Course:Harris, Fall 07: Diary Week 12

Mon:

• 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

Wed:

• 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

Fri:

• 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