# Course:Harris, Fall 07: Diary Week 14

Mon:

• Handed back Exam II.
• Most difficulty was with explaining why there are only three regular tessellations of the plane.
• Biggest hint: You have to know what the formula is for the angle in a regular n-gon (or derive the formula from what you know for the angle-sum of an n-gon).
• For final exam: Explain why there are only five regular tessellations of the sphere.
• Biggest hint: Same as above--except it's an inequality you get.
• examined concept of "how much infinity is out there" in the three geoemtries, using triangular tessellations
• Sphere, with {3,5} regular tessellation (or any other!), eventually has no neighbors at large enough distances from a given point.
• Plane, with {3,6} regular tessellation, has 4n neighbors at distance n from a given point.
• Hyperbolic plane, with {3,7} regular tessellation, has 7, then 21 neighbors at distances 1, then 2, from a given point.
• Groups started in on Three Geometries Exploration, didn't finish.
• Discussion for next time: What is the shape of the universe?
• possibilities:
• spherical?
• flat?
• hyperbolic?
• other? (what other?)
• How might we tell?
• Is this a sensible question?
• How could you tell?

Wed:

• Groups finished up Three Geometries Exploration (20-25 minutes).
• Class discussed, "What is the shape of the universe?"
• issue of dimension: What does 3-dimensional spherical or hyperbolic geometry mean?
• This can be answered in terms of how much is "out there" at a given distance from a given point:
• For spherical, far enough away, the universe shrinks back to a point
• The universe plus (divine) heavens is given a three-dimensional spherical geometry in Dante's Paradiso)
• For hyperbolic, the universe expands more quickly than Euclidean space does.
• Could experiments help us to tell?
• If the universe is finite in extent, then it can't be flat (i.e., Euclidean) or hyperbolic.
• But what experiment would reveal finiteness of extent?
• Experiments with triangles could distinguish among spherical, flat, and hyperbolic by measuring angle-sum.
• Bigger triangles are better (more sensitive measurement).
• What can be used for geodesics?
• Maybe lasers.
• Maybe experiments with parallel geodesics could be done:
• Spherical geometry has no parallel geodesics.
• Flat geometry has parallel geodesics that stay a constant distance apart.
• Hyperbolic geometry has parallel geodesics, but none of them stay a constant distance apart.
• Exercise (due Monday):
• 1-2 page paper exploring these issues:
• What are possible answers to "What is the shape of the universe?"
• What are possible experiments that could help us know?
• Is this necessarily a sensible question, and what might that mean?
• If any student is interested in doing so, this could be expanded to something more substantial.
• Bring Visions of Symmetry for Dilation Exploration Friday.

Fri:

• Mentioned similarity transformations:
• preserves angles
• multiplies distances between points by a constant factor
• has a central point of expansion or shrinkage
• can incorporate isometries (rotations, reflections, etc.) in addition to shrinking/expanding
• Fractals are "self-similar" diagrams: Some similarity transformation leaves the diagram unchanged.
• approximate examples in nature:
• wrinkliness in sea shore (expanding the picture by multiple factors keeps it still looking just as wrinkled)
• structure of a cloud (at all scales, it has same degree of fuzziness)
• structure of a tree or leaf (repeated division and branching at finer and finer levels)
• alveoli in the lungs (repeated division and branching at finer and finer levels)
• Fractal images are extremely easy to create with computer graphics, hence, an easy way to make invented landscapes or trees look natural.
• Groups did Similarity Exploration and nearly finished Iteration Exploration