even works on three dimensions.  If I stir my cup of coffee, at any given time there will be at least one coffee molecule that is in exactly the same position it was in before I stirred the cup.

Speaking of which, all this is turning my brain to mush.  I think I need to get more coffee before I go on to...

4. The types of infinity

You might think that infinite is infinite.  If something goes on forever, it just... does.

Turns out that's not true.  There are countable infinities, and uncountable infinities, and the latter is much bigger than the former.

Infinitely bigger, in fact.

Let's define "countable" first.  It's simple enough; if I can uniquely assign a natural number (1, 2, 3, 4...) to the members of a set, it's a countable set.  It may go on forever, but if I took long enough I could assign each member a unique number, and leave none out.

So, the set of natural numbers is itself a countable set.  Hopefully obviously.

So is the set of odd numbers.  But here's where the weirdness starts.  It turns out that the number of natural numbers is exactly the same as the number of odd numbers.  You may be thinking, "Wait... that can't be right, there has to be twice as many natural numbers as odd numbers!"  But no, because you can put them in a one-to-one correspondence and leave none out:

1-1
2-3
3-5
4-7
5-9
6-11
7-13
etc.

So there are exactly the same number in both sets.

Now, what about real numbers?  The real numbers are all the numbers on the number line -- i.e. all the natural numbers plus all of the possible decimals in between.  Are there the same number of real and natural numbers?

Nope.  Both are infinite, but they're different kinds of infinite.

Suppose you tried to come up with a countable list of real numbers between zero and one, the same as we came up with a countable list of odd numbers above.  (Let's not worry about the whole number line, even.  Just the ones between zero and one.)  As I mentioned above, if you can do a one-to-one correspondence between the natural numbers and the members of that list, without leaving any out, then you've got a countable infinity. So here are a few members of that list:

0.1010101010101010...
0.3333333333333333...
0.1213141516171819...
0.9283743929178394...
0.1010010001000010...
0.13579111315171921...

And so forth.  You get the idea.

German mathematician Georg Cantor showed that no matter what you do, your list will always leave some out.  In what's called the diagonal proof, he said to take your list, and create a new number -- by adding one to the first digit of the first number, to the second digit of the second number, to the third digit of the third number, and so on.  So using the short list above, the first six decimal places will be:

0.242413...

This number can't be anywhere on the list.  Why?  Because its first digit is different from the first number on the list, the second digit is different from the second number on the list, the third digit is different from the third number of the list, and so forth.  And even if you just artificially add that new number to the end of the list, it doesn't help you, because you can just do the whole process again and generate a new number that isn't anywhere on the list.

So there are more numbers between zero and one on the number line than there are natural numbers.  Infinitely more.

5. Russell's Paradox

I'm going to end with one I'm still trying to wrap my brain around.  This one is courtesy of British mathematician Bertrand Russell, and is called Russell's Paradox in his honor.

First, let's define two kind of sets:

• A set is normal if it doesn't contain itself.  For example, the "set of all trees on Earth" is normal, because the set itself is not a tree, so it doesn't contain itself.
• A set is abnormal if it contains itself.  The "set of everything that is not a tree" is abnormal, because the set itself is not a tree.

Russell came up with a simple idea: he looked at "the set of all possible normal sets."  Let's call that set R.  Now here's the question:

Is R normal or abnormal?

Thanks, I'll show myself out.

****************************************

Weird math

When I was in Calculus II, my professor, Dr. Harvey Pousson, blew all our minds.

You wouldn't think there'd be anything in a calculus class that would have that effect on a bunch of restless college sophomores at eight in the morning.  But this did, especially in the deft hands of Dr. Pousson, who remains amongst the top three best teachers I've ever had.  He explained this with his usual insight, skill, and subtle wit, watching us with an impish grin as he saw the implications sink in.

The problem had to do with volumes and surface areas.  Without getting too technical, Dr. Pousson asked us the following question.  If you take the graph of y = 1/x:

And rotate it around the y-axis (the vertical bold line), you get a pair of funnel-shapes.  Not too hard to visualize.  The question is: what are the volume and surface area of the funnels?

