# Course:Harris, Fall 08: Diary Week 10

Mon:

• Discussion: How are planar and spherical geometry different?
• One interpretation: What are the different subject matters of the two geometries?
• Example:
• Planar geometry examines lines.
• Spherical geometry examines great circles.
• Second interpretation: When considering analogous terms, how are the terms related to one another differently in planar geometry than in spherical?
• Example of analogous terms:
• lines in the plane <--> great circles in the sphere
• extent:
• Lines are infinite.
• Great cirlces are finite.
• parallel:
• Parallel lines exist in the plane.
• Parallel great circles do not exist in the sphere.
• The second interpretation is the deeper one, and that is the one we will most be concerned with.
• But to get to that, you have to first know which terms in each geometry are to be considered analogous.
• point <--> point
• line <--> great circle
• Both are called geodesics.
• between <--> ???
• We still need to consider what "between" ought to mean on a sphere.
• line segment <--> great circle arc
• Both could be called a geodesic segment.
• polygon <--> polygon
• Just be sure to express it in terms of geodesic segments joined at their endpoints, etc.
• In particular, "rectangle" can be used for any quadrilateral with four right angles, in any geometry.
• In the plane it is also a parallelogram, but that need not be part of the definition of the word.
• Can there be rectangles in a sphere?
• Consider first a triangle on a sphere:
• The spherical segments connecting the three points bow outwards (on the sphere) from the three line segments that connect the points going through the interior of the sphere.
• Therefore, each of the angles on the spherical triangle is greater than the corresponding angle of the corresponding line-segment-triangle of those same three points.
• Therefore, the angle-sum of the spherical triangle is greater than the angle-sum of the line-segment-triangle.
• In other words, the angle-sum(spherical triangle) > 180 degrees.
• And the further apart the points on the sphere are, the more bowed-out the spherical segments, and the greater the deviation from 180 degrees.
• Since a quadrilateral can be divided into two triangles, this means
• angle-sum(spherical quadrilateral) > 360 degrees.
• Therefor, a spherical rectangle is impossible, as four right angles would mean angle-sum = 360 degrees.
• We had about 20 minutes left to do the Spherical Polygon Exploration.
• Preliminary sketches for the Escher Art Project were handed in, and the first Spherical Exercises were collected.

Wed:

• Preliminary sketches for the Escher Art Project were returned.
• We discussed defects of spherical triangles:
• For a spherical triangle,
• defect = angle-sum - 180 (for degrees)
• defect = angle-sum - $\pi$ (for radians)
• On a given sphere, defect is larger for larger triangles:
• For smaller triangles, there is less difference between the spherical segments and the line segments between the vertices, so less bowing out of the spherical sides, so a closer correspondence between spherical and linear angles, so a closer correspondence between spherical and linear angle-sum.
• More specifically, defect is proportional to area of the triangle.
• We can see what the proportionality constant is by concentrating on biangular triangles:
• A biangle can be considered a triangle by using the two vertices of the biangle (call them North and South Pole) and a third vertex V in one of the two sides.
• This yields the three angles as the same angle A at North Pole and South Pole and 180 degrees (or $\pi$ radians) at the vertex V.
• Thus, angle-sum = 2A + 180 (degrees) or 2A + $\pi$ (radians).
• So defect = 2A (in either case).
• The ratio of the biangle to the entire sphere is the same as the ratio of the angle A to one revolution (best seen by looking from the vantage point of North Pole).
• In other words,
• $\frac{area(triangle)}{area(sphere)} = \frac{A}{360}$ (degrees) or
• $\frac{area[triangle]}{area[sphere]} = \frac{A}{2\pi}$ (radians)
• Continuing in radian measure: Use $A = \frac{1}{2}defect$:
• $\frac{area[triangle]}{area[sphere]} = \frac{defect}{4\pi}$
• $defect = 4\pi\frac{area[triangle]}{area[sphere]}$
• Now use the fact that for a sphere of radius R, $area[sphere] = 4\pi {R^2}$:
• $defect = 4\pi\frac{area[triangle]}{4\pi {R^2}}$ or
• $defect = \frac{1}{R^2}area[triangle]$
• We've shown this only for biangular triangles, but it's true generally.
• We considered isometries of spheres:
• Isometry means a rigid motion, preserving
• distances and
• angles.
• Isometries of the plane:
• translations (by some distance along a direction)
• rotations (around a point)
• reflections (across a line)
• glide-reflections (by some distance along an axis, a line)
• Isometries of the sphere:
• translations = rotations (by some angular amount around a point = along a great circle)
• reflections (across a great circle)
• gilde-reflections (by some angular amount along an axis, a great circle)
• Groups started the Spherical Isometries Explorations, not quite finishing it in 20 minutes.

Fri:

• Art Project due in one week (Friday, Nov. 7)
• Finished art work:
• no grid lines
• excellence of execution counts
• Report:
• What group of symmetries was used?
• Include a listing of types of symmetries in the work.
• Comparisons with specific Escher tessellations are good.
• How did your choice of group of symmetries effect the finished artwork?
• How did your artistic vision effect your choice of group of symmetries?
• Why are great circles the geodesics of spheres?
• Finishing up the Spherical Isometries Exploration
• Betweenness on spheres.