Day 29: Heart of Mathematics

• Last time: The Platonic Solids Turn Amorous! Duality; the fact that the solids "pair up". Plotting the solids in 2-d, using the stereographic projection.

• Today: The Fourth Dimension
  • Reminder:
    • Problem assignment #3 due Friday (problems from section 4.3).
    • Volunteers for sections should step forward right away - I'm pidgeonholed out! I'm make assignments on Wednesday, otherwise....

  • Further stereographic projections

  • Slices of a cube (Problem II.1, p. 285)

  • What thoughts and questions do you have about the fourth dimension? What is the "fourth dimension"?

  • Sweeping out additional dimensions ("inked points")
    • point (imprisoned!)
    • point -> line (imprisoned by two points, each on one side)
    • line -> plane (imprisoned by a circle)
    • plane -> three-space (imprisoned by a sphere)
    • three-space -> the fourth dimension!

  • Slicing and dicing: Problems I.1, I.2

  • If there's time, we'll look at the other problems:
    • I.7
    • I.9
    • II.1
    • II.3

• Next time: More on the fourth dimension

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)")
• The Fourth dimension and its relationship with the Platonic solids
• Archimedean Solids
• Some aids to the fourth dimension
• On the Fourth Dimension