Exam 2 Informal Geometry, Spring 2004  Name:____________________

 

 

  1. Compare and contrast Euclidean, and spherical geometry.

a.     Compare and contrast the geodesics in both geometries

b.     Compare and contrast the polygons (which ones exist, which ones donÕt),

c.     Compare and contrast the theorems about the sum of the angles in a triangle.

d.     Compare and contrast the number of regular and semi-regular tessellations  in both geometries(how many?).

e.     Compare and contrast the isometries in both geometries

f.      Compare and contrast the area formulas for triangles in both geometries.

 

  1. Symmetry Groups for rosette symmetry groups.
    1. What is the symmetry group for a triangle? List the elements in this symmetry group.
    2. What is a subgroup? List some subgroups for this symmetry group.
    3. Show how to ÒmultiplyÓ isometries. Explain how this is done in general, illustrate with an example.
    4. What is a cyclic subgroup? Does this symmetry group have a cyclic subgroup? If so, what is it?

 

  1. Symmetry Groups.
    1. What is the symmetry group for the MM border pattern?
    2. List the elements in this symmetry group.
    3. What are infinite cyclic subgroups?
    4. Does this symmetry group have any other cyclic subgroups?

 

4.     a. Draw a triangle on this sphere with three 90¡ angles.

 

 

 

 

 

 

 

 

 

 

 

 

 


   

 

 

               b. Draw a triangle with one angle larger than 180¡.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


5.     All spherical triangles have angles adding up to more than 180¡.  We called the amount over 180¡ the defect of the triangle.  We found that a triangle covers a fraction of the sphere equal to .

     Fill in the empty places in this table:

Angles

Defect

Area Fraction

90¡-90¡-90¡

 

 

90¡-90¡- ?

45¡

 

150¡-150¡- ?

 

1/4

 

Show your computations below:

 

 

6.     Use your knowledge of triangles to explain why the sum of the angles of a quadrilateral on a sphere is always larger than 360¡.

 

7.     A "bi-angle" is a polygon with two sides and two angles.  They donÕt exist in Euclidean geometry, but they do on the sphere.  Draw some (at least two different)  bi-angles.

 

 

  1. Identify the following Wallpaper Patterns found in the Saint Louis Basilica .

 

Rotational Symmetry? (What degree?, Where?)

Reflectional Symmetry? (YES or NO)

 

Wallpaper Group?

 

 

 

Rotational Symmetry? (What degree?, Where?)

Reflectional Symmetry? (YES or NO)

Wallpaper Group?