Course:Harris, Fall 07: Diary Week 10

Mon:

• groups finished up all Spherical Explorations
• looked at how Platonic solids (hand-held examples, Escher-patterned) lead to regular tessellations of the sphere

Wed:

• collected second set of Spherical Exercizes
• began talking about regular tessellations of the sphere:
  • regular tessellation of the sphere means
    • made up of regular (equi-angular, equilateral) spherical polygons
    • all the polygons are the same
    • all the vertices look the same
  • Platonic solid means
    • closed solid made up of regular planar polygons
    • all the polygons are the same
    • polygons meet vertex to vertex
  • each regular tessellation of the sphere corresponds to a Platonic solid, and vice versa (except for "degenerate" regular spherical tessellations):
    • degenerate:
      • 2 1-gons (no-angles): point
      • any number of 2-gons (biangles): line segment
    • non-degenerate:
      • 4 triangles: tetrahedron
      • 6 quadrilaterals: cube
      • 8 triangles: octahedron
      • 12 pentagons: dodecahedron
      • 20 triangles: icosahedron
  • for next time: Why are there only 5 (non-degenerate) regular tessellations of the sphere?
    • Hint: look at what happens at a vertex. For instance: Is it possilble to have a vertex surrounded by regular hexagons?
• looked more closely at defects [I mistakenly called it "deficit" in class]:
  • for a triangle, defect = angle-sum - 180
  • defect is proportional to area (this is not easy to show why it's true, but examples should be somewhat convincing):
    • defect/4x180 = triangle-area/sphere-area
  • an n-gon into can be decomposed into n-2 triangles:
    • angle-sum of n-gon = sum of angle-sums of the n-2 triangles (from looking at picture)
    • defect for n-gon = sum of defects for the n-2 triangles (we define n-gon-defect this way)
    • area of n-gon = sum of areas for the n-2 triangles (from looking at picture)
  • as a result of the triangle analysis:
    • n-gon-defect = angle-sum - (n-2)x180 (this makes the definition above come out right)
    • n-gon-defect is proportional to area (because it works with triangles):
      • n-gon-defect/4x180 = n-gon-area/sphere-area

Fri (Dr. Bart subbed):

• Gave a general idea why there are only 5 platonic solids.
  • Consider the interior angles of the regular n-gons; note that in order to get a solid we need 3 or more polygons around a vertex and the angle sum cannot be ≥360.
  • When n = 3 we can put 3, 4, or 5 regular 3-gons around a vertex and we get a tetrahedron, octagon and icosahedron respectively.
  • When n = 4 we can put 3 squares around a vertex, but no more. This gves a square.
  • When n = 5 we can put 3 pentagons around a vertex, but no more. This gives a dodecahedron.
  • When n = 6 we start running into obstacles. With 3 or more hexagons the sum of angles around one vertex is ≥ 360.
  • When n > 6 we get that the sum of angles around a vertex is > 360 even with just 3 polygons.

• Intro to Hyperbolic geometry
  • Reviewed the axioms for Euclidean and spherical geometry. I gave somewhat imprecise statements, but tried to convey the idea that they concern lines, extension of segments, circles, (right) angles and parallel lines.
  • I stated that there is a third geometry in 2-D and that this geometry has similar properties when it comes to the first four axioms, but is radically different with respect to the parallel axiom. I wrote down a simple version of the axioms.
  • I mentioned models. Euclidean geometry is drawn on a plane, spherical geometry is drawn on a sphere. I mentioned that there are many models for Hyperbolic geometry, but that we will use the disk model.
  • Mentioned that our choice of the disk model was somewhat influenced by the Escher's use of the disk model in the construction of the Circle Limit series.
  • Stated what geodesics in the Poincare Disk model look like (line segments through the center and semi circles perpendicular to the "boundary of the disk". Did point out that the boundary is not part of the "hyperbolic universe".
  • Mentioned the scaling we see in the poincare disk model. "Objects are larger than they appear". I compared this to perspective as it is used in art. The object nearer to the boundary are not really smaller, they are far away.
  • The "boundary" is sometimes called the circle at infinity.
  • Showed the first part of the circle limit exploration. Showed them that the shaded (red) 4-gon is actually a rhombus (sides are equal) and it is also a parallelogram.