Assignments for Last Week

READ Sved Chapters 4 and 5

I. Written assignment due Wed 3/8 (see below)

II. Construction Portfolio due Fri 3/10 (see link)

III. Study problem to prepare for Final Exam (see bottom of this page)

Written Assignment Due Wed 3/8 (Problems 1 and 2)

Problem 1 (midpoint triangles and simplest geodesic dome)

NOTATION:  I = (1, 0, 0), J = (0, 1, 0), K = (0, 0, 1).  O = (0,0,0).

Let S be the sphere of radius 1 with center O. Notice that the points +I, -I, +J, -J, +K, -K are all on S. The are also the vertices of an octahedron inscribed in the sphere.

Consider the 90-90-90 spherical triangle IJK. Construct the midpoints I', J', K' of the sides. (I' is on side JK, etc., as usual.)

Then the triangle IJK can be divided into four triangles, the midpoint triangle I'J'K' and the 3 corners triangles, IJ'K', etc.

Draw a circle e with center S on a sheet of paper. Then we may assume that e is the image by stereographic projection of the plane with equation (z = 0) in (x,y,z) space, i.e., e is the equator if +K is the North Pole. Since I is on e, draw a point I on the circle and then construct J so that ISJ is a right angle.

1. Now construct the stereographic images of I', J', K', if the projection center (north pole) is at K. (You may use Sketchpad or a straightedge and compass for this, but print out the result.)
2. Tell what are the equations in x,y,z of the plane of the great circles through I'J', J'K', K'I'.
3. Compute precisely using algebra the spherical angle I'J'K' (i.e., the angle in the spherical triangle I'J'K').
• What is the spherical excess of this triangle? What is the area, if the area of the sphere is S?
• What is the spherical excess of any of the corner triangles? What is the area?
• How are these excesses related to the spherical area of IJK?
4. Compute precisely using algebra as needed:
   • The spherical distance (in degrees) from K to I'.
   • The euclidean distance in 3-space from K to I'.
   • The spherical distance from I' to J'.
   • The euclidean distance in 3-space from I' to J'.

Problem 2: P-model

A: Suppose m is a P-line.

• If p and q are two P-llines orthogonal to m, must p and q be ultraparallel?
• Explain why the support circles of the P-lines orthogonal to m form a coaxal family.
• Tell what kind of family of circles this is, and tell what are the special points that define the family.

B. Draw a P-triangle ABC so that one vertex A is at the center of h, the boundary circle of the P-disk. Prove that the angle sum of this triangle is less than 180 degrees.

Study Problem for Final Exam

Angular Defect in Hyperbolic Geometry

• Show that the angular defect is additive. This means that if a triangle is partitioned into two triangles, then its angular defect is the sum of the angular defects of the sub-triangles.
• Does this proof also work on the Euclidean plane and the sphere? What does this say in each case?
• Explain why, if one rectangle exists in a plane, then arbitrarily large rectangles exist.
• Explain why, if one triangle with positive angle defect exists in a plane, then no rectangles can exist.

Read Sved Chapter 4 to learn about angular defect.