- Today we're having a review for our exam on Thursday.
- You will be allowed a page of hand-written notes (front and back).
- If you use a phone as a calculator, you will need to keep it in
front of you on the desk, in calculator mode.
- Don't forget your readings.
- Assignment to turn in: Use the Fibonacci
Spiral Fractal maker, with your own image, to make a photo Fibonacci spiral.
You might try some of the "art" features.
Email me this, and we'll hold an art show of spirals, with "get out of quiz free" prizes....
(Due Wednesday, 3/5)
I already have some entries!
- We built another Fibonacci spiral, by building up more beautiful rectangles by attaching more and more squares.
- After building a 34x21 rectangle, with ratio of side lengths
1.619, we turned around and built a golden rectangle by removing the
largest square possible from a golden rectangle.
- The rectangle that results has exactly the same shape as the original
rectangle: therefore the ratio of its side lengths is
the same as the original: that means that
\[
\frac{\varphi}{1} = \frac{1}{\varphi - 1}
\]
- This becomes the quadratic equation \[ \varphi^2-\varphi-1 = 0 \] (cross multiply and take everything to the left side).
- Pull that quadratic formula out of the dim recesses of your mind and you'll find two solutions: \[ \varphi=\frac{1 \pm \sqrt{5}}{2} \approx 1.618 \] (Why do we choose the "+" solution?)
- The rectangle that results has exactly the same shape as the original
rectangle: therefore the ratio of its side lengths is
the same as the original: that means that
\[
\frac{\varphi}{1} = \frac{1}{\varphi - 1}
\]
- We talked a little about fractals -- a preview of "A world within a world."
- I received one request:
"Can we go over probability in Pascal's triangle and do examples of picking donuts out of certain numbers. Also factorials, Fibonacci numbers and primitive counting"
So, as we hit those topics, I'll stop to do an example.
- What is six? (and a six update -- what could we add to that, that we've studied since then?)
- 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
- Fibonacci spirals, and the golden rectangle and golden mean or ratio
- Fibonacci
Spiral Fractal maker, to make images like these:
- How to win at Nim
- Logo Design using the Golden Ratio
- A review of the Mayan calendar we study, Unearthing the Heavens: Classic Maya Murals and Astronomical Tables at Xultun, Guatemala , including the reference to this article from the journal Science: Ancient Maya Astronomical Tables from Xultun, Guatemala, and this more "popular" interpretation of the calendar.
- Do math like an Egyptian! (Walk Like An Egyptian/Michael Jackson Mash-up!)
- Hexagonal Paper
- Pascal's triangle (wikipedia)
- American Bandstand dances to Gimme Some Lovin', a hit by The Spencer Davis Group: here's their version.
- List of musical works in unusual time signatures
- Blue Rondo a
la Turk, by Dave Brubeck: illustrating
how to play a piece in 9/8.
Watch Dave Brubeck's leg, counting out the rhythm....