Course:Harris, Fall 07: Diary Week 11

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  • reminded class that Tessellation Art Projects are due Wednesday
    • full Project includes a written report; but that may permissably be delayed till Friday
  • groups worked on first hyperbolic exploration, Escher's Circle Limit, mostly finishing
    • angles are to be measured same as Euclidean angle measure (but it helps to draw a Euclidean tangent line to a hyperbolic geodesic, to measure angles well)
    • hyperbolic distances use "Poincare rulers" which get Euclidean-shorter, the closer they get to the boundary
    • the boundary is actually at infinite hyperbolic distance
  • briefly explored questions 11-13 (about Circle Limit III)
    • the smaller white circles are clearly not hitting the boundary at right angles to it
    • trying to see if it's possible to reconcile knowledge of hyperbolic polygons (angle-sum is less than Euclidean angle-sum) with what purport to be polygons in this drawing


  • Art Projects due today (reports may wait till Friday)
  • looked again at Circle Limit III:
    • if we assume that all the fish are congruent to one another, then all the angles--both the triangle angles and the quadrilateral angles--are congruent to one another (since each consists of half a head and half a tail)
    • since every vertex has 6 angles around it, that means all the angles are 60 degrees
    • while that's not a problem for the quadrilaterals, it is a problem for the triangles:
      • that would make the triangles have angle-sums of 180 degrees, which is impossible in hyperbolic geometry
    • conclusion: this really is not a tessellation of hyperbolic space by geodesics
      • that agrees with the previous conclusion, from Monday, that the smaller white circles do not hit the boundary at right angles, which is required for hyperbolic geodesics in the Poincare disk
  • groups finished up (if necessary) the first Hyperbolic Exploration, Escher's Circle Limits
  • groups completed the second Hyperbolic Exploration, Hyperbolic Geometry
  • we looked at the two other programs mentioned in that Exploration:
    • Hyperbolic Applet shows how to shove around the Poincare disk ("by hand") using hyperbolic translations
    • Hyperbolic Animations shows a number of "pre-programmed" motions via hyperbolic translations
    • the import of these is to familiarize us with what it means to move things in the Poincare disk in a way that keeps hyperbolic size and shape the same (no matter how different it looks to Euclidean eyes)


  • Borders Bookstores currently has two interesting Escher books on remainder:
    • $5: "M. C. Escher: The Graphic Work", published by Taschen: This has been available before, but this is an especially attractive price.
    • $8: "M. C. Escher", by Sandra Forty: This is a large-format book, very pretty. It has not previously been much available, well worth collecting at this time by those who like Escher in general (not just tessellations).
  • gave a revised description of the Written Report of the Tessellation Art Project, with revised due date of Monday, November 12 (see main page)
  • groups largely finished 3rd hyperbolic exploration (Non-Euclid II)
  • Hyperbolic Exercises I due Monday (more will be assigned next week)
  • propounded major question for the balance of the class:
    • What is the shape of space?
      • (meaning: what is the geometry of the actual, physical universe)
      • obvious options
        • Is it Euclidean (3-dimensional)?
        • Is it spherical (3-dimensional version)?
        • Is it hyperbolic (3-dimensional version)?
      • not so obvious options
        • It is some other sort of geometry?
        • Is this necessarily a sensible question?
        • Is this an experimental question that only astronomers can answer?