Original Math Notes
All Seasons
Season 5
Episode 502: The Decoy Effect
Four recursively subdivided triangles rotate around their edges and form a regular tetrahedron
Interactive Computations
Use the free Wolfram Mathematica Player to interact with the math behind NUMB3RS.
Scene 3:
CHARLIE goes through a thick pile of files with LARRY --

                    CHARLIE
          Wideband Mixer-Circulator Retro-Reflector.

                    LARRY
          Obviously not...

                    CHARLIE
          Ternary computing.

                    LARRY
          Mmm... no.

Each rejected file going on a tall stack, next to a second,
very small one, as AMITA and ALAN enter.

Balanced Ternary Notation »

A number represented in binary is a sum of the powers of 2 (1, 2, 4, 8, 16, ...) multiplied by 0 or 1. For example, 60 in binary notation is 111100 = 1×32 + 1×16 + 1×8 + 1×4 + 0×2 + 0×1, using six "bits". Balanced ternary notation multiplies each power of 3 (1, 3, 9, 27, ...) by -1, 0, or 1. In balanced ternary, 60 is 11110 = 81-27+9-3+0, with 1 indicating -1; 60 requires five "trits". With weights 1, 3, 9, 27, and 81, the notation can be used to balance any unit amount from 1 to 121 by putting the weights on either side of the balance pan.
Scene 3:
                    CHARLIE
          With my NSA clearance suspended,
          sometimes it feels like that's all I
          get to do -- look.
            (beat)
          My next paper might very well be "Our
          Friend the Triangle."

                    AMITA
          Actually, John Conway and Steven Sigur
          already wrote a great book on the
          subject.

Triangle Altitudes and Inradius »

If a triangle has inradius r and altitude α, β, and γ, then 1/r = 1/α + 1/β + 1/γ.
Scene 44:
                    LARRY
          I suspect I'm looking at a combinatorics
          problem.  Would you mind --

                    AMITA
          Sure --

Heilbronn Triangles in the Unit Square »

For n points in a unit square, find the three points that make the triangle with minimal area. Finding the placement of n points that produces the largest such triangle is known as the Heilbronn triangle problem. The point placements shown here are the best known. Minimal triangles are colored red. All solutions above 12 points are due to Mark Beyleveld and David Cantrell, with optimization and exact solutions found by Mathematica.
Scene 49:
The SAME FEEDS that were in the War Room are visible on
screens here, now -- as DON brings up the SPIDER program for
Amita and Larry --

                    DON
          SPIDER is a --

                    AMITA
          -- real-time ATM tracking program...
          Charlie and I did work on its
          Distributed Neural Network -- along
          with half the Calsci math and
          computer departments...

Cellular-Automaton-Like Neural Network in a Toroidal Vector Field »

This Demonstration shows a neural network evolving under rules similar to those for a four-neighbor outer-totalistic cellular automaton. You can sample a variety of evolution rules exhibiting integrate-and-fire-like behavior. Red indicates cellular activity (a neuronal spike), while blue indicates inactivity. Color intensity encodes the value of a binary internal state variable.
Scene 60:
                    COLBY
          Then why risk Herman on the
          kidnappings in the first place?

                    CHARLIE
          Because he was running his own
          version of a scheduling algorithm.

                    AMITA
          (to others)
          Computers use them to weigh the
          duration or difficulty of different
          tasks against the system's resources
          and allocate them accordingly.

                    DON
          So what does that tell us?

                    CHARLIE
          The interesting thing about
          scheduling algorithms is that no one
          has come up with a perfect one.

                    AMITA
          There are thousands -- the Smith
          Rule, the O2, the Beam Search... any
          programmer can design one, name it,
          test it...

                    CHARLIE
          And just like a programmer's
          design... or a guitarist's signature
          riff, or a painter's brush stroke...

Optimal Transport Scheduling »

This Demonstration shows the optimal transport scheduling for two depots that are responsible for supplying building materials to seven construction sites. Given the amount of supply available at each depot and the demand at each site, the optimal scheduling minimizes the transport cost, assuming that the distance between a depot and the site is the Euclidean distance.
 
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