Test 2 on Polyhedra, Metric Perspectives, Parallelism, and
This test will be closed to notes/books, but a calculator will be allowed (but no cell phone nor other calculators bundled in combination with additional technologies) and you may also bring your ball. There will be various types of questions on the test and your grade will be based on the quality of your responses in a timed environment (turned in by the end of class). You may also bring a straight edge and compass or circle, but any construction sketches I ask you to create will specify "roughly sketch" so a sketch by-hand without tools will be fine too.
Big Picture Questions: Briefly skim the information on the
class highlights page and your notes from
the day we began Polyhedra up through the day before the test.
You will be asked to choose and discuss in detail an example or topic from
class that illustrated concept development [summarize the development and how
the example/topic illustrated this] and [a separate question]
connections among multiple mathematical perspectives
[which perspectives, and how the example/topic
Content Questions: Review the following and be sure that you could
answer related questions on these topics. Notice that Euclidean and
non-Euclidean has been woven throughout these:
Polyhedra in Various Geometries:
Content from the class slides, specifically:
Tiling 3-space with dodecahedral polyhedra-why Euclidean tilings
don't work for dodecahedron but do work for cubes,
and why spherical and hyperbolic ones do, as well as the
connection to a finite universe with no edges.
- 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
- The proof that there are only 5 regular polyhedra using Euler's formula
- What goes wrong with the proof on the sphere and one example of a spherical
polyhedron with no flat equivalent.
Metric Perspectives in Various Geometries:
The formulas for measuring the distance between 2 points in Euclidean
and taxicab geometry.
The Pythagorean theorem in Euclidean geometry, spherical geometry, taxicab
geometry, and hyperbolic geometry [be able to answer the following questions:
is the theorem always, never, or sometimes true,
briefly summarize a method we used to demonstrate and/or prove
the answer in class
Bhaskara's proof of the Pythagorean theorem in Euclidean geometry
from test 1, and what goes wrong in spherical and
hyperbolic geometry and (sometimes) in taxicab geometry.
Parallels in Various Geometries and Applications:
Euclid's 5th postulate in Euclidean, spherical and hyperbolic geometry
Playfair's postulate in Euclidean, spherical and hyperbolic geometry
and the number of parallels in these geometries.
The proof of the
existence portion of Playfair in Euclidean and hyperbolic geometry and
what goes wrong on the sphere.
Finding lines of symmetry
in Escher's drawings and using this to identify the
geometry via the number of parallels (if any).
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].
Specific Examples of Types of Questions
Question types include short answer/short essay, like:
What goes wrong with the proof in hyperbolic geometry?
Does this always, never, or sometimes (but not always)
work on the sphere? Explain.
How did we explore this in class?
Sketch counterexamples for...
In the following proof, write out the
reasons using Book 1 of Euclid (which I will hand out to you)
Discuss in detail an example or topic from class that illustrated
connections among multiple mathematical perspectives
[which perspectives, and how the example/topic illustrated the connections]
Be familiar with the language and organization of the appendix of Sibley The Geometric Viewpoint p. 287-292 - I will give you a copy of this appendix 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:
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
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