1.1 When do you call a line straight? Describe
each symmetry, include practical interpretations for each, and
non-straight paths that satisfy them. Work to pull this
together into a definition that can work on all surfaces. If
you have any thoughts from children, I'd also be happy to see
it here.
2.1 What is straight on a sphere?
a.
Imagine
yourself to be a bug
crawling around on a sphere. Imagine you live
on the earth. What is
“straight” for you? What will you see or experience as
straight? How
can you convince yourself of this? Use the properties of
straightness (such as
symmetries) that you talked about in Problem 1.1.
b.
Show
(that is, convince
yourself, and give an argument to convince others) that the
great circles on a
sphere are straight with respect to the sphere, and that no
other circles on
the sphere are straight with respect to the sphere. Talk
about other paths also.
3.1 What is an angle?
Give some
possible
definitions of the term “angle.” Do all of these definitions
apply to the plane
as well as to spheres? For
each definition, what does it mean for two angles to be
congruent? How can we
check? Be sure to include object, measurement, and
motion. The order you do them may matter
logically. Do you have another interpretation?
3.2 Vertical Angle Theorem (VAT)
Prove:
Opposite angles
formed by two intersecting straight lines are congruent. What
properties of straight lines
and/or the plane are you using in your proof? Does your proof
also work on a
sphere? Why? Prove it using each of the definitions in
3.1.
4.1/4.2 Straightness on Cylinders
a.
What
paths are straight with
respect to the surface of a cylinder Why? Why are others not?
b.
Examine:
Can geodesics intersect
themselves on cylinders?
Can there be more than one
geodesic joining two points on cylinders?
c.
How do
we determine the
different geodesics connecting two points? How many are there?
Is there always at least one geodesic joining each
pair of points? How can we justify our conjectures?
d.
How
many times can a geodesic
on a cylinder intersect itself? How can we justify these
relationships?
4.1/4.2 Straightness on Cones
a.
What lines are
straight with
respect to the surface of a cone? Why? Why not?
b.
Examine:
Can geodesics intersect
themselves on cones?
Can there be more than one
geodesic joining two points on cones?
What
happens on cones with
varying cone angles, including cone angles greater than 360?
c.
How do we determine
the
different geodesics connecting two points? How many are there?
How does it
depend on the cone angle? Is there always at least one geodesic
joining each
pair of points? How can we justify our conjectures?
d. How many times can a geodesic on a cone intersect itself? How are these self-intersections related to the cone angle? At what angle does the geodesic intersect itself? How can we justify these relationships?
5.1 What is Straight in a Hyperbolic Plane?
a.
On a
hyperbolic plane, consider
the curves that run radially across each annular strip. Argue
that these curves
are intrinsically straight. Also, show that any two of them
are asymptotic, in
the sense that they converge toward each other but do not
intersect.
b.
Find
other geodesics on your
physical hyperbolic surface. Use the properties of
straightness (such as
symmetries) you talked about in problems 1.1, 2.1, and 4.1.
c.
What
properties do you notice
for geodesics on a hyperbolic plane? How are they the same as
geodesics on the
plane or spheres, and how are they different from geodesics on
the plane and
spheres?
5.4 Rotations and Reflections on Surfaces
a.
Let l
and m be two geodesics that intersect at the point P. Look at
the composition of
the reflection Rl through l with the reflection Rm through m.
Show that this
composition RmRl deserves to be called a rotation about P.
What is the angle of
rotation?
b.
Show
that Problem 3.2 (VAT)
holds on cylinders, cones (including the cone points), and
hyperbolic planes.
c.
Define
“rotation of a figure
about P through an angle theta” without mentioning reflections
in your
definition. What does a rotation do to a point not at P?
d.
One of
high school textbooks
defines a rotation as the composition of two reflections. Is
this a good
definition? Why or why not?
Do not write up 6.1.
6.2 a. Prove ITT (on three surfaces), b. prove
corollary, c. prove converse (also give counterexamples on the
sphere and reconcile this by indicating for which triangles on the
sphere your proof works).
In between we proved: If two isosceles triangles have the
same base, then the segment joining their top angles is also an
angle bisector to each of them and a perpendicular bisector to
their base. Feel free to use this; do not reprove it.
6.3 a. Describe and demonstrate a compass and straightedge
perpendicular bisector construction. Justify it.
Analyse it on all three surfaces. Strengthen your work by
using a Euclidean compass, which closes when you lift it off the
paper. b. same for angle bisectors. Try to see
the two as related.
6.4 Consider different Side-Angle-Side configurations on our three
surfaces. How many triangles do each determine? When
1, why? When 0, why? When 2, why? When more than
two, why? For situations in which there are two or more, can
you restrict to a subset of triangles on that surface so that
there is only one?
6.5 Same as 6.4, but for Angle-Side-Angle.
7.1 Derive a formula for the area of a triangle on a
sphere. Justify all steps. (I recommend following the
path we did in class.)
7.2 Derive a formula for the area of a triangle on a
hyperbolic plane. Justify all steps. (I recommend
following the path we did in class.)
