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This website is set up to provide examples of some of the 3D manipulatives (such as balloons and paper polyhedra) and labs which my students and I have found to be useful in my Modern Geometry course. I presented this topic on March 27th 2004 at the SSHEMA conference in East Stroudsburg. (Doc format of paper, HTML format of paper) These toys enhance undergraduate exploration of non-euclidean geometry by strengthening students' intuition of non-euclidean spaces and augmenting their enjoyment of the subject in a laboratory-style and teambuilding setting.

The labs below build up to a discrete form of the Gauss-Bonnet theorem. To quickly summarize the main points of the labs:
1) In elliptic geometries/surfaces (such as a balloon), the total turning angle of a closed curve's tangent is less than 360 degrees
2) In hyperbolic geometries/surfaces (such as a horse's saddle), the total turning angle of a closed curve's tangent is more than 360 degrees
3) Triangular areas (hyperbolic or elliptic) are proportional to the difference between 180 degrees and the angle sum.
4) The net effect of the Gauss curvature enclosed within a closed curve can be determined from the total turning angle of the curve's tangent. (Gauss-Bonnet) The sum of the two quantities must equal 360 degrees (or 2 Pi). As a note: this concept parallels contour integration in Complex Variables around pole points!!!
5) The total Gauss curvature on an object which is topologically a sphere is 4 Pi. The Euler characteristic of such an object is 2.
6) The total Gauss curvature on an object which is topologically a torus is 0. The Euler characteristic of such an object is 0.
Lab Number HTML format DOC format Comments
1 Lab 1 Balloons 1 Lab 1 Balloons 1 Lab1: 1 day. Some assistance was given. This lab was fine, but I think the others were better. Obejective: Discover the triangle's angles add to more than 180 degrees. Get an idea that the amount by which the sum differs from 180 degrees correlates roughly with the area of the triangles. There was much error in the results, which I expected would cluster more closely around 16. But, balloons are inaccurate, and bh/2 is not an accurate approximation to the area of the triangle for a sphere. We even got the ratio A/Q as a negative number on one triangle!
2 Lab 2 Balloons 2 Lab 2 Balloons 2
Lab2: 2-3 days. Much assistance was given, especially with part 5. Objectives: Recall or learn that the area of a sphere is 4Pi R^2. Learn that a triangle on a sphere comes from 3 great circles. Derive formula for area of triangle on a sphere. Generalize to derive area of any polygon on a sphere.
3 Lab 3 Paper Models of Cone Points Lab 3 Paper Models of Cone Points Lab3: 1 day. Some assistance was given. Objectives: Draw and share hyperbolic and elliptic models with 1 to 2 cone points. Find out that the angles cut out or added to form cone points re-manifest themselves in the difference of angular sum (versus 180 degrees).
4 Lab 4 Paper Models of Polyhedra Lab 4 Paper Models of Polyhedra Lab4: 1 day. Some assistance was given. Objectives: Angles cut out or added are called angle deficits. An object like a sphere topologically has angular deficits adding to 4Pi.
5 Lab 5 Paper Models: Proof of Theorem using Euler's Characteristic Lab 5 Paper Models: Proof of Theorem using Euler's Characteristic Lab5: 2-3 days. Much assistance was given. Model of paper "square donut" was provided. Objectives: Proof of why topologically spherical objects have angular deficits of 4Pi. Also, if topology is like a torus, angle deficit is zero. Euler characteristics are used. in the proof.
6 Lab 6 Gauss Bonnet Lab 6 Gauss Bonnet Lab6: 2-3 days. Much assistance was given. Objective: Learn that the sum of angle deficits within a curve and angle defects on the curve sum to 2Pi. (Gauss-Bonnet)

LAB SOLUTIONS:

Answers to Balloon Two Day
Gauss-Bonnet Day

OTHER USEFUL SITES FOR USE WITH THE LABS:

Euclid's Elements Online
Where to find the paper polyhedra
Where to find the equilateral triangular grid paper

Two "Parallel" Lines on a Weird Surface