Exam 2

At the Exam

  • You may have your child's ball with you.
  • You may have out food, hydration, ear plugs, or similar if they will help you (however any ear plugs must be standalone--no cell phone, internet or other technological connections).
  • Partial credit will be given, so (if you have time) showing your reasoning or thoughts on questions you are unsure of can help your grade.
  • This exam is closed to notes/books and closed to technology, but I will give you a copy of Euclid's Elements Book I.
  • Any IGS explorations I ask you to describe or create will specify "roughly sketch" so a sketch by-hand without tools will be fine, however, if you prefer, you may bring a straight edge and/or a compass or circle.
  • Your grade will be based on the quality of your responses in the timed environment.
  • Because there is more material on this exam than exam 1, you may make yourself some reference notes on the small card I hand out (additional cards are on my door if you need to rewrite it). The reference card must be handwritten. Think of the card as a way to include some important content that you aren't as comfortable with. You won't have room for everything, and you should try to internalize as much as you can.

    Topics: Review the following and be sure that you could answer related questions on these topics. Polyhedra in Various Geometries:
  • The names, numbers of vertices, edges, and faces of the 5 regular polyhedra [Euler's formula can help you determine the edges if you remember the vertices and faces, and you can also use the dual polyhedra to help you remember] from class.
  • The proof that there are only 5 regular polyhedra using Euler's formula from class.
  • What goes wrong with the proof on the sphere and one example of a spherical polyhedron with no flat equivalent from class.
  • Without assuming that you already know what the 5 Euclidean Platonic solids are, use Euler's Formula to prove that a polyhedron with three triangular faces meeting at each vertex must have a total of four faces. Identify underlying assumptions from Project 5.

    Metric and Axiomatic Perspectives in Various Geometries:
  • Given a diagram being able to compute the distance between 2 points in Euclidean and taxicab geometry from class.
  • The Pythagorean theorem in Euclidean geometry, spherical geometry, taxicab geometry, and hyperbolic geometry [be able to answer the following questions: is the theorem true and briefly summarize a method we used to demonstrate and/or prove the answer in class or Project 5.
  • The diagram from the Euclidean proof of the Pythagorean theorem from Euclid's Elements and goes wrong in spherical and hyperbolic geometry and (sometimes) in taxicab geometry.
  • The diagram from the Zhou Bi Suan Jing or Chou Pei Suan Ching proof and puzzle of the Euclidean Pythagorean theorem, and what goes wrong in spherical and hyperbolic geometry and (sometimes) in taxicab geometry from class and project 5.
  • The proof of SAS in Euclidean geometry and what goes wrong in spherical and taxicab geometry and what goes right in hyperbolic and projective geometry from class.
  • Euclidean proof that the sum of the angles in a triangle is 180 degrees and what goes wrong in spherical and hyperbolic geometry.
  • Finding angle sums of a triangle formed by 3 vertices in an Escher artwork.
  • Know that we proved the sum was always greater than 180 degrees on a sphere by using lunes on a ball to compare areas along with a little algebra [but no need for the details of this] and obtain congruence for nondegenerate triangles satisfying AAA on the sphere (rather than similarity).

    Parallels in Various Geometries and Applications:
  • Euclid's 5th postulate in Euclidean, spherical and hyperbolic geometry.
  • A reason why Euclid may have chosen to state the 5th postulate the way he did from Project 6.
  • Playfair's postulate in Euclidean, spherical and hyperbolic geometry.
  • The proof of the existence portion of Playfair in Euclidean and hyperbolic geometry and what goes wrong on the sphere.
  • The proof that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles in Euclidean geometry and what goes wrong on the sphere and in hyperbolic geometry from Project 6.
  • Finding lines of symmetry in Escher's drawings and using this to identify the geometry via the number of parallels (if any).
    • The vast majority of the exam will come from a variety of types of questions related to some of these topics. However, exams are not only an opportunity for you to demonstrate your mastery of the material, but are also an opportunity for you to be challenged with new material in order for you to make new connections.

    Specific Examples of Types of Questions Question types include short answer/short essay, like:
  • Sketch the construction or diagram...
  • Does this construction or diagram always, never, or sometimes (but not always) work on the sphere? Explain.
  • Sketch or provide counterexamples...
  • In the following proof, fill in the blank using reasons from Book 1 of Euclid's Elements (which I will hand out to you) and identify any additional underlying assumptions.
  • Write a paragraph proof and identify underlying assumptions/limitations.
  • What goes wrong with the proof in hyperbolic geometry?
  • Identify applications.
  • Give reasons why or why not.
  • Describe one of the Interactive Geometry Software explorations.
  • How did we explore this in class?

    Euclid's Element's Book I: I will give you a copy to use on the test. I may give you a proof and ask you to fill in the reasons with the Postulates and/or Propositions. For example, you should be familiar with the statements of the five postulates, and roughly know where some of the propositions are located, as follows:

    Create a line segment: Postulate 1 [fails in spherical geometry and taxicab geometry since uniqueness of lines was used in Prop 4, even though it is not explicitly stated here]
    Extend a line: Postulate 2
    Create a circle: Postulate 3
    All right angles are equal: Postulate 4
    How to tell that two lines intersect: Postulate 5 [fails in hyperbolic geometry]

    Construct an Equilateral triangle: Prop 1
    Bisect an angle: Prop 9
    Construct perpendiculars: Prop 11 or 12
    Congruence Theorems: Prop 4: SAS, Prop 26: ASA and AAS
    Exterior angle is larger than each remove angle: Prop 16
    Recognizing a parallel: Prop 27
    Statements that use the parallel postulate begin with Prop 29, so if you have "if parallel then ..." generally you will want to look at 29 and beyond.
    If parallel then alternate interior angles...: Prop 29
    Construct parallels: Prop 31
    Sum of the angles in a triangle is 180 degrees: Prop 32
    Pythagorean Theorem: Prop 47 and 48