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

Polygons and polyhedra 3-D Scanned pictures

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

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

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:

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

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

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

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

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

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

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

This function F, by the construction, satisfies: ∂F(x)/ ∂x i = ln(tan(y i /2)).

Thank you