Day 44, MAT115R

1. A mobius cheat sheet (made from a single sheet of 8.5x11 paper)
2. The "RCO" (Really Cool Object), made from index cards.

There is new material, however, on bands, knots and links; and on infinity.

• 1/3 new material
• 1/3 exam 1 material
• 1/3 exam 2 material

I had a visit from Kamora, who insisted that the knot on the first page is tricolorable. How could it be tricolorable when it was a figure-8 knot?

Then she showed me that it is indeed tricolorable, and so proved that it is, in fact, a trefoil knot! So you all will get points back on your quiz, up to 2 points (with a maximum of 10 for the quiz), thanks to Kamora.

Nice work Kamora!

• Logo winners (each had 5 or more votes):
  • Amalie
  • Sean
  • Finley
  • JB

• Project winners (each had 8 or more votes!):
  • Josh
  • Sam (Fraudini and Cat gear)
  • Meghan (lyrics)
  • JB -- interpretive dance

• Natalie gets the Coach's award!

• Infinity
  • Zeno's paradoxes
  • Hilbert's Hotel
  • cardinality, and measuring infinite sizes via one-to-one correspondences
  • Little infinities, like the natural numbers, the primes, the powers of two, the Fibonaccis, the evens, etc. etc.
  • power sets are always bigger! And, if finite, always have a power of 2 number of elements. Remember their connection to Pascal's triangle.
  • Bigger infinities, like the real numbers, the power set of the natural numbers, etc.
  • In fact, there are an infinite number of bigger and bigger infinities, as shown by: \[ Card(N) \lt Card(P(N)) \lt Card(P(P(N))) \lt Card(P(P(P(N)))) \lt Card(P(P(P(P(N))))) \lt \ldots \]
  • Adding up infinite things, found in Pascal's triangle (everything is found in Pascal's triangle!)

• Bands
  • Untwisted bands (wedding ring, whose edges are the unlink)
  • Once-twisted (Mobius) bands (recycling symbol, whose edge is an unknot)
  • Twice-twisted band (edges are a link, the Hopf link)
  • Thrice-twisted band (edges are a knot, the trefoil)
  • Four-times-twisted band (edges are a link, the Solomon's "knot" (link))
  • Five-times-twisted band (edges are a knot, the cinquefoil)

  These objects, derived from twisted bands, are the torus knots and links.

• Other Links
  • unlink
  • Olympic rings
  • Borromean rings

• Other Knots and knot/link ideas:
  • Twist knots (figure eight and \(5_2\))
  • Distinguishing knots:
    • Tricolorability
    • Reidemeister moves
  • Be sure to check out Natalie's beautiful video!

• Do you know where the golden mean comes from? How it's derived? How it can be used to create a (fractal!) spiral? Fibonacci spirals, and the golden rectangle and golden mean or ratio

• Do you understand Symmetry?
  • rotational
  • reflective
  • rotational and reflective symmetry?
  • of scale (e.g. fractals)
  • Other places where we've noticed and perhaps even used symmetry in our work? e.g.
    • complete graphs,
    • the Platonic solids;
    • all simple graphs of five vertices,
    • or with four vertices which we drew together;
    • discovering the true recycling symbol from the fraudulent one)

• Have you done your readings, e.g. this short summary of some of our graph theory material, or this short summary of fractals? Or the The Enemy Of My Enemy reading?

• Graphs:
  • Do you know your graph definitions? Planar, simple, complete? Euler paths, and Euler's formula (R-A+N=2)?
  • The Konigsberg bridge problem, and the story of the origin of graph theory?
  • Which are the two simplest non-planar graphs?
  • some graphs are fractals (Fibonacci tree)
  • Platonic solids can be drawn as graphs (e.g. the tetrahedron and K4 are equivalent in some sense)
  • Facebook graphs
  • Trees are useful organizational devices!

• Fractals
  • A world within a world...
  • Do it again;
  • initiator and generator
  • Can you name some of the common fractals?
  • Can you make of the classics, like the Cantor's middle third fratcal, Koch fractal, Sierpinski triangle, etc.
  • Did you try your fractal homework?
  • Do you understand the concept of exponential growth or decay of the number and length/area of these fractals?

• Platonic Solids
  • Regular polygons
  • the symmetry of each solid
  • their duality
  • Can you draw 2-dimensional projections of each of the Platonic solids, which are all clearly planar?
  • Do you know which are duals of each other?
  • Do you know your table of their characteristics?
  • Can you identify them if you see them?
  • Do you know which are traversible (have Euler paths)?
  • Where the Platonic solids appear in our world?

• What is six? (and a six update -- what could we add to that, that we've studied since then?) Fibonacci number decomposition, for example.
• Counting numbers (natural numbers -- zero isn't natural)
• First factorization: Prime factorization and the sieve of Eratosthenes
• Testing for "primality" -- is a number prime? -- by first taking square roots of the number, to see how many primes we needed to check
• Special primes (twin primes, only even prime, triplet primes); 1 is not prime.
• One-to-one correspondences: tallies, body counting, cairns, knotted strings, etc.
• Triangular numbers, and squares.
• Figurate numbers: e.g. 'rectangular numbers" (e.g. squares as sums of consecutive odd numbers), triangular numbers as the number of vertices in a complete graph, tetrahedral numbers
• Graphs and trees; simple graphs (Facebook graphs), complete graphs (can you draw them?) \(K_n\)
• Primitive counting
• Second factorization: Binary factorizations and binary representation (as a sum powers of two)
• Can you do the binary card trick (Fraudini's trick)?
• Yang Hui's triangle and bamboo counting rods
• Pascal's triangle (and variations, and all the number patterns contained therein.... and its uses.... and symmetry....)
• Counting combinations in Pascal's triangle, and probabilities; also the factorials.
• Egyptian multiplication (no division!); but you still need to know about Egyptian division.
• Examples of Mayan and Babylonian numbers, and our hero zero. Facets of each system (base, zero, vertical versus horizontal, etc.)
• Fibonacci Nim, and winning strategy
• Third factorization: Fibonacci factorization, and Fibonacci numbers defined