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Episode 409: Graphic--Explore the Math
Episode 409: Graphic
Interactive Computations
Use the free Wolfram Mathematica Player to interact with the math behind NUMB3RS.
Scene 2:
Charlie turns to the board and underlines a book title:

                    CHARLIE (cont'd)
          For next week, chapter 15 of Theory of
          Games and Economic Behavior by John von
          Neumann and Oskar Morgenstern.

Macroeconomic Effects of Interest Rate Cuts »

Major Federal Reserve rate cuts can have wide and varying effects on macroeconomic variables.
Scene 7:
COLBY shows Charlie the map as they walk the floor.

                    CHARLIE
          Fourteen locations in a five-mile
          radius -- excellent.  I'll get you a
          probable base for the seller.

Basic Statistics of Movable Points »

See how various location measures change as you move points around. Note that the median jumps as points change their relative coordinates. The harmonic mean is never to the right of the geometric mean, which is never to the right of the arithmetic mean.
Scene 18:
                    CHARLIE
          Fractal Number Estimate.  It's
          based on Mandlebrot's use of
          fractal dimension to measure the
          jaggedness of a coastline.  It's
          been used to detect forged
          handwriting, but we're applying it
          because hand-drawn art can be
          evaluated with the same process.

Julia Sets and the Mandelbrot Set »

Each Julia set on the right corresponds to a quadratic polynomial of the form f(z) = z2 + c. You can use the locator to choose the complex number c from the picture of the Mandelbrot set on the left. The Julia set will be a connected set precisely when c lies in the Mandelbrot set.
Scene 19:
                    CHARLIE (V.O.) (cont'd)
          The slower the pen, the longer the
          contact with the paper, allowing
          more ink to be absorbed.  Creating a
          more irregular or wrinkly edge.

The INK bleeds into the PAPER, leaves a JAGGED EDGE.  GRAPHIC
FREEZES, the previous GRAPHIC of the REAL SIGNATURE SLIDES
over to be compared.  A GRAPHIC OVERLAY.  The SLOWER SIGNATURE
has a more 'Jagged' INK EDGE.  MATH overlays and measure.

                    CHARLIE (V.O.) (cont'd)
          Fractal Dimension allows us to
          compare the wrinkliness -- and
          detect which is the fake.

RETURN TO SCENE:

                    AMITA
          These fractal comparisons are
          telling us the same thing.

Curlicue Fractal »

Choose an irrational number s and a horizontal unit segment with angle φ0. Define φn+1 = φn+ 2πs (mod 2π), with φ0 = 0. To the previous segment, add a new unit segment with angle φn + 1 = θn + φn (mod 2π). The resulting series of line segments is the curlicue fractal. The temperature of these fractals measures the boundedness of these curves. If these curves were infinitely extended, their fractal dimension would be a measure of how well they covered the plane.
Scene 23:
                    DON
          And you can prove this?

                    CHARLIE
          No, but I have a theory, and it's
          too elegant not to be true. 

Einstein's Most Excellent Proof »

According to his autobiography, a preteen Albert Einstein devised a new proof of the Pythagorean theorem based on the properties of similar triangles. Many known proofs use similarity arguments, but this one is notable for its elegance, simplicity, and the sense that it reveals the connection between length and area that is at the heart of the theorem.
Scene 24:
                    CHARLIE
          Maybe, with time.  But listen, I had
          a hunch about the Eppes Convergence
          before I had the math.  Einstein
          had a 'hunch' about relativity
          decades before it could be proven.

Russell's Thought Experiment in Special Relativity »

In Bertrand Russell's book The ABC of Relativity, Russell explains the following thought experiment: Suppose an observer is on a platform at rest, a second observer is on a second platform that moves with a speed that is a fraction of the speed of light relative to the first, a third observer moves with respect to the second with the same speed, and so on.
Scene 38:
                    CHARLIE
          A genuine good commingled with
          false goods.  In Auction Theory we
          talk about equilibria or symmetry.
          All bidders should have the same
          exact information.  But in this
          case, Seth knows something the
          other bidders don't know.  This is
          asymmetrical information.  And this
          can create asymmetrical bidding.

Location Theory - 3D Bid Rent Curves Become Urban Form »

Just as the bid rent curve is not linear, neither is the earth upon which it is projected flat. The initial view shows the cone shape of an economic topographical map arising from the bidding process as it occurs in all directions from the center of activity. Note that the rent gradient is the rate at which rents decline as one moves away from the center in a particular direction. The more shallow slope indicates the slower rate of rent decay and would be considered the "path of progress." Using the secondary slider, you can see the emergence of secondary centers of activity (there need not be three of them and they need not be equidistant from each other), creating new high-rent districts away from the center.
 
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