Title Bar - Cosmology

Hypersheet Directions

Try making a hyperbolic sheet as described here by William Thurston in his book Three-dimensional geometry and topology.

" (a) Cut out a number of equilateral triangles from paper...and join them so that there are seven around each vertex. Alternatively, sew pieces of cloth together.

" (d) These polyhedral models [described in (a)] have negative curvature concentrated at their verticies. You can make smoother models by diffusing the curvature out along the edges: replace each side of the equilateral triangles by an arc of circle such that the three sides of the resulting curvilinear triangle are 2/7. (The radius of the circle should be approximately 6.69 times the side of the triangle.)"

If you choose not to draw your own template, here is a model curvilinear triangle that you may choose to print and cut out out. Glue, tape or sew 7 of these triangles together at each vertex to form a hyperbolic sheet. footnote33

The first model is somewhat simpler even though it may not work as well for the Curvature of the univere activities, you can still use the sheet as a good example to show the class and the students will probably enjoy helping to construct the sheet. You can save some time when constructing this model by creating six equilateral triangles at once on the interior of a regular hexagon and then inserting an extra (seventh) triangle. To do this make a cut from one of the corners of the hexagon to the center of the hexagon and tape or glue the to sides of this slit to the sides of another equilateral triangle that is the same size as the six in the hexagon. You should notice that the paper will no longer lie flat, this is what causes the negative curvature.footnote38

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