The Mathematics of Rivers

Dave Richeson, who writes the Division by Zero blog is an Associate Professor of Mathematics at Dickinson College and also the author of a really good math book, " Euler's Gem: The polyhedron formula and the birth of topology" from Princeton University Press.


He also just introduced me to a word I didn't know, potamology, from the Greek ποταμός, river. The word is the technical name for the study of Rivers. Incredibly, there is some really cool math and statistics involved in the study of rivers. For one thing, they are sort of fractal, or as Dave explained it, " the size of a river cannot be determined by its shape on a map. In particular, if you looked at an aerial snapshot of a meandering river, you would not be able to tell whether it is the Amazon or a small neighborhood stream!"

Dave goes on to relate how a the distance between two meanders in a river are related to its width. If we let the width be w, and lamda be the distance between the beginning and ending of one not-quite-sinusoidal period of the meander, then lamda = 11w.
For stats kids, he also posts a regression plot of the actual ratio between meander length and channel width....stats in action baby.

Go to Dave's site and read the whole thing.... he has cool pictures for examples also, including the one above.

A Simple Geometry Exploration

It is a simple geometry question... if you hold the perimeter of a convex polynomial constant, what are the possible limits on the sum of the diagonals, in particular, what is the maximum. I realized I was pretty unsure about the solution, even for a quadrilateral, the simplest case. I thought about a rectangle, and realized that the diagonals get longer as the sides become less equal. If we assume the perimeter is p, then any two sides of the rectangle would be x, and p/x-2. The two equal diagonals would each be the square root of (x^2 + (p/2 -x)^2)) which is largest when the sum of the squares is largest. But the sum is 2x^2 - px + p^2/4 . This is a positive quadratic, and so it is greatest at the ends of its domain, and smallest at the vertex (p/4)... a square has the smallest sum of the diagonals of any rectangle. This is easy to confirm if you take a 4x4 square which has two diagonals of 4 sqrt(2) for a total of 8 sqrt(2) (more or less 11.314), but if we put make it more oblong, say a 7x1 rectangle, the diagonals are each sqrt(50) or 5 sqrt(2) for a total of 10 sqrt(2) (a larger 14.142)... if we extend this to the limiting behavior, we see that each of the diagonals would be close to p/2 so the total sum of the diagonals would approach the perimeter as a limit.. Trying different shapes led me to think (but not prove) that p is probably the limit for a quadrilateral.

But what happens if we let the number of sides go beyond four.... I had no idea how to approach the problems except to experiment. I began by drawing five points on a circle and taking the ratio of the sum of the diagonals to the perimeter. I quickly realized that if I moved two of the points close together on one place and three others near the opposite side of the circle the ratio approached two.. the diagonals were nearly twice the perimeter. This made sense, three of the edges would be nearly zero, and two would be almost the length of the diagonal of the circle, so the total perimeter would approach 2d. And what of the diagonals? Well, there would be four diagonals that went from the two points on one side to the opposite three points that would each be approximately the length of the diagonal, for a total of 4d.

Would this be better with four on one side and one on the other? The perimeter would be the same, but there would only be two long diagonals. It seems that splitting the points up evenly increased the total sum of the diagonals. It seems the diagonals would sum to 2p in the limit. I couldn't imagine exceeding this (correct me if I have failed to visualize something here) for a pentagon...

So, what about a hexagon......? six edges, and 9 diagonals.. . Using our previous insights, we could try putting 3 points close together on opposite sides of the circle, This would mean that would mean that the total perimeter was again approaching 2d, but there would be 7 diagonals which were also approaching the limit of a diameter. The other two diagonals would approach zero, and the total sum of the diagonals would be 7d, for a ratio of 3.5......

Hmmmmm could I see a pattern here? for n=4, the ratio of diagonals to perimeter was 1, for n=5 sides, the ratio was 2, and at n=6 the sum would be 3.5. Well, for a four sided figure, there were two diagonals (and essentially all the perimeter was in two sides, 2/2 =1 ) . For a pentagon, there were 5 diagonals, but one of them went to essentially zero... the ratio was 4/2 or 2. With a hexagon there were 9 -2 long diagonals, so a ratio of 7/2. Could I extend this?

If we went to seven sides, there would be 7 choose 2 - 7 diagonals, or 14 of them. We would put four on one side and three on the other. By my count there woud be 10 long diagonals and four that diminished to zero (one on the end with three points connecting the outside two, and three on the end with four points) so the ratio would grow to 5 times the perimeter

lets tabulate what we know (or think we know)

n-sides... total diag.. short diag...long diag.. ratio --- change
4----------2----=======--0-------------2------------2/2=1---------0
5------------5------------1--------------4-----------4/2=2--------1
6------------9-------------2-------------7-------------7/2--------1.5
7-----------14-------------4------------10------------10/2=5------1.5
8-----------20------------6-------------14-----------14/2=7 -----2
9-----------27------------9 ------------18-----------18/2=9-------2
10----------35-----------12-------------23----------23/2=11.5----2.5

Ok, the recursive pattern seems to be r(n+1) = r(n) + floor(n/2)/2... I think I did that right....

and it is late, so I will let you write the explicit function...I'm just thankful that the holiday is at hand. I'm off to London with my sweetheart for the weekend to see a west-end show and have a good Japanese meal... and hold hands and walk through the neighborhood markets... hope you have a great holiday too.

Scientists Say the Stupidest Things

I was in Cambridge recently with my beautiful sweetheart, and we were browsing through Oxfam when I came across the book, Foolish Words, The most stupid words ever spoken by Laura Ward. It is gut wrenchingly funny in places (and also terribly sad at the same time), but I especially treasured some of the bold predictions about the future from people who would be expected to have a better than average insight into the topic. It reminds me of the wisdom of a quote on my classroom wall, "Never miss a good opportunity to shut-up."

Here are a few of them:
"The energy produced by the breaking down of the atom is a very poor kind of thing. Anyone who looks for a source of power in the transformation of the atom is talking moonshine."
Lord Ernest Rutherford after splitting the Atom in 1933

Einstein had said only a year earlier, "There is not the slightest indication that atomic energy will ever be attainable. It would mean that the atom would have to be shattered at will."


Here is the Astronomer Royal of Great Britain, Sir Richard Woolley, in 1957, "I cannot see any nation or combination of nations producing the money necessary to put a satellite in outer space or to circumnavigate the moon." He was actually expanding earlier remarks (not in the book) when, on his appointment as Astronomer Royal, he reiterated his long-held view that 'space travel is utter bilge'. Speaking to Time in 1956, Woolley noted "It's utter bilge. I don't think anybody will ever put up enough money to do such a thing . . . What good would it do us? If we spent the same amount of money on preparing first-class astronomical equipment we would learn much more about the universe . . . It is all rather rot". Woolley's protestations came just one year prior to the launch of Sputnik, five years before launch of the Apollo Program, and thirteen years before the first landing on the moon. (from Wikipedia)

Thomas Edison in 1928, "I have determined that there is no market for talking pictures."

German Physicist/chemist Johann Poggendorf proudly announced, "It is impossible to transmit speech electrically, the 'telephone' is as mythical as the unicorn."

Simon Newcomb, one of the most brilliant men of his period, a polymath and perhaps the first discoverer of Benford's law, among other things; stated boldly, "Aerial flight is one of that class of problems with which man cannot cope", 1903. (for my students who know so little history they make me cry, it was in December 17, 1903 that the Wright Brothers made their famous flight at Kitty Hawk.)

In March of 1949 an article in Popular Mechanics wrote, "..the Eniac is equipped with 18,000 vacuum tubes and weighs 30 tons, computers in the future may have only 1000 vacuum tubes and weigh only 1.6 tons." (or perhaps, 1.6 pounds?)

Lord Kelvin, President of the Royal Society, and so esteemed that he was buried in Westminster Abbey next to Isaac Newton, said, "X-rays will prove to be a hoax" and is also known to have said, "Heavier-than-air flying machines are impossible" in 1895. Then he went on to doom another budding invention, "Radio has no future." Perhaps he had anticipated the birth of television.

Lee DeForest was an inventor whose work ushered in the electronic age. The inventor of the tube which made audio amplification possible and bringing sound to the motion picture went on to proclaim, "While theoretically and technically television my be possible, commercially and financially it is an impossibility."

Finally, a quote from Dr. William Clark, president of the Arthritis Foundation in 1966, "We just won't have arthritis in the year 2000." (I have it on good authority from my aching shoulder that it is back by 2009).

I guess I leave the last word to that Great Statesman, George Bush, who explained it all as an educational problem when he stated in 2000, "What's not fine is, rarely is the question asked: Is our children learning?".

What Can You Say About a Polynomial with All Coefficients Equal

My lunch math kids and I came across a problem about a function with all its coefficients equal (hereafter called PWEC, polynomials with equal coefficients). I had to admit to them that I had never thought about such a function, so we developed an interactive graph on geogebra and played around with it, looking for what we might say about such a function.. So here is your chance to show how clever you are...

I took a couple of the kids observations about PWEC functions and intentionally violate at least one of them in each graph. Each of the functions is NOT a function with all coefficients equal; and your task is to say WHY? Some have more than one reason...

Game on.. here we go:

Can you get it? If not, scroll down a little to find at least one reason...
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Ok, observation number one, Joe S picked up on this one right away... Every odd powered PWEC will have a zero at x=-1... easy to confirm because the alternating powers will be negative and positive. Descarte's rule of signs proves that there can be no real zeros that are positive. My experience is that it always has just one; the one at x=-1.

Did you get it?? Well, how about another...

Well? What about this one?....
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This time you can't get much from Descarte, but it seems from observation that an even PWEC can never have a zero. I played with this a little and haven't succeeded in proving it analytically for anything but a few simple polynomials.. any offers?

Ok..just one more... another even function... what do you think...


OK, no easy stuff on this one... what do you think?
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This time there are no tricks with the zeros... Ok, maybe one more.... notice that the horizontal tangent occurs with x>0...but that is a no-no as well. The derivative will also have all positive coefficients, so by Descarte's rule of signs again, it can not have a zero when x>0.

I admit I had never thought of any of these things, and noticed them all by working with interactive graphs on Geogebra.. one reason I am thankful for the power of modern computing... I know more math because of the technology available.. and I want my kids to see that it can be more than an easy way to get the answer, it can help you learn more about the ideas of mathematics.