Well, calculating volumes and surface areas is pretty much the point of integral calculus, so it's not such a hard problem.  One issue, though, is that the tapered end of the funnel goes on forever; the red curves never strike either the x or y-axis (something mathematicians call "asymptotic").  But calc students never let a little thing like infinity stand in the way, and in any case, the formulas involved can handle that with no problem, so we started crunching through the math to find the answer.

And one by one, each of us stopped, frowning and staring at our papers, thinking, "Wait..."

Because the shapes end up having an infinite surface area (not so surprising given that the tapered end gets narrower and narrower, but goes on forever) -- but they have a finite volume.

I blurted out, "So you could fill it with paint but you couldn't paint its surface?"

Dr. Pousson grinned and said, "That's right."

We forthwith nicknamed the thing "Pousson's Paint Can."  I only found out much later that the bizarre paradox of this shape was noted hundreds of years ago, and it was christened "Gabriel's Horn" by seventeenth-century Italian physicist and mathematician Evangelista Torricelli, who figured it was a good shape for the horn blown by the Archangel Gabriel on Judgment Day.

There are a lot of math-phobes out there, which is a shame, because you find out some weird and wonderful stuff studying mathematics.  I largely blame the educational system for this -- I was lucky enough to have a string of fantastic, gifted elementary and middle school math teachers who encouraged us to play with numbers and figure out how it all worked, and I came out loving math and appreciating the cool and unexpected bits of the subject.  It's a pity, though, that a lot of people have the opposite experience.  Which, unfortunately, is what happened with me in my elementary and middle school social studies and English classes -- with predictable results.

So math has its cool bits, even if you weren't lucky enough to learn about 'em in school.  Here are some short versions of other odd mathematical twists that your math teachers may not have told you about.  Even you math-phobes -- try these on for size.


1.  Fractals

A fractal is a shape that is "self-similar;" if you take a small piece of it, and magnify it, it looks just like the original shape did.  One of the first fractals I ran into was the Koch Snowflake, invented by Swedish mathematician Helge von Koch, which came from playing around with triangles.  You take an equilateral triangle, divide each of its sides into three equal pieces, then take the middle one and convert it into a (smaller) equilateral triangle.  Repeat.  Here's a diagram with the first four levels:

And with Koch's Snowflake -- similar to Pousson's Paint Can, but for different reasons -- we end up with a shape that has an infinite perimeter but a finite area.

Fractals also result in some really unexpected patterns coming out of perfectly ordinary processes.  If you have eight minutes and want your mind completely blown, check out how what seems like a completely random dice-throwing protocol generates a bizarre fractal shape called the Sierpinski Triangle.  (And no, I don't know why this works, so don't ask.  Or, more accurately, ask an actual mathematician, who won't just give you what I would, which is a silly grin and a shrug of the shoulders.)

2.  The Four-Color-Map Theorem

In 1852, a man named Francis Guthrie was coloring in a map of the counties of England, and noticed that he could do the entire map, leaving no two adjacent counties the same color, using only four different colors.  Guthrie wondered if that was true of all maps.

Turns out it is -- something that wasn't proven for sure until 1976.

Oh, but if you're talking about a map printed onto a Möbius Strip, it takes six colors.  A map printed on a torus (donut) would take seven.

Once again, I don't have the first clue why.  Probably explaining how it took almost a hundred years to prove.  But it's still pretty freakin' cool.


3.  Brouwer's Fixed-Point Theorem

In the 1950s, Dutch mathematician Luitzen Brouwer came up with an idea that -- as bizarre as it is -- has been proven true.  Take two identical maps of Scotland.  Deform one any way you want to -- shrink it, expand it, rotate it, crumple it, whatever -- and then drop it on top of the other one.

Brouwer said that there will be one point on the deformed copy of the map that is exactly on top of the corresponding point on the other map.

It even works on three dimensions.  If I stir my cup of coffee, at any given time there will be at least one coffee molecule that is in exactly the same position it was in before I stirred the cup.

Speaking of which, all this is turning my brain to mush.  I think I need to get more coffee before I go on to...


4.  The types of infinity

You might think that infinite is infinite.  If something goes on forever, it just... does.

Turns out that's not true.  There are countable infinities, and uncountable infinities, and the latter is much bigger than the former.

Infinitely bigger, in fact.

