An Introduction to Polyhedral Geometry
Feng LuoRutgers undergraduate math club
Thursday, Sept 18, 2014New Brunswick, NJ
The 2 most important theorems in Euclidean geometry
Pythagorean Theorem
Area =(a+b)2 =a2+b2+2abArea = c2+4 (ab/2)=c2+2ab
Gauss-Bonnet theorem
distances, inner product, Hilbert spaces,….
Theorem. a+b+c = π.
HomeworkCurvatures
The 3rd 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: http://www.math.brown.edu/~res/
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 = 0E = 12F = 6V = 8V-E+F = 2
genus = 0E = 15F = 7V = 10V-E+F = 2
genus = 1E = 24F = 12V = 12V-E+F = 0
4 faces
3 faces
A link between geometry and topology:Gauss Bonnet Theorem
For a polyhedral surface S,
∑v Kv = 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 E3.
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 Fh 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);
Fh(e) = +.
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.
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) = ∫x w
This function can be shown to be convex in x.