# Course:Harris, Fall 08: Diary Week 10

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- 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.

- Example:
- 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.

- extent:

- lines in the plane <--> great circles in the sphere

- Example of analogous terms:
- 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.

- One interpretation: What are the different subject matters of the two geometries?
- 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.

- Consider first a triangle on a sphere:
- 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 - <math>\pi</math> (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 <math>\pi</math> radians) at the vertex V.
- Thus, angle-sum = 2A + 180 (degrees) or 2A + <math>\pi</math> (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,
- <math>\frac{area(triangle)}{area(sphere)} = \frac{A}{360}</math> (degrees) or
- <math>\frac{area[triangle]}{area[sphere]} = \frac{A}{2\pi}</math> (radians)

- In other words,
- Continuing in radian measure: Use <math>A = \frac{1}{2}defect</math>:
- <math>\frac{area[triangle]}{area[sphere]} = \frac{defect}{4\pi}</math>
- <math>defect = 4\pi\frac{area[triangle]}{area[sphere]}</math>

- Now use the fact that for a sphere of radius R, <math>area[sphere] = 4\pi {R^2}</math>:
- <math>defect = 4\pi\frac{area[triangle]}{4\pi {R^2}}</math> or
- <math>defect = \frac{1}{R^2}area[triangle]</math>

- We've shown this only for biangular triangles, but it's true generally.

- For a spherical triangle,
- 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)

- Isometry means a rigid motion, preserving
- 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.

- What group of symmetries was used?
- 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?

- Finished art work:
- Why are great circles the geodesics of spheres?
- Finishing up the Spherical Isometries Exploration
- Betweenness on spheres.