7.3 Show all work to derive the possible values for angle
sum on all three surfaces. Pay particular attention to
justifying the planar case. Use the approach we did in
class.
7.4 Show all visual work to derive the holonomy of triangles
on the sphere, hyperbolic plane, and Euclidean plane. Be
careful about which prior results you are using.
9.1 Same as 6.4, but for Side-Side-Side.
9.2 Same as 6.4, but for Angle-Side-Side. Find as many
particular cases as you can where you limit to only 1 triangle.
9.3 Same as 6.4, but for Side-Angle-Angle.
9.4 Same as 6.4, but for Angle-Angle-Angle. Explain why the
situation is unsalvagable for the Euclidean plane. Justify
your results for the other two surfaces.
8.1 Prove Euclid's exterior angle theorem for all cases on all surfaces for which it is true, and for those which it is not, prove that it is not. (EEAT: any exterior angle of a a triangle is greater than either remote interior angle.)
8.2 Consider two lines that are parallel transports along a third line. Discuss any symmetries of this entire figure of three lines. What can you say about the two parallel transported lines? Do they intersect? If not, why not? If so, where? On all three surfaces.
8.3 [I believe David's questions are well written here, mostly this is just rewriting them.] Prove on all surfaces: if two lines are parallel transports along a third line, then they are also parallel transports along any other transversal (i.e. that crosses both lines) through the midpoint of the segment between the two lines. Are there other parallel transport lines on the sphere and hyperbolic plane? Prove: two lines are parallel transports if and only if they have a common perpendicular. Is the common perpendicular unique?
8.4 First six T/F questions: Parallel transport lines on H^2/S^2 do not intersect; any transversal has congruent corresponding angles for parallel transport lines on H^2/S^2; parallel transport lines on H^2/S^2 are everywhere equidistant. Fully justify your six T/F answers. Then show that there are pairs of lines on H^2 that do not intersect but are not parallel transports. Do NOT write up 8.4c.
10.1 Prove on the Euclidean plane: If two lines are parallel transports, then they are parallel transports along any transversal. Use as few assumptions as possible, and identify your assumptions explicitly. Since this is not true on the other surfaces, why not? This question is better if your assumption is more interesting. Thoughtful creativity is appreciated.
10.2 Logically prove the equivalence of the following: PT!, triangle angle sum=π, H = 0, and parallel transported lines are equidistant. Include your assumption in 10.1 if it was not about intersecting lines. There should be many steps to this to get it all logically equivalent.
10.3 Logically prove the equivalence of the following: Euclid's Fifth Postulate, the high school postulate, and your assumption in 10.1 if it was about intersecting lines. Then logically prove the equivalence of the group in this question to the group in 10.2. In the end you will have a list of six or seven statements which are all logically equivalent. They are different versions of what makes the Euclidean plane special or weird.
10.4 This question returns to H^2 and S^2 and asks what we can
say about parallelism there.
a. Show Euclid's fifth postulate is true on the
sphere, and that if the sum of the interior angles is less than a
straight angle on one side that the lines intersect closer on that
side. Be precise about what you mean by closer.
b. Show that given any point not a line in H^2,
there is a minimal angle of non-intersection, i.e. the smallest
angle to the perpendicular so that at that angle and any greater
than it will produce a non-intersecting line.
c. Use parallel transport to change the high
school postulate to something true on S^2 and H^2. Change as
little as you can, and keep uniqueness.
d. Adapt your 10.1 assumption to H^2 or S^2.
12.1 Both problems on the Euclidean plane: show that every triangle is equivalent by dissection to a parallelogram with the same base (for all bases), and show that every parallelogram is equivalent by dissection to a rectangle with the same base and height (for either choice) [This will require AP].
12.2 [We may switch top and bottom from the book:] Prove that the top angles of a Khayyam Quadrilateral are congruent. Prove that the perpendicular bisector of the base of a Khayyam Quadrilateral is also the perpendicular bisector of the top. Show that the top angles are greater than a right angle on sphere and less than a right angle on a hyperbolic plane. Finally show that a Khayyam Quadrilateral on the Euclidean plane is a rectangle and a Khayyam Parallelogram on the Euclidean plane is a parallelogram.
12.3 This is the analogue of 12.1 after learning 12.2 - so now on all surfaces: show that every triangle is equivalent by dissection to a Khayyam parallelogram with the same base (for all bases), and show that every Khayyam parallelogram is equivalent by dissection to a Khayyam quadrilateral with the same base (use David's definition of base) and height [This will require AP].
13.1 On the Euclidean plane, show that every rectangle is equivalent by dissection to a square.
13.2 On the Euclidean plane, show that the (disjoint) union of two squares is equivalent by dissection to another square.
13.4 Prove: If two triangles have congruent corresponding angles, then the corresponding sides of the triangles are proportional. If two triangles on a plane have an angle in common and if the corresponding sides of angle are in the same proportion to each other, then the triangles have corresponding angles congruent. Finally prove SSS similarity - if two triangles have proportional sides then the corresponding angles are congruent.
Constructions: Complete TWO of
20, 25, 30, and 34. Include computer constructions for each
one. Also include justifications for claims.