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.

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.

                    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.

Choose an irrational number s and a horizontal unit segment with angle φ₀. Define φ_n+1 = φ_n+ 2πs (mod 2π), with φ₀ = 0. To the previous segment, add a new unit segment with angle φ_{n + 1} = θ_n + φ_n (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.

                    DON
          And you can prove this?

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

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.

                    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.

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.

                    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.

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.