Why the three-halves Power

One of the Christmas presents from my sweetheart was John D. Barrow's "One Hundred Essential Things You Didn't Know You Didn't Know." Perhaps it is a little "Math-Lite", but it is still a nice read.
Anyway, reading a section on "A sense of Proportion" in which he points out that if you plot world-record weight lifting records against the weights of the lifters, they fall into a line along the 3/2 power rule, that is (weight of athlete)²=(weight lifted)³. OK, that isn't THAT surprising, but it got me thinking about other things that fall into a square to cube relation, that don't seem to be as easy to understand.

The obvious first choice, is Kepler's third law... The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit, or as I think of it to make it easy, year²=(distance from sun)³. That has always seemed unexpected to me. One is dependent on the circumference (sort of) and one on the radius, so they might be expected to be essentially equal.. but that inverse square effect of gravity, somehow makes it work.

From there I wondered about other examples I had heard of..For instance the Beaufort scale gives a relationship between wind speed and wave height on the oceans as v = 0.836 B^3/2 m/s .. ;

There is another that shows up in forestry... I couldn't remember it offhand, so I found a note that says, "The 3/2 power law states that the relative rate of self-thinning with respect to biomass growth per unit area is a universal constant" so maybe there is more of a geometric relationship here that would make sense.. Maybe you can explain it to me.

ADDENDUM:  I got a tweet from tom@liverbubble that told me about a similar result with the 3/4 power called Kleiber's Law.  The principal is named after a agricultural chemist named Max Kleiber.  In 1932 he came to the conclusion that the ¾ power of body weight was the most reliable basis for predicting the basal metabolic rate (BMR) of animals and for comparing nutrient requirements among animals of different size.  (Send more examples please)

Tic Tac Toe.... Looks Like the Number 15...

Playing around with the mathematics of Tic-Tac-Toe with some kids to talk about symmetry and counting... got me thinking..... and searching....

I found the following (slightly edited) interesting history notes at the ADIT sight..

"It may be that the ancient Roman game of Terni Lapilli was an identical game although the evidence is somewhat mixed. It is certainly true that identical grids to the noughts and crosses grid have been found scratched and etched into surfaces all over the ancient Roman empire. However not a single Nought or Cross has been found to confirm the link. It seems probable that Terni Lapilli was played with simple pieces and may have been played with the same rules but in my mind it’s sheer popularity casts doubt upon the connection.

The first software program designed to play Noughts and Crosses (Which is how the English describe T-T-T,... and I recently was told that many Irish folk call the game Boxin' Oxen) was written by A.S. Douglas as part of his PhD dissertation on Human-Computer interaction. The computer was the EDSAC machine built at Cambridge University in 1949. The EDSAC machine was the first true programmable computer as we would understand it today." Here is what the output looked like on the old Edsac..

There are 255,168 possible games of T-T-T, if you label the positions on the board (from 1-9 for instance), but if you allow for symmetry, there are only 138 possible outcomes. If one (or both) of the two players play the game poorly then it may end in a win on the fifth, sixth, seventh, eighth or ninth move....91 wins by the first player (X) and 44 by the second (o)..... and if you are good at arithmetic, then you now know that out of all the games ever played that ended in a draw.... there are only three different final boards. Somehow I think that is incredible. Can you find them all? It is easiest if you just figure out how to place the four " o " so that X can't win...

You can play against an online computer here...
This is a breakdown of all the endings as presented at Wikipedia...
"Ignoring the sequence of Xs and Os, and after eliminating symmetrical outcomes (ie. rotations and/or reflections of other outcomes), there are only 138 unique outcomes. Assuming once again that X makes the first move every time:
91 unique outcomes are won by (X)


* 21 won by (X) after 5 moves

* 58 won by (X) after 7 moves

* 12 won by (X) after 9 moves


* 44 unique outcomes are won by (O)


* 21 won by (O) after 6 moves

* 23 won by (O) after 8 moves


* 3 unique outcomes are drawn"


Now for the 15 part... you can play TTT without the board... Players alternate picking numbers from the integers one to nine inclusive. The first who has picked three numbers that add up to fifteen wins...How is it Tic-Tac-Toe? Well label the TTT board like this....Ohhhh, like MAGIC...

