An Introduction to Polyhedral Geometry

1. An Introduction to Polyhedral Geometry Feng Luo Rutgers undergraduate math club Thursday, Sept 18, 2014 New Brunswick, NJ

2. Polygons and polyhedra 3-D Scanned pictures

3. The 2 most important theorems in Euclidean geometry Homework Curvatures Gauss-Bonnet theorem Theorem. a+b+c = π. Area =(a+b)2 =a2+b2+2ab Area = c2+4 (ab/2)=c2+2ab Pythagorean Theorem distances, inner product, Hilbert spaces,….

4. The 3rd theorem is Ptolemy For a quadrilateral inscribed to a circle: It holds in spherical geometry, hyperbolic geometry, Minkowski plane and di-Sitter space, … Homework: prove the Euclidean space version using trigonometry. It has applications to algebra (cluster algebra), geometry (Teichmuller theory), computational geometry (Delaunay), ….

5. Q. Any unsolved problems for polygons? Triangular Billiards Conjecture. Any triangular billiards board admits a closed trajectory. True: for any acute angled triangle. Best known result (R. Schwartz at Brown): true for all triangles of angles < 100 degree! Check: http://www.math.brown.edu/~res/

6. Polyhedral surfaces Metric gluing of Eucildeantriangles by isometries along edges. Metric d: = edge lengths Curvature K at vertex v: (angles) = metric-curvature: determined by the cosine law

7. the Euler Characteristic V-E-F 4 faces 3 faces genus = 1 E = 24 F = 12 V = 12 V-E+F = 0 genus = 0 E = 12 F = 6 V = 8 V-E+F = 2 genus = 0 E = 15 F = 7 V = 10 V-E+F = 2

8. A link between geometry and topology:Gauss Bonnet Theorem For a polyhedral surface S, ∑vKv = 2π (V-E+F). The Euler characteristic of S.

9. Q: How to determine a convex polyhedron? Cauchy’s rigidity thm (1813) If two compact convex polytopes have isometric boundaries, then they differ by a rigid motion of E3. Assume the same combinatorics and triangular faces, same edge lengths Thm Dihedral angles the same. Then the same in 3-D.

10. Thm(Rivin) Any polyhedral surface is determined, up to scaling, by the quantity F sending each edge e to the sum of the two angles facing e. F(e) = a+b h =0: a+b; h=1: cos(a)+cos(b); h=-2: cot(a)+cot(b); h=-1: cot(a/2)cot(b/2); So far, there is no elementary proof of it. Thm(L). For any h, any polyhedral surface is determined, up to scaling, by the quantity Fh sending each edge e to : Fh(e) = +.

11. Basic lemma. If f: U R is smooth strictly convex and U is an open convex set U in Rn, then ▽f: U  Rn is injective. Proof.

12. Eg 1. For a E2 triangle of lengths x and angles y, the differential 1-form w is closed due to prop. 1, w= Σi ln(tan(yi /2)) d xi. Thus, we can integrate w and obtain a function of x, F(x) = ∫xw This function can be shown to be convex in x.

13. This function F, by the construction, satisfies: ∂F(x)/ ∂xi = ln(tan(yi /2)).

14. Thank you