# Tag Info

## Hot answers tagged triangulations

29 votes

### Can one determine the dimension of a manifold given its 1-skeleton?

You cannot hope to find the dimension exactly from the 1-skeleton alone. The complete graph on seven vertices is both the 1-skeleton of a triangulation of the two dimensional torus and of the five ...
• 83.2k
24 votes
Accepted

### Critical dimensions D for "smooth manifolds iff triangulable manifolds"

All smooth manifolds are triangulable, as you say. This follows from Morse theory, which dictates that you only need to know how to triangulate (PL) handle-attachments, which one can do by hand. The ...
• 8,850
24 votes
Accepted

### Acute triangles in "obtuse" polygons?

Take a very obtuse isosceles triangle and chop its acute angles.
• 41.2k
17 votes
Accepted

• 9,439
7 votes

### Minimum weight triangulation of lattice points in a circle

I would like to propose a suggestion for finding some asymptotic bound, I think it should be $$\frac{1}{2}\cdot (2+\sqrt{2})\cdot \pi r^2.$$ Namely, the ratio of this number to the actual weight will ...
• 28.4k
7 votes

### Can triangulations (or some related combinatorial structure) distinguish smooth structures on $RP^4$?

• 6,626
7 votes
Accepted

### Is every (not necessarily PL-) triangulation of a manifold pure, non-branching and strongly-connected?

Suppose that $M$ is a connected $d$-dimensional topological manifold without boundary. (We make the last assumption to simplify matters.) Let $\Delta$ be the given pseudo-triangulation. So the ...
• 22.4k
7 votes
Accepted

### Refining a triangulation

Suppose that $S$ is a closed, connected surface with negative Euler characteristic. Suppose that $T$ is a triangulation of $S$. Define "refine" to mean "replace each triangle by four ...
• 22.4k
7 votes

### Ideal triangulations of $3$-manifolds with "cusps" of genus $\ge 2$

A nice class of examples are the (generalized) triangulations with only one edge. The manifolds obtained by removing an open neighborhood of the vertex have totally geodesic boundary (or a cusp in the ...
• 63.2k
6 votes

• 41.2k
3 votes

### Complexity of Random Delaunay Triangulation in 3D

Yes, at least for periodic boundary conditions so we don't have to worry about what happens near the boundary. See Dwyer, Rex A. Higher-dimensional Voronoĭ diagrams in linear expected time. Discrete ...
• 18.3k
3 votes

### Geometric realization of an abstract triangulation of the plane

Yes. For finite complexes this is completely standard, for infinite graphs, this follows by a compactness argument (in particular, there is an infinite circle packing, though it is not unique in ...
• 94.5k

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