Day 17: Mat115

• Hand in outlines/references of your projects.

• Don't forget our up-coming exam (next Wednesday, 3/26). It should be hard to forget, since you're getting a take-home component from sections 4.3-4.7, worth 40% of your exam grade: The following mindscapes should be done, and typed up each on a separate page:
  • 4.3: p. 244, #3
  • 4.4: p. 263, #2 and 3
  • 4.5: p. 287, #22
  • 4.7: p. 323, #16

  In preparation for the exam, you might want to look at the following additional Mindscapes:

  • 4.3: #2, 9, 11, 12, 14
  • 4.4: #1, 4, 5, 6, 17
  • 4.5: #1, 2, 4, 5, 11, 14, 15, 16
  • 4.7: #5,6, 7, 9, 12

• A little more on the Platonic solids
  • Why the symmetry in the following table of data for the five Platonic Solids (What is this thing called duality?)?

    	# of Vertices	Edges	Faces	faces at each vertex	sides at each face
    Tetrahedron	4	6	4	3	3
    Tetrahedron	4	6	4	3	3
    Cube	8	12	6	3	4
    Octahedron	6	12	8	4	3
    Dodecahedron	20	30	12	3	5
    Icosahedron	12	30	20	5	3

  • Why do you think that there are only five Platonic solids?

  • Which of the Platonic solids can be used to tile space?

  • For those which tile space, what are the rigid symmetries?

• The Fourth Dimension:
  • What thoughts and questions do you have about the fourth dimension? What is the "fourth dimension"? Where have you already encountered 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 II.6, II.7

  • What if there were creatures confined to two dimensions? How would we mess with their heads?
    
    • Safes aren't safe (p. 312)
    • 3-D surgery (p. 311)
    • 3-D rope tricks (p. 313)
    • How might a cube dipped into two-space appear to creatures stuck in two-space?
  • Now: what if there are four dimensional creatures living with us? How might they mess with our heads?
    • Safes aren't safe!
    • 4-D surgery
    • Hypercubes
    • Slicing and dicing: Problem II.8

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