# Questions tagged [triangulations]

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### Approximation of connected set by triangluation / covering by simplices

Good afternoon. I have two distinct questions: If I have connected compact in $\mathbb{R}^n$, how much $(n+1)$-simplices are needed to fill its interior such that diameter of maximal uncovered part ...
1 vote
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### What is the difference between a simple polyhedron and a triangulated graph?

On a famous website I've seen the following: The skeletons of the simple polyhedra correspond to the triangulated graphs, the smallest of which are illustrated above. That "illustration above&...
123 views

### Two ears polygon in a maximal planar hamiltonian graph

Given a maximal planar graph (+6vertices) without separating triangles. Then it can have many Hamilton cycles°. Such a cycle divides the graph into two triangulated polygons. Is it always possible to ...
286 views

### Does every triangulable manifold have a vertex-transitive triangulation?

Does every triangulable manifold have a vertex-transitive triangulation? When I talk about a vertex-transitive triangulation of a manifold, I mean in the sense of realizing a manifold homeomorphically ...
146 views

### Handle attachment information from Morse function and triangulation

First, allow me to setup the relevant information. It is well known that a Morse function $f:M\to\mathbb{R}$ induces a handle decomposition of $M$. For simplicity, let's restrict for now to the ...
183 views

### Proof of Lemma 37.5 in Pak's Lectures on Discrete and Polyhedral Geometry

I am staring at the proof of Lemma 37.5 in Lectures on Discrete and Polyhedral Geometry, see page 331. I cannot understand why the required triangulation exists. In the first paragraph it says "...
1 vote
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### Relation of MSTs in the Euclidean plane to Delaunay triangulations

It is known that the Minimum Spanning Tree (MST) of a finite set of points in the Euclidean plane is contained in the point set's Delaunay triangulation, but is that all that can be said about their ...
185 views

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

Typically when one thinks about ideal triangulations of a $3$-manifold the link of each ideal vertex is a circle, so the ideal points correspond to toroidal cusps; alternatively, one can truncate the ...
1 vote
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### What are some other methods for partitioning an n-dimensional space based on a set of points in that space?

So this is a very general question, but I'm curious if there are any other methods for partitioning an n-dimensional space based on the location of a set of points, either randomly chosen or specified,...
177 views

### Distance between two points using triangulation

Suppose we have two points $p_1$ and $p_2$ in a metric space with unknown dimensionality, with no way to directly compute the distance between them, e.g. no coordinates. Say we can randomly sample a ...
83 views

### A question on relation of different triangulations of a triangulable space

Suppose we get two triangulations of a manifold with boundary $M$ such that the triangulation is compatible with boundary, i.e. the restriction on the boundary is itself a triangulation, is it these ...
232 views

### Do combinatorially equivalent polytopes have the same triangulations?

A triangulation of a convex polytope $P\subset\Bbb R^n$ is a partition of $P$ into $n$-simplices $\{\Delta_1,...,\Delta_m\}$ each of which has all its vertices among the vertices of $P$. A polytope ...
1 vote
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### Triangulations with discrete metrics and conformal equivalence

A discrete metric for a triangulation of a 2-dimensional manifold is a map associating $\mathbb{R}_+$-valued lengths to all edges, such that the triangle inequality holds on every triangle. In many ...
1 vote
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### Influence of the degenerate Delaunay tiles on the Voronoï diagram

About three or four years ago, I implemented the Delaunay and Voronoi tessellations in Haskell, with the help of the Qhull C library. Now I reimplement it in R. I have noticed that including or not ...
1 vote
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### Triangulations of conformal manifolds

I'm seeking for a discrete representation of $2$-manifolds with a conformal structure (i.e. metric modulo scalar prefactor). The topology of a $2$-manifold is determined by the combinatorics of a ...
150 views

### (Non-)Orientability of non-triangulable manifolds

We heard and learned from Mike Miller's answer to Not all manifolds can be triangulated: In which dimensions? that "All orientable 5-dimensional manifolds are triangulable. In dimensions at least ...
1k views

### On Thurston's triangulations of sphere

I have two questions from Thurston's paper . In the paper , Thurston talks about classifying certain classