Original Math Notes
All Seasons
Season 5
Episode 510: Frenemies
Visualization of the spectra of random, parametrized 5x5 matrices
Scene 16:
MARSHALL PENFIELD in front of a large group of graduate
students -- lecturing.  Charlie quietly slips in the back.

          From set theory's inception, some
          have argued it's a game that
          includes elements of fantasy.
          Wittgenstein questioned how
          Zermelo-Fraenkel set theory handled
          infinities.  Errett Bishop dismissed
          it as "God's mathematics, which we
          should leave for God to do."
Marshall spies Charlie in the back.

                    MARSHALL (cont'd)
          Next week's talk will be on the
          Spectral Theory of non-Hermitian
          Random Matrices.  Thank you.

The students file out.  Charlie approaches. 

Spectral Realizations of Polyhedral Skeleta »

Each eigenvalue in the spectrum of a combinatorial graph's adjacency matrix gives rise to a "spectral realization" of the graph. Such a realization is both eigenic (replacing each vertex with the vector sum of its neighbors is equivalent to scaling the figure) and harmonious (each automorphism of the graph is realized by an isometry of the figure). In many cases, they are also just plain fascinating. This Demonstration reveals the spectral realizations of (the skeleton of) each named polyhedron in Mathematica's PolyhedronData[] collection.
Scene 20:
          What if it's not just computer code
          ... but a cipher code?

          If it is, could take months to
          figure out.

          They painted this on a wall -- I
          doubt they made it hard to crack.

          Maybe a simple Vigenere cipher?

          With the first 13 alphabet letters
          matched to these 13 code symbols --

          And the key word is "Vanguard"--

Amita brings up a Vigenere software program on a second
screen -- and it translates the computer code into TEXT. 

Cipher Encoder »

Substitution ciphers are one of the simplest forms of encryption, where replacement rules are provided that map each letter of the alphabet onto a different letter. This tool will generate randomly chosen keys that map the 26 letters of the English alphabet onto themselves. It also generates snippets of sample text from a range of English sources.
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