Save each Sketchpad file (control/click and then download it)
and then open it up from Sketchpad and follow the directions:
- What are the shortest distance paths in hyperbolic geometry?
Once you have answered the question in the sketch, drag point L and compare the shortest
path to the symmetric paths on
Escher's Heaven and Hell.
- What was the sum of the angles in the triangle we formed in
Escher's Heaven and Hell?
- What is the sum of the angles in a hyperbolic triangle? How
large can the sum of the angles get? How small can the sum of the angles
get? What kind of triangles must you form to get a large/small angle sum?
Sketch pictures in your notes.
Sketchpad Sum of Angles
Image of Sum of Angles
- Turn to Appendix A in Sibley and (in your notes)
write down the statement of
Euclid's 5th postulate.
- Is Euclid's 5th postulate ever (ie sometimes), always or never true in
hyperbolic space? Drag C in the sketch to answer this question and sketch
pictures in your notes.
Euclid's 5th Postulate
Image of Euclid's 5th Postulate
Playfair's axiom says: given a line and a point not on it, exactly
one line parallel to the given line can be drawn through the point.
Show that the existence portion of Playfair's axiom works
in hyperbolic geometry via Sketchpad:
From Sketchpad, use Help/Sample Sketches and
Tools/Advanced Topics/ Poincare Disk Model of Hyperbolic Geometry
Click on the Script View Tool
. Notice that
this sketch comes with pre-made hyperbolic tools, which is what we will
use for hyperbolic constructions and measurements.
Under Hyperbolic Tools, choose Hyperbolic Line and then
Choose Hyperbolic Perpendicular,
click on A, then B, and then on a point off of the line to create a perpendicular [line through D is
perpendicular to AB].
The hyperbolic perpendicular is still selected, so now click on two points of your perpendicular and a point off of it [line through F is
perpendicular to DE and parallel to AB].
Create points and use the Hyperbolic Angle to measure the angles
and verify that they are right angles. Click on Hyperbolic Angle
each time you wish to measure.
Compare your work with the following image
of the existence portion of Playfair's axiom to verify
that you have
the correct diagram and measurements. If not, try again!
Recall that this part of the Euclidean proof only required up to I-27,
which did not require Euclid's 5th postulate, so it is not surprising that the
construction still works in hyperbolic geometry.
- Once your Sketchpad work in #2 matches the image above:
Use a Hyperbolic Segment to connect I and B.
Use the Hyperbolic Angle to measure the alternate interior angles
Notice that the angles are not equal.
Compare your work with the
image of the hyperbolic measurement of the
alternate interior angles. If it does not match, then repeat
the construction and the measurements.
Review I-29 in Sibley's The Geometric Viewpoint. This
proposition does not hold in hyperbolic geometry.
This should not be surprising, since I-29
required Euclid's 5th postulate, and we showed above that Euclid's 5th
doesn't always hold in hyperbolic geometry.
- Review our Euclidean proof that the
sum of the angles in a triangle is 180 degrees (I-32).
What goes wrong in the Euclidean proof for hyperbolic geometry?
Use the above to help you answer this question.
- The Hyperbolic Parallel Axiom states that
if m is a hyperbolic line and A is a point not on m, then there exist
noncollinear hyperbolic halflines AB and AC which do not intersect m and such
a third hyperbolic
halfline AD intersects m if and only if AD is between AB and AC.
Try to make sense of this axiom by creating a hyperbolic sketch that
illustrates it. Be sure to use the
Hyperbolic Line and Hyperbolic Segment tools.