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 are the torus knots and links.
Other Knots and knot/link ideas:
Twist knots (figure eight and \(5_2\))
Distinguishing knots:
Tricolorability
Reidemeister moves
Other Links
unlink
Olympic rings
Borromean rings
Infinity (let's start by looking over a key to the
infinite quiz....)
Zeno's paradoxes
power sets
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.
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....
Then, looking back to exam 2, we were mostly studying geometry:
Fractals
A world within a world...
Do it again;
initiator and generator
Some classics, like the Mandelbrot fractal, Koch
fractal, Sierpinski triangle, etc.
Symmetry
rotational
reflective
of scale (e.g. fractals)
Seventeen Wallpapers of Polya
Platonic Solids
Regular polygons
the symmetry of each solid
their duality
where they appear
Exam 1 materials:
one-to-one correspondences (what is six? What is infinity?)
The Fraudini trick -- powers of two
prime factorizations, and their sieving in the "Sieve of
Eratosthenes"; primes helped us fit folks into the
Hilbert Hotel, where we invoked the prime factorization
theorem.
binary factorization (the power set grows by powers of
two!); there's another theorem!
Babylonian and Mayan numbers -- different bases
Our hero zero
Primitive counting (back to binary)
Fibonacci factorizations (The
golden rectangle and golden ratio are intimately
related to Fibonacci numbers, the ratio being the limit
of consecutive Fibonaccis as they march off to
infinity); the last of our factorization theorems.
Fibonacci nim (how Aiden is paying for his schooling!)
Fibonacci spirals, and their appearance in our forests,
gardens, and grocery stores
Graphs and trees; complete graphs, simple 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
"figurate numbers": 'rectangular numbers" (e.g. squares as
sums of consecutive odd numbers), triangular numbers,
tetrahedral numbers, etc.
Yanghui's (Pascal's) triangle
Its historical ambiguity
and all the number patterns contained therein:
powers of two, natural numbers, triangular numbers,
tetrahedral numbers, Fibonacci numbers, .... and its
uses as a calculator of "how many ways?"
We can use it as a model of Sierpinski's triangle (fractal)
Plus it goes on forever in two directions! The
backwards version....
Website maintained by Andy Long.
Comments appreciated.
