Flappy Triangles or How Curved is Your Surface?

MCTM Lesson Plan

Don Hickethier

Abstract: This lesson has absolutely nothing to do with the popular app Flappy Birds!Flappy Triangles do however provide a very nice hands-on activity to introduce students to curved surfaces and non-Euclidean geometry through the familiar sum of the angle measures of a triangle.

Background:

Euclid’s Parallel Postulate is unique to flat surfaces. One consequence of the Parallel Postulate is that the sum of the interior angles in a triangle equals two right angles (Proposition 32 of Book I of Euclid’s Elements).On non-flat, i.e. curved, surfaces the Parallel Postulate fails and hence the Angle Sum Proposition fails. On such curved surfaces one must introduce non-Euclidean geometries. The big question now is: How does one determine if a surface is flat or curved? Another big question: How would an ant on a smooth surface determine if it is living on a flat or curved surface? The answers to both questions are beautifully addressed in the Gauss-Bonnet Theorem. Loosely stated, for a triangle on a surface,the difference between the angle sum and 180° is equal to the Gaussian curvature over the triangle. (Don’t worry. You need to know nothing about Gaussian curvature.)

The Gauss-Bonnet Theorem gives us an intrinsic (ant on the surface) method to determine when a surface is curved. If the angle sum equals 180°, the surface is called flat. If the angle sum is greater than 180°, the surface is called spherical. If the angle sum is less than 180°, the surface is called hyperbolic. The difference, called the defect, determines how “curved” the surface is on the inside of the triangle.

Objective:

Students will learn to use the sum of angle measures from a “flappy” triangle to identify flat or curved surfaces. In addition students will be able to determine when a curved surface is spherical or hyperbolic.

Needed Supplies:

  • Protractor
  • Pencil
  • Straightedge
  • Tracing paper
  • Scotch tape
  • Smooth curved surfaces, e.g. any balls, banana, bowls, flower vase, paper towel roll, pumpkin, squash, coffee cup, …
  • Flappy Triangles, provided

Procedure:

For each object identify a relatively smooth area you wish to place your triangle.

Placing the Flappy Triangle

  1. Tape the tab Q to the surface.
  2. Fit the strip QR smoothly to the surface, no wrinkles. Tape in the middle and at R.
  3. Lay strip BP smoothly on the surface and tape at P.
  4. Place SC over BR so the line segments coincide and CT overlaps BP. Tape at S.
  5. Making sure line segments SC and BR coincide, tape the overlapping “flappy” sides. Label the intersection of BP and CT with Z.

Measuring the Defect

  1. Gently remove the flappy triangle taking great care not to change the angle BZC. You may want to remove the tape at S prior to removing the triangle.
  2. Place your “triangle” on a flat surface. Copy angle BZC onto the tracing paper. (It is okay if BR and CS no longer align.)
  3. Use your protractor to measure angel BZC.
  4. Determine the defect for your surface.
  5. Determine if your surface is flat, hyperbolic or spherical.

Conclusion

Determining where a surface is curved is just the beginning. If you happen to be fortunate enough to have one of those students or classes that “knows” all of the answers, this activity should open a few eyes. There are many more questions that can now be asked:

  • What would a surface of constant curvature look like?
  • Identify flat surfaces that don’t “look” flat.
  • Are there surfaces that have both spherical and hyperbolic regions?
  • Is it possible to do trigonometry on a curved surface? How?
  • What is a “straight” line on a curved surface?

This activity should also let students know that Euclidean geometry is more of a special exception (flat) to our reality. The fact of the matter is geometry is all around us and very little of it is on a flat surface.

This activity was taken for Using a Surface Triangle to Explore Curvature, The Mathematics Teacher, Vol 87, No 2, February 1994.

Shameless plug: I plan to offer a session during next Fall’s MCTM meeting in Missoula where we actually measure curvature with Flappy Triangles.