Let's define "countable" first.  It's simple enough; if I can uniquely assign a natural number (1, 2, 3, 4...) to the members of a set, it's a countable set.  It may go on forever, but if I took long enough I could assign each member a unique number, and leave none out.

So, the set of natural numbers is itself a countable set.  Hopefully obviously.

So is the set of odd numbers.  But here's where the weirdness starts.  It turns out that the number of natural numbers is exactly the same as the number of odd numbers.  You may be thinking, "Wait... that can't be right, there has to be twice as many natural numbers as odd numbers!"  But no, because you can put them in a one-to-one correspondence and leave none out:

1-1
2-3
3-5
4-7
5-9
6-11
7-13
etc.

So there are exactly the same number in both sets.

Now, what about real numbers?  The real numbers are all the numbers on the number line -- i.e. all the natural numbers plus all of the possible decimals in between.  Are there the same number of real and natural numbers?

Nope.  Both are infinite, but they're different kinds of infinite.

Suppose you tried to come up with a countable list of real numbers between zero and one, the same as we came up with a countable list of odd numbers above.  (Let's not worry about the whole number line, even.  Just the ones between zero and one.)  As I mentioned above, if you can do a one-to-one correspondence between the natural numbers and the members of that list, without leaving any out, then you've got a countable infinity.  So here are a few members of that list:

0.1010101010101010...
0.3333333333333333...
0.1213141516171819...
0.9283743929178394...
0.1010010001000010...
0.13579111315171921...

And so forth.  You get the idea.

German mathematician Georg Cantor showed that no matter what you do, your list will always leave some out.  In what's called the diagonal proof, he said to take your list, and create a new number -- by adding one to the first digit of the first number, to the second digit of the second number, to the third digit of the third number, and so on.  So using the short list above, the first six decimal places will be:

0.242413...

This number can't be anywhere on the list.  Why?  Because its first digit is different from the first number on the list, the second digit is different from the second number on the list, the third digit is different from the third number of the list, and so forth.  And even if you just artificially add that new number to the end of the list, it doesn't help you, because you can just do the whole process again and generate a new number that isn't anywhere on the list.

So there are more numbers between zero and one on the number line than there are natural numbers.  Infinitely more.


5.  Russell's Paradox

I'm going to end with one I'm still trying to wrap my brain around.  This one is courtesy of British mathematician Bertrand Russell, and is called Russell's Paradox in his honor.

First, let's define two kind of sets:

• A set is normal if it doesn't contain itself. For example, the "set of all trees on Earth" is normal, because the set itself is not a tree, so it doesn't contain itself.
• A set is abnormal if it contains itself. The "set of everything that is not a tree" is abnormal, because the set itself is not a tree.

Russell came up with a simple idea: he looked at "the set of all possible normal sets."  Let's call that set R.  Now here's the question:

Is R normal or abnormal?

Thanks, I'll show myself out.

**************************************

In keeping with Monday's post, this week's Skeptophilia book recommendation is about one of the most enigmatic figures in mathematics; the Indian prodigy Srinivasa Ramanujan.  Ramanujan was remarkable not only for his adeptness in handling numbers, but for his insight; one of his most famous moments was the discovery of "taxicab numbers" (I'll leave you to read the book to find out why they're called that), which are numbers that are expressible as the sum of two cubes, two different ways.

For example, 1,729 is the sum of 1 cubed and 12 cubed; it's also the sum of 9 cubed and 10 cubed.

What's fascinating about Ramanujan is that when he discovered this, it just leapt out at him.  He looked at 1,729 and immediately recognized that it had this odd property.  When he shared it with a friend, he was kind of amazed that the friend didn't jump to the same realization.

"How did you know that?" the friend asked.

Ramanujan shrugged.  "It was obvious."

The Man Who Knew Infinity by Robert Kanigel is the story of Ramanujan, whose life ended from tuberculosis at the young age of 32.  It's a brilliant, intriguing, and deeply perplexing book, looking at the mind of a savant -- someone who is so much better than most of us at a particular subject that it's hard even to conceive.  But Kanigel doesn't just hold up Ramanujan as some kind of odd specimen; he looks at the human side of a man whose phenomenal abilities put him in a class by himself.

[Note: if you purchase this book using the image/link below, part of the proceeds goes to support Skeptophilia!]