Geometry Explorer Lab F2. Edited F2 after class.
>>>Note, edited portions are marked by a >>> so you don't have
>>>to check every line.
Choose group members (3-4). Exchange email/phone coordinates.
Choose name of group: Thales, Pythagoras, Euclid, Archimedes,
Pappus, Klein, Poincare, Hilbert, Birkhoff.
Elect Presenter, Secretary, Demonstrator.
Record your experiments and prepare to write up a report.
A. Euclidean Geometry
In this exercise we explore Euclid's Five Postulates in all the
models GEX 2.0 gives us. See p403 of Hvidsten
E1: To draw a straight line from any point to any point.
E2: To produce a finite straight line continuously in a straight line.
E3: To describe a circle with any center and distance.
E4: That all right angles are equal to one another.
E5: That, if a straight line falling on two straight lines makes the
interior angles on the same side less than two right angles, the
two straight lines, if produced indefinitely, meet on the
side side on which are the angles less than two right angles.
Our translation:
E1: Two points determine a (unique) line (segment) between them.
E2: This segment lies on an infinite line.
E3: Draw a circle with given a radius (center and point on circle)
>>> E4: ????????
E4: (Perpendiculars) From a point outside a line drop the perpendicular.
E5: (Playfair's Postulate). Given a line and a point not on the
line there is a unique line through the point parallel to
the given line. (Lines with no common points are called parallel.)
>>> I've decided to substitute Euclid's Proposition 12 for E4. Thus our
>>> working versions of the postulates all say you can construct something.
Models:
Euclidean: Plane, Central Ortho, Central Stereo, Quad Model
Suggestion: Start with Central Stereo, and switch to Quad.
You can continue to construct in the top left view, and see
what it looks like in all three.
Hyperbolic: Poincare Disk, Klein Disk, Upper Half Plane, Quad
Elliptic: (This part of GEX 2.0 is not complete yet. For now
there is only one view
Vocabulary:
Wiggle: Choose an object (e.g. a point) and move it around.
Produce:
Choose endpoints of a segement > Transfrom Mark > Vector > 2 Pts
Choose segment > press Translate button.
B. Exterior Angle Theorem
Make this construction:
1. Choose an arbitrary triangle ABC with exterior angle DBC by
2. producing base AB one length beyond B to D..
3. Construct the median (midpoint of the opposite side) from A to A' on BC
>>> and produce is one length into the exterior angle.
4. Complete connect A"B to form exterior triangle A"A'B and prove that it
is congruent to triangle AA'C by SAS. (Watch the order of vertices!)
5. Conlude the angle ACB is congruent to angle A"BC.
6. Observe that the opposite interior angle at C has a congruent image
inside the exterior angle DBC. So it is smaller.
7. Repeat the construction by extending median CC', where C' is the
midpoint of AB.
8. What is your conclusion in Eucliean, Hyperbolic and Elliptic geometry?
A report on labF2 is due F3.