Hyperbolic Flexors
Summary:
Craft: Build flexible cardbord models of the hyperbolic, and see how they contort to fit into three dimensional space
Math: Examine the consequences of many hyperbolic verticies attached together. Look into the size and shape of a section of the hyperbolic plane.
summary
Last class we examined how different tilings can force nonzero curvature. Today we’ll build a more robust model of a negatively curved surface using cardboard and tape. These make very nice twiddle toys, because the dihedreal angle between cardboard planes is very flexible. By placing many hyperbolic vertices together, we will see how the hyperbolic plane contorts itself in three dimensional space.
Activity
Part 1: drawing the hyperbolic plane
From Annie Perkins: Question: On the blackboard, draw as manny connected triangles as you can, with seven edges at each vertex. What do you notice happens? Could you keep going forever?
Hyperbolic cardboard
I first saw this concept from Daniel Piker’s tweet. Here is a single vertex with 8 squares:
Here’s a single vertex with six hexagons
Here’s four vertices together, with 7 triangles each. Notice how it contorts into the familiar saddle shape.
The hyperbolic plane: Lots of triangles
From Annie Perkins: Question: On the blackboard, draw as manny connected triangles as you can, with seven edges at each vertex. What do you notice happens? Could you keep going forever?