Week 5: Polygons and Tilings on the SPhere Math 653 Fall 2012

Plan for Today Talk about spherical polygons (angle sum and area) Talk about tilings in the plane and on the sphere Make some connections to Euclidean geometry Briefly discuss the role of the sphere in earth maps.

Reminder Last week we discovered/developed “Girard’s Theorem”, which relates the angle sum and area of a spherical triangle. We wrote: 𝐴𝑟𝑒𝑎 Δ = 𝐴𝑛𝑔𝑙𝑒𝑆𝑢𝑚 Δ −180 720 ∙4𝜋 𝑟 2 How would this look if we used radians instead of degrees for the angles?

Angle Sum of a Polygon We also decided that the angle sum of an n-sided spherical polygon would range between 180(n – 2) and 180(n + 2) degrees. How could we characterize the angles in a regular spherical polygon? We can define a regular spherical polygon as a spherical polygon with all angles and all sides congruent.

Regular Tessellations in the Plane Take about the next 5 minutes and work with your partners to identify and justify which regular polygons can tessellate the plane. (A tessellation is a continuous tiling of the shape which can cover the plane with no gaps or overlaps.)

Regular Tessellations on the Sphere What arrangements of regular spherical polygons can tessellate the sphere? Take about the next 20-30 minutes with your group (or individually) to decide and justify your conclusions. Remember, worry about regular tessellations only (using only one kind of regular polygon).

Regular Tessellations on the Sphere For each of these tessellations, imagine what you would see if you connected each vertex, not with great circles on the sphere itself, but with straight lines that connect the points in 3 dimensions, cutting through the interior of the sphere. Notice anything interesting?

Polyhedra Let’s take the next few minutes and investigate a relationship between the faces, vertices, and edges in a polyhedron. A polyhedron is a simple, closed 3-D figure composed entirely of flat (planar) faces. Consider the quantity “Faces + Vertices – Edges”. What do you notices about this quantity for several different examples of polyhedra? We can prove this using the sphere!

Area of Spherical Polygons Let’s start by seeing what we can find out about the area of spherical polygons. Because we know that the area of spherical triangles is directly related to their angle sum, we can use a “partitioning into triangles” strategy to investigate the area of spherical polygons. Can you find a formula for the area of a general polygon on the sphere, in terms of n (the number of sides), the polygon’s interior angle sum, and the radius of the sphere? (I suggest using Radians.)

Proving Euler’s Formula Let V be the number of vertices in a polyhedron, F the number of faces, and E be the number of edges. Imagine placing the polyhedron inside the sphere and shining a bright light at the very center, then tracing the shadow of each edge onto the sphere. What would you see? We call this the spherical projection of the polyhedron.

Proving Euler’s Formula – An Alternative (Inductive) Approach Call χ (chi) the “Euler Characteristic.” We want to show that χ = 2 Start with a sphere with one point on it. Since there is one face (the sphere) and one vertex (the point) χ = v − e + f So, χ = 1 − 0 + 1 = 2 in this case. (The basis step)

Proving Euler’s Formula – An Alternative (Inductive) Approach Now (the inductive step) we build whatever tessellation we desire by using one of the following two moves: Move I: Add a new point and an edge connecting it to an existing point. Move II: Add an edge connecting two (different) existing point. What happens to χ when you use either of these two moves?

Proving Euler’s Formula – An Alternative (Inductive) Approach The key point is that neither Move I or Move II changes χ. Move I adds one vertex and one edge, which cancel each other in v – e + f. Move II adds one new edge, and cuts one face into two, creating a net increase of one face. Again, e increasing by 1 and f increasing by 1 cancel in v − e + f. To make this argument into a rigorous mathematical proof, we would need to argue that any spherical tiling can be built from the one-point-sphere via a series of Moves I and II.

Spherical Geometry & the Earth Although the earth is a sphere, we often act as if it were a plane. How big must a triangle on the earth be before we notice the spherical distortion effect? The radius of the earth is approximately 4000 mi. Colorado is roughly rectangular-shaped and has an area of 104,094 sq mi. What is the angle sum of the state of Colorado? How big would a region have to be (in sq miles) to have a detectable defect?