Hyperbolic surfaces

You are probably familiar with the hexagonal tiling of the plane:
400px-Tiling_Regular_6-3_Hexagonal.svg
There is something special about hexagons that makes this successful. In this discussion we asked what would happen if we tried to tile using hexagons together with other polygons. For example, if at every corner you place two hexagons and one pentagon, you will get a polygonal “soccer-ball”.
Comparison_of_truncated_icosahedron_and_soccer_ball
Instead of tiling an unbounded plane, the figure has curvature to it which causes it to be bounded.

If, on the other hand, at each vertex you place two hexagons and one heptagon, you get a figure which does not lie flat and also does not close up:
soccerball
Because of this, the surface is said to have negative curvature. In fact it is a polygonal approximation to a “hyperbolic plane”.

We each made planes like this and then used a straightedge and protractor to observe several geometric phenomena that occur in this plane.

  • We drew a pair of parallel lines (creating an ‘H’ shape). Participants noticed that the two parallel lines seemed to diverge from one another
  • We drew three straight lines to form a triangle. My triangle had its angles sum to just 156 degrees!
  • When drawing a triangle, many students noticed that even though they tried to make the third side hit the other two, it would diverge. This means there are many lines parallel to a given one through any point! (That is, Euclid’s parallel postulate doesn’t hold.)
  • Some participants observed that the larger a triangle they drew, the smaller the interior angle sum. In hyperbolic space, non-congruent triangles are automatically non-similar!

This activity was borrowed from Frank Sottile. Check it out here!

Photos of us working

IMG_0003

IMG_0004

IMG_0005

IMG_0001

IMG_0002