$1.7 Trillion

This is a repost from my Cary Academy blog;

During US Government Politics & Political Theory, we watched a West Wing episode from the first season, Six Meetings Before Lunch.  (Supposedly, that title is a reference to Lewis Carroll’s Through the Looking Glass, where the White Queen says to Alice, “Why, sometimes I’ve believed as many as six impossible things before breakfast.”)  We finished in Chapter 3, entitled $1.7 Trillion, about slavery reparations, and it reminded me of a poem that I had written almost forty years ago:

fellow honky
  A big jump we got
    lived two and three centuries dragging chains
    they never lost
    forty acres and two ass
    they never got
    no landrush lot
  Big jump we got
  Ten decades and more it’s built up
    prosperity for our children
    Inherent right? Right.
    we’ve inherited gift and debt
brother ofay
  Pretty soon
  we have to pay.
                                 –1972

Benjamin Banneker and the square root of 3

At the Teaching Contemporary Mathematics Conference held Jan 21-22, 2011, at the North Carolina School of Science and Mathematics, teacher John Mahoney from the Benjamin Banneker Academic High School in Washington, DC, presented the plenary session talk Saturday morning.  The talk was about the math of Benjamin Banneker, who helped survey the District of Columbia, and John's work with Banneker's journals (but I was also very inspired by his description of his experience in moving from the Sidwell Friends school to the public school where he now teaches.)

John has studied and electronically restored copies of Banneker's journal entries (an example is at the top of this blog post), and he has been intrigued by one approximation that he found in the journals, that the square root of 3 is approximately 5/9 of pi.  Even back in 2003, John had asked about the approximation, in a post in the math forum of Drexel University.  In that post, he mentioned another Banneker approximation, which happens to be the same as approximating the square root of 3 by 26/15--which is actually better than 5/9 of pi.  A good question would be why use one approximation when another is better, and, even simpler, but that still leaves the question of where the approximation came from. I think I have found a possibility.

If you take the Fourier series expansion of arcsine of x divided by the square root of one minus x squared (you can plug "arcsin x / sqrt(1-x^2)" into WolframAlpha to find it), you get a series that starts with these first three terms: x+(2 x^3)/3+(8 x^5)/15.  If you let x be 1/2 in this series, using those first three terms, you get Banneker's formula!

Rhomaleosaurus?

The crude panorama above is taken from Sand Turn, on Highway 14 in the Big Horn Mountains of Wyoming, facing east on the left,  and south on the right.  You can see how the strata to the south has been uplifted by the mountains, almost as if they were waves lapping at the shore.  I've stopped at Sand Turn every time I've passed that way, to look out over the vast plains to the east.  One of my geology professors noticed the view, and remarked that he'd spent a summer geology camp there, and that one of the campers had picked up a large fossilized femur one day.  My brother's company was doing some surveying there, and mentioned that to the rancher.

In the summer of 1999, the rancher was using a cat to move rock from a limestone ledge, so that he could build a fence there, and he discovered this fossil (the coin is a USA Sacajawea dollar), approximately where the red v points in the middle of the panorama.  I carried it back to North Carolina, and showed to to a professor at the University of North Carolina, who referred me to a researcher at North Carolina State University, who referred me to Dr. David Martill of the University of Portsmouth in England.  I furnished photos, and a geological map of the area (prepared, partly, during the summer geology camps).  He suggested that it was a humerus with radius and ulna, possibly from an early Jurassic pliosauru such as Rhomaleosaurus, which would be the first such found in the United States.

Stars in the daytime

This is a sunrise over the Gulf of Mexico, from a beach near Destin, Florida (05:46 CST 22 Nov 2010).  The white dot up in the clouds is Venus, barely more than a pixel, about 40% from the left, 80% up.  Venus, when it is visible, is always very bright, fifteen times brighter than the brightest star.  That morning I watched it disappear behind the clouds, but the clouds burnt off later in the morning.  The sun was fully visible when I mentioned over breakfast that Venus was still in the sky and I looked up and found it almost immediately.  I'd never done that before, although I'd found Venus in the day sky many times, usually by first using binoculars and calculating its position in the sky.  Sometimes I've showed it to groups of elementary school kids--who could usually see it much easier than I can.  Views of Venus may have contributed to the myth that stars can be visible during the daytime.

To me, one of the most remarkable coincidences in astronomy is the consistency of the brightness of Venus.  Brightness of a planet varies as the inverse square of the distance, and our distance to Venus varies more than any other planet, because Venus gets so close to Earth.  However, when it is close, we see its dark backside, and very little of its sunlit side.  The two opposing effects neatly cancel each other, and the net result is that our perception of Venus is that it changes very little in brightness.  It is always close to the sun in our view, because it is closer to the sun than the earth, but whether it is on the other side of the sun, or on this side of the sun, there is very little difference in its brightness, the total range is less than one magnitude.  Mars varies by over four magnitudes, Mercury over eight, and even far away Jupiter and Saturn vary by over a magnitude.