Almost Pythagoras

For some reason I really like little simple geometric relations that remind me of the Pythagorean Theorem. I still remember the first time I saw 3² + 4²=5² set next to 3³+4³+5³=6³and wondered.... Could it be????... Is it possible???? Oh go on, you know you are going to check... how could you resist?


Just reminded of some other "almost Pythagorean" relations in geometry from an old (1932) Mathematics Teacher article...(Thank you, again, Dave Renfro)

The Two Triangles formed when the median is drawn to any side are almost Pythagorean
Treat the two sides not cut by the median as if they were the hypotenii (hypotenuses?) of a right triangle. Both are wrong, but in sum, they are right..

That is, while neither of the following are true, AM²+MC²= AC²

AM²+MB²=AB²
Adding the two equations produces a true relation.

AM²+MC²+AM²+MB²= AC²+AB²

The proof is pretty easy using the Law of Cosines.

A second, and about as easy to prove.... Let G be the centroid of Triangle ABC, then AB²+BC²+CA² = 3 (GA²+GB²+GC²)


A third, which is, I believe, both necessary and sufficient to prove a parallelogram is that the sum of the squares of the diagonals is equal to the sum of the squares of the sides.

And if it is not a parallelogram, but is a trapezoid, then the sum of the squares of the diagonals is equal to the sum of the squares of the non-parallel sides, plus twice the product of the parallel bases (sort of a law-of-cosines look-a-like).

Recently saw the old Wizard of Oz mis-spoken statement of the Theorem by Ray Bolger as the Scarecrow posted at 360, so I thought this was a good time to include it for my students (and anyone else who has never seen it, or just wants to chuckle one more time). [] Play it for your students and see if they can catch all the mistakes...

Zeros and More Zeros,

An Interesting Relationship Between the Zeros of a Polynomial and the Zeros of Its Derivative.

Just reread an article from Dan Kalman at American University a while back which is here if you want to read the original inspiration for what I am blabbing on about.

Now with real valued polynomials, it is easy to show that between any two zeros of the function, the derivative will have a zero as well, this is essentially what Rolle's theorem says... (Strange that he is remembered most for a "calculus" theorem when he spent much of his life speaking out against calculus). I don't think I knew before I first read this, though, that the average value of the zeros of the polynomial is the same as the average of the zeros of the derivative. Dan's paper shows that this property is true, even if we extend to complex polynomials.

Now I don't spend a lot of time in my high school classes on complex polynomials, but the geometry involved was really nice. It seems that if you plot the zeros of the polynomial (real or complex) on a coordinate plane [complex zero a+bi is plotted as the point (a,b)] and then plot the zeros of the derivative, they will always be inside the boundary determined by the derivatives of the original function (This is called Lucas' Theorem... "For an arbitrary not identically constant polynomial, the zeros of its derivatives lie in the smallest convex polygon containing the zeros of the original polynomial."

For a cubic polynomial for instance, if the roots are not all real, then there will be two complex roots so on the complex plane the three zeros will fall in a triangle. What is really clever about the theorem that Dan has called "Marden's Theorem", is that the two zeros of the derivative of the cubic will not just be inside the triangle formed by the three, but it will be the focus of the ellipse inscribed in the triangle touching the midpoints of the three sides. This means they are symmetric about the centroid of the triangle. The image below shows my rough interpretation of the zeros of x³-7x²+24x-18 which are at 3+3i, 3-3i, and -1. The two zeros of the derivative would then be at 7/3 (+/-) i sqrt(23)/3. And since the second derivative is linear (6x²-14) then the average of the roots will be at x=7/3, which is the centroid of the triangle. Now when the roots are all real, this confirms what most good calculus student's observe pretty quickly, the two zeros of a cubic's derivative are symmetric about the point of inflection (they have the same average).

SOOOoooooo, he wonders... what happens if we go to a fourth degree polynomial. Time to play.

Ok, so if the polynomial has real coefficients (and if everyone hasn't already dozed off in boredom by now) I just leaned about something called Jensen's Theorem...
which seems to narrow the location of possible zeros of the derivatives a little more. If the polynomial coefficients are real, then all the complex zeros will occur in conjugate pairs (Alg II, you remember!). It seems that if the derivative has any complex zeros, they will have to occur in a circle whose diameter has endpoints on a pair of these conjugate zeros.

Willing to learn more..HELP!