MA-3329 Modern Geometry Name: ____________________________

Fun Paper Model Day #1

So far, we know that in hyperbolic geometries, triangles' angles add up to less than 180 degrees, and

in elliptic geometries, triangles' angles add up to more than 180 degrees. In both cases, the difference

between that sum and 180 degrees gives the area of the triangle, up to a constant. For the following models,

however, only the sums of the angles will be useful to us. Our elliptic or hyperbolic areas are compressed

down to single points of a surface, and the remainder of the surface is flat everywhere. Note: any surface which can be built up from flat pieces of the plane is called "developable". Spheres are not developable, but cones are, for example. A cone is flat everywhere except the tip. Spheres are rounded equally everywhere and cannot be built up from pieces of the plane. These paper models are special types of developable surfaces, as are most of your clothing: jeans, shirts, skirts.

Instructions:

1) Look at the models in the front of the room. What do you notice?

2) Take a couple pieces of triangular graphing paper and cut darts into them. Either add paper to the

darts, or remove paper, creating ruffles and/or points. (Ruffles have hyperbolic curvature while

points have elliptic curvature.) (You may also use one of the extra models already created.)

3) Draw a triangle on your surface. You may choose to use the handy 60 or 120 degrees already marked

or you may make the angles however you like. Measure any remaining angles and find their sum.

How do your results support or contradict your predictions based on your observations of the original

models?