Geom/Top RTG 2017 Polygonal Surfaces and Euler Characteristic Author: Prof. Caine Notre Dame Name:

Directions: In this activity you will build models of polygonal surfaces out of paper and study some of their features. You will need scissors and tape in order to construct the models. Work with a partner or small group through each of the exercises. This will set up the story in the final lecture of the workshop.

1 Polygonal Surfaces

We are familiar with convex polygons in the plane such as triangles, squares, parallelograms, trapezoids, pentagons, and so on. These basic two-dimensional shapes can be lifted into space, rotated, and assembled by taping edges to edges to form polygonal surfaces, like the surface of a jewel.

3 Definition 1. An polygonal surface Σ in R is a union of a finite collection of planar convex polygons in 3 R with the property that the intersection of any two polygons in the collection is either empty or an edge of both, and no more than two polygons meet along any given edge. For simplicity, we will assume that 3 the union is connected as a subset of R . The vertices of Σ are the common vertices of all the polygons. The edges of Σ are the common edges of all the polygons. The faces of Σ are the polygons themselves. A polygonal surface is closed if every edge is adjacent to two faces of the surface.

Example 1. Convex polyhedra such as the tetrahedron, cube, or cuboctahedron are examples of closed 3 polygonal surfaces in R .

Example 2. Other examples can be made by removing matching faces from two polyhedra and gluing them together along the corresponding edges. For example, this polygonal surface which represents the surface of a barbell is obtained by gluing together two octahedra and a cube.

Exercise 1. Using the attached handout, scissors, and tape, build paper models of a cube, tetrahedron, and cuboctahedron. Each member of the group should construct at least one model of each shape, but it will be better if each person can make two. Later, you will use these models to experiment with what kinds of polygonal surfaces you can construct by gluing such pieces together. The more basic ingredients you produce, the larger and potentially more complicated surfaces you will be able to construct. Remark 3. There is no requirement that the polygons comprising a polygonal surface be regular, but many of the examples you will play with will have this feature (i.e., the edges of the surface all have the same length). To imagine other examples, consider stretching the surface by pulling on one or more vertices while leaving the others fixed. For example, a cube can be stretched into a rectangular box by pulling the four vertices of the top face upward by the same amount.

2 Vertices, Edges, and Faces

Exercise 2. Work with a partner to count the numbers of vertices, edges, and faces of each of the polygonal surfaces you made and enter them in a table as shown below. Be careful in your counts. It is imperative that you are accurate. Surface Σ V E F Tetrahedron 4 6 4 Cube Cuboctahedron . . Add to this table surfaces you can make with the basic models that you and your partner have by matching faces together. For example, how would the counts change if you glued a cube onto a square face of the cuboctahedron? Or a tetrahedron onto a triangular face of the cuboctahedron? Create at least 4 additional surfaces this way and enter their counts into your table. Devise some method of describing the surfaces you create, either by picture or some a verbal description of how it was assembled. You’ll want to be able to reconstruct them later. Exercise 3. Now, for each of your surfaces, return your table and compute the alternating sum χ(Σ) = V − E + F for each surface. What do you observe? The Greek letter χ we write as chi where the ch is pronounced as in the word “character” and i is pronounced “eye.” The quantity χ(Σ) is called the Euler characteristic of the polygonal surface in honor of the great Swiss mathematician Leonhard Euler. Exercise 4. It is very likely that you computed the same number χ(Σ) for each of the surfaces you built. What number was it? Work with a larger group to build a polygonal surface having a different Euler characteristic than the one you found. To help your problem solving, suppose that Σ is a polygonal surface built out of cuboctahedra, cubes, and tetraheda and consider how the Euler characteristic is updated when you glue on a single cube, tetrahedron, or octahedron. What would need to be true for this update not to produce the same number? With a larger group, you will have more paper models to use as building blocks and more minds to work the problem.

Exercise 5. What numbers can occur as the Euler characteristic of a closed polygonal surface made out of tetrahedra, cubes, and cuboctahedra in this way? Can you describe in some way the fundamental difference between such surfaces with different Euler characteristics? What does the Euler characteristic of a surface actually measure?

3 Discrete Curvature for Polygonal Surfaces

For oriented regular curves, the sign of the curvature at a point indicated whether the curve was turning left or right and the absolute value was the reciprocal of the radius of the best fitting circle to the curve at that point. For oriented polygonal curves, the curvature was defined at each vertex with sign indicating whether the next leg turned left or right and absolute value equal to the exterior angle at formed by the segments at that vertex. This discrete definition reflected the continuous one, matched our intuition, and more importantly allowed us to define a discrete version of the rotation index for closed oriented polygonal curves matching the continuous one for closed oriented regular curves. What do we mean by curvature at a point on a surface? What is the corresponding discrete concept for polygonal surfaces? In a course on the differential geometry of curves and surfaces, a considerable amount of time is spent on the careful definition of curvature for surfaces. In these notes, we will take a more relaxed approach in order to motivate a discrete definition of curvature for polygonal surfaces. Without explaining precisely what the curvature of a regular surface at a point is, here are three basic examples to get a feel for what it measures.

negative curvature at X zero curvature at X positive curvature at X

Saying that the middle example has zero curvature at the point X may not match your intuition. After all, the surface certainly does not appear to be flat like a plane. But in fact, the curvature of such a surface is zero at every point. A model of that surface can be constructed by taking a flat sheet of paper and bending it upward into the shape of a parabola while it is not possible to bend a sheet of paper into either of the other two surfaces. So perhaps there is something intrinsically flat about the middle example and the curvature reflects this. For the non-zero cases, imagine the curve of intersection of the surface with a plane through X containing the line perpendicular to the surface at X. There is a one parameter family of such curves obtained by rotating a given such plane through all the possibilities about the line through X perpendicular to the surface. Positive curvature at X reflects the fact that all of these curves bend toward the same side of that line where as negative curvature indicates that some bend towards while others bend away.

Definition 2. Let Σ be a polygonal surface and v be a vertex of Σ. The discrete curvature κΣ(v) of Σ at v is defined to be 2π minus the sum of the measures of the angles about v, i.e., X κΣ(v) = 2π − m(α) α where α ranges over the angles with vertex v from the polygons comprising Σ.

Example 4. The tetrahedron has four vertices and three equilateral triangles meeting at each vertex. π Therefore, the sum of the measures of the angles meeting a given vertex is 3 · 3 = π. Hence, the discrete curvature is κΣ(v) = 2π − π = π > 0 for each vertex v of the tetrahedron Σ.

Exercise 6. Follow the instructions on the attached sheet to build the “Curvature Simulator.” By ma- nipulating the simulator, can you make models of simple polygonal surfaces resembling the three basic examples? How does the sign of the discrete curvature of your models compare with the stated sign of the continuous curvature?

Exercise 7. For the cube, the tetrahedron, and the octahedron, compute the value of the discrete curvature at each vertex. Let V denote the set of vertices of the surface Σ. For each of these surfaces, compute the quantity 1 X κ (v). 2π Σ v∈V What do you observe?

Exercise 8. Carefully label the vertices of the surface in Example 2. Work with a partner to compute the discrete curvature at each vertex. Again, accuracy is important here so work together. Then compute the quantity 1 X κ (v). 2π Σ v∈V What do you observe? Try this for the other polygonal surfaces you constructed. What patterns do you observe? Directions: 1. Cut out the perimeter of each net of polygons. Cube 2. Fold along each interior edge and crease like a mountain ridge. 3. Join remaining edges with tape to form the indicated polyhedron.

Cuboctahedron

Tetrahedron Curvature Simulator

Directions: 3

1. Cut along the perimeter of Join 3

each both gures. π 2. Along each interior edge fold and π crease as a mountain ridge, then reverse

the fold and crease as a valley, then return to π 3 at. 3. Connect the two pieces along the edge marked π “join” so that the angle labels all surround a common vertex and the two 3 pieces overlap.

Join

3 π 3 π

π π 3 3