Geometry for Geometry Teachers

Math 896 Summer 2011

Day 1

Course Information.

YouTube video of telescopes in Pirates of the Caribbean. Looking through tubes.

Webpage of GeoGebra.

Euclid's Proposition 1 on constructing equilateral triangles. Construction of regular quadrilaterals (squares).

Homework 1.

An excerpt of Flatland by Edwin A. Abbott. The full text can be read here and here.

Day 2

Discussion of Homework 1; presentation of tube-at-an-angle problem; Tom's GeoGebra worksheet.

Definitions in geometry. Discussion of Euclid's definitions and postulates from The Elements. Comparison with the University of Chicago School Mathematics Project axioms for Euclidean geometry.

Discussion of what makes a line straight. Clip from Flatland The Movie; also there is Flatland The Film which is closer to the original text.

Introduction to spherical geometry. Use of Lénárt spheres as manipulatives.

Homework 2.

Day 3

Discussion of Homework 2 (circumference and radius of circles on spheres, and bug on box problem).

Continued discussion of spherical geometry.

Continued discussion of lowest point of tilted square between two unit squares. Solution of two-dimensional problem. Modeling in GeoGebra. Posed problem of path of other corners of the square, and the three-dimensional cubes version.

Discussed definition of triangle in the plane. Checked definition against a menagerie of triangles. Made a Venn diagram categorizing types of triangles. Discussed spherical triangles.

Spherical Easel, a Java applet that models spherical geometry.

Homework 3. The Encyclopedia of Triangle Centers has a list of over 3000 triangle centers.

Day 4

Discussion of Homework 3 (GeoGebra construction of triangle centers, and cake cutting problem).

More discussion of the definition of triangles on the sphere.

SAS congruence of triangles in the plane; Euclid's and Henderson's proofs. SAS congruence of triangles on the sphere.

Minimizing distance carrying a bucket from the car to the river to the campfire.

Homework 4.

Day 5

Discussion of Homework 4 (range of angle sums of spherical triangles, and area of star).

Area of a spherical triangle.

Discussed solutions to bucket-carrying problem. Connection to Snell's Law on refraction.

Construction of quadrilaterals on the sphere; properties of angles.

Motion in GeoGebra of other points on the rotating square.

Homework 5.

Day 6

The Vanishing Leprechaun, created by Pat Lyons and published in 1968 by the W. A. Ellott Co. in Toronto.

Discussion of Homework 5 (wine bottles in GeoGebra and angle sums).

The spherical law of cosines and application to calculating distance on the Earth.

Models of a minimal set of geometric axioms. Finite geometries. A worked-through version.

Trucks going under bridges; video 1 and video 2.

Homework 6.

Day 7

Discussion of Homework 6 (moving sofa around corner, conveying card hand, height of right pyramid). Further discussion of the dropped cube problem.

Proving the Saccheri-Legendre theorem, using only the axioms of neutral geometry. A worked-through worksheet.

Homework 7.

Day 8

Discussion of Homework 7 (central angle theorem and circles at right angles).

Showing that the angle sum of triangles in the Euclidean plane is 180 degrees.

Building the soccer ball model of a hyperbolic plane. Angle sum of triangles in the hyperbolic plane.

Homework 8.

Day 9

Discussion of Homework 8 (Legendre's false proof of the parallel postulate, and Playfair's Axiom). Writeup for Problem 1.

Finished the worksheet on angle sum of triangles in the hyperbolic plane. Discussion of the critical angle for parallel lines.

GeoGebra model of the sofa problem. Discussion of the truck problem.

Poincaré disk model of hyperbolic geometry. A Java applet for experimentation on the hyperbolic plane in the disk model.

Empirically determined a relationship between area of triangles on the hyperbolic plane and their angle sums.

Homework 9. End of course assignment.

Day 10

Discussion of Homework 9 (testing whether space is Euclidean or hyperbolic). Discussion of relativity and the types of 3-dimensional spaces.

Derived the formula for area of triangles on the hyperbolic plane in terms of the angle sum defect.

Discussion of The Vanishing Leprechaun.

Regular shapes in 3-dimensional Euclidean space: Platonic and Archimedean solids.

Remember that the End of course assignment is due on Tuesday June 28, 2011. Homework problems re-worked for the Fabulous Five will be removed from the homework grade and instead will be counted as part of the End of course assignment.

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