Day 15: Heart of Mathematics

• Announcements:
  • Welcome back!

  • Homework, Section 4.7 due Wednesday:
    #5, 6, 7, 9, 10, 12; and do a really good job with #19.

  • For today you were to read section 4.3 (the Sexiest Rectangle!)

  • New Homework: Read section 4.5 for next time. Problems, section 4.3, pp. 244-, #2, 3, 9, 12, 14, 15 (due Wednesday, 3/25)

• The Fourth Dimension: big issues of dimension
  • We use lower dimensions to make sense of higher dimensions, and our world: in two dimensions, that is "Flatland",
    
    • Safes aren't safe (p. 312)
    • 3-D surgery is possible (p. 311)
    • 3-D rope tricks (p. 313)
    • Activity:
      How might a cube dipped into two-dimensional space appear to creatures stuck in two-space?
      Draw four distinctly different impressions of the cube.
      What's the most dramatically strange representation of a cube we can come up with?
    • How might we appear to creatures in two-dimensional space? What might we look like to them?

  • If four-dimensional creatures live among us, then
    • Safes aren't safe!
    • Miracles would include
      • 4-D surgery
      • Angels
      • Ghosts
    • Hypercubes (or Tesseracts)
    • How might four dimensional creatures look to us in three-dimensions? (This reminds me of the Blind men and the elephant.)

  • Let's imagine a one-dimensional world moving through time, and examine the "lives of one-dimensional creatures".
    • Where would any eyes, mouths, etc. be on one-dimensional creatures?
    • Suppose a creature grows from the union of a sperm and an egg, grows, and then dies.

      Activity:
      Draw a picture of its "lifeline".

    • What issues arise?

  • Can you now relate your "lifeline" to the one-dimensional creature's? How are they the same? How different?

• A little fun (that will tie in to golden rectangles in a moment):

  The sixth stanza of Amazing Grace:

  When we've been here ten thousand years...
  bright shining as the sun.
  We've no less days to sing God's praise...
  than when we've first begun.

• "The Sexiest Rectangle" - or is it just cute?
  • Take a look at the rectangle contest, page 233. What do you think?

  • What's wrong with squares?!

  • We'll start with a nice Fibonacci spiral (see this site)

    Activity: draw a Fibonacci spiral. Where would you start to make the biggest one possible?

  • Fibonaccis give rise to what is known as the "golden mean":

    Activity:
    Compute the ratios of successive Fibonacci numbers: 1 1 2 3 5 8 13 21 34 55

  • Golden Rectangle (definition): a rectangle whose side lengths are in the ratio

  • The Golden Section in Art and Architecture

    or not? Many people think that if you look hard enough, you'll find them - or any rectangles you're particularly interested in!

    The Parthenon

  • Interestingly, one of the canvasses by Turner claimed to show the Golden Rectangle is of Slavers Throwing Overboard the Dead and Dying -- so much for Amazing Grace!

  • What sizes do U.S. flags come in? (How about Togolese Flags?)

  • Where might you find a golden rectangle? (Hint: what's in your wallet?)

Links:

• Abbot, Edwin A. Flatland, A Romance of Many Dimensions. New York: Dover, 1992.
• Signifying Nothing: The Fourth Dimension in Modernist Art and Literature, cites Abbot, above, and has a scan of Duchamp's "Nude Descending a Staircase, No. 2 (1912)")
• Some aids to the fourth dimension