An Introduction to Polyhedral Geometry

Feng LuoRutgers undergraduate math club

Thursday, Sept 18, 2014New Brunswick, NJ

Polygons and polyhedra

3-D Scanned pictures

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.

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

Thank you