Original Math Notes
All Seasons
Season 5
Episode 505: Jack of All Trades
Nonlinear warping applied to a set of parallel stripes
Scene 6:
                    LARRY
          Ah, Charles, my ambulatory
          reference book.  Chaitin's Omega 
          Constant...?

                    CHARLIE
          Omega equals .00787499699.  Why,
          what're you working on?
            (sees the file, reacts)
          Oh.  FBI file.

Busy Beaver »

Chaitin's constant measures the probability that a Turing machine will halt. The busy beaver game is a competition to find an n-state Turing machine that either leaves the largest number of 1s on its tape or makes the largest number of head movements before halting. A busy beaver is a Turing machine that, when provided with a blank tape, does a lot of work.
Scene 22:
Larry points. Colby grabs the remote, zooms in.  In the
corner of the driver's area, a shiny, CHROME HANDLE.  On it,
a WILDLY DISTORTED, fun-house mirror version of the Suspect.

                    DAVID
          We see it.  What're we supposed to
          do with it?  It's too distorted...

                    LARRY
          Certainly from the limited
          perspective of the human eye it
          isn't what we normally comprehend
          as a face...

                    AMITA
          The image's been altered by the
          refracting nature of the surface, 
          but the information is still there.
          It's called Non-Linear Warping.

Nonlinear Warping »

The figure shows a nonlinear warp, which maps a regular grid onto a cylinder or hemisphere. The effect is the same as wrapping a flat image around an object and viewing it from afar. The image transformation is a function of the radius of the cylinder or hemisphere.
Scene 27:
                    AMITA
          Contemplation, or second thoughts?

                    LARRY
            (iffy)
          Working cell by cell will suffice, 
          certainly, in producing Splines... 
          But there must be a more elegant
          approach...

Simple Spline Curves »

In the full Demonstration, each of the four control points can be moved to see the effect on the form of the spline curves. Either Bézier or cubic splines can be experimented with.
Scene 32:
                    CHARLIE
          Belief Propagation.  Algorithms 
          that convert scattered information 
          into a coherent picture, even if 
          the information is incomplete. 

Charlie moves toward the living room FLAT SCREEN -- 

                    CHARLIE (cont'd) 
          It's like... say you're a TV 
          network on election night... 

AUDIENCE VISION - Charlie appears INSIDE the TV, with a 
flashy, Network-like chyron - 'Election 2008: America's 
Decision.'  As Charlie moves to the TV -- 

                    CHARLIE (cont'd)
          You want to be first to predict a 
          winner, so you collect information 
          from all over the country... 

Voter Model »

The voter model shows how the views of neighbors can influence a neighborhood. On a given day, several voters will abandon their current views and assume the position of one of their four neighbors. When run, strong groupings appear almost immediately, with conflict at the borders. Eventually, some positions vanish entirely.
 
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