Course:Harris, Fall 07: Diary Week 12

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  • collected remaining written reports for Tessellation Art Project
  • collected first set of Hyperbolic Exercises
  • announced Exam II for Monday of next week
  • introduced concept of ideal polygons in the Poincare disk, using "ideal points" on the boundary of the disk
  • groups finished work on last week's hyperbolic explorations and worked on Hyperbolic Tessellations Exploration, largely finishing it
    • had to revise assignment in Hyperbolic Tessellations to avoid duality questions and add questions on which regular tessellations are possible


  • looked at how much information about a regular tessellation can be obtained just from the Schlaefli information {n,k}:
    • k = number of polygons around a vertex:
      • if all angles are congruent (due to regular tessellation), and k angles fit around a vertex, then, since they add up to 360 degrees, each angle must be 360/k
    • n = number sides in each polygon
      • then angle-sum for each is n x 360/k
  • looked at how angle-sum for n-gons is different for the three geometries we've been looking at:
    • planar (Euclidean) geometry: angle-sum = (n-2) x 180
    • spherical geometry: angle-sum > (n-2) x 180
    • hyperbolic geometry: angle-sum < (n-2) x 180
  • groups finished Ideal Tessellation Exploration


  • took a look at Circle Limit II, in respect of Hyperbolic Exercise 15:
    • not evident how to find an underlying geometric tessellation of that picture with polygons meeting vertex-to-vertex
    • since the exercise assumes a regular tessellation--and that includes being vertex-to-vertex--we'll omit #15
  • tried to complete Hyperbolic Exercise 19:
    • found number of neighbors which are 1 edge away, 2 edges away, etc., from a given vertex for
      • plane ({3,6} tessellation):
        • 4n neighbors are n edges away
      • sphere (icosohedron, {3,5} tessellation):
        • 5 are 1 edge away, 5 are 2 edges away, 1 is 3 edges away, nothing is more than that
    • worked on building a model of the hyperbolic plane from triangles ({3,7} tessellation), according to Hyperbolic Paper Exploration:
      • small triangles don't work too well
      • larger triangles seem to have promise, but within one class period, we couldn't build anything large enough to count all neighbors 3 edges away
    • used PoincareApplet (from Hyperbolic Tessellations Exploration) to make a {3,7} tessellation (with layer = 4, to produce many triangles):
      • found 7 neighbors 1 edge away, 21 are 2 edges away