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Topics in Geometry |
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Topics in Geometry |
Don taught a Topics in Geometry course for fifteen students this past fall. He used Greenberg's Non-Euclidean Geometry, and made extensive use of Sketchpad. Students completed fifteen to twenty Sketchpad projects, and used Sketchpad for homework. Don also used Sketchpad interactively in class in a mode in which he drove the program and students told him what to do in order to pursue their conjectures. He showed six sketches including some very interesting student work. Below are the notes taken from the six sketches.
inv = ref
This sketch illustrates that, for the Poincaré disk, hyperbolic reflection corresponds to Euclidean inversion. Namely, assume that the blue circle (= hyperbolic line) is given and that P is a given point. Then P' is constructed as the inverse of P in the given circle. The green circle is the hyperbolic line through P and P'. One observes that the green and blue circles intersect orthogonally and that P and P' are equidistant from the blue circle (where distances are measured hyperbolically). Hence, P' is the hyperbolic reflection of P.
Rotation?
This takes place in the upper half-plane. The dark red line is the x-axis, and the magenta ray is the positive y-axis. The sketch examines the transformation f(z) = -1/z (denoted z# above), which can be decomposed as inversion in the unit circle (black) followed by reflection in the y-axis, i.e., a composition of two hyperbolic reflections. Since the lines of reflection intersect orthogonally, one might suspect that the composition is a rotation by 180 degrees. By dragging point z, one sees some evidence that this is true (the blue circle is the hyperbolic line through z and z#).
Translation?
Now we are back in the Poincaré disk. This sketch examines a composition of reflections in parallel lines (first the red line, then the blue). By analogy with Euclidean geometry, one might suspect that the composition is a translation. The sketch suggests that this is false; the hyperbolic distance between S and its double refection S'' does not remain constant as S varies. On the other hand, If S is constrained to lie on the common perpendicular to the lines of reflection (magenta), then the distance between S and S'' does remain constant and equals twice the separation between the lines.
Pentagon
This was done by a student, Ken Williams, on a take-home exam. The problem was to construct a pentagon in the Poincaré disk, all of whose angles are right angles. To save space, I have deleted from the sketch the sound and music that was part of the demonstration in Cincinnati.
Euclidean and Hyperbolic Morley
This was again done by a student, Stephen Sample, as part of a paper on Morley's Theorem. The theorem states that, for any given triangle ABC, the angle trisectors intersect to form an equilateral triangle (labelled PQR above). This sketch gives empirical evidence that this is true. In the sketch "Hyperbolic Morley," Stephen goes on to collect evidence that the thoerem remains true in hyperbolic geometry (using the Poincaré disk), though the usual proofs no longer apply.
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