What are the shortest distance paths in hyperbolic geometry?
Sketchpad Shortest Distance Paths
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.
Image of Shortest Distance Paths
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.
Sketchpad 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 create AB.
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 FIB and IBG.
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 postulate 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 exactly two noncollinear hyperbolic halflines AB and AC which do not intersect m and such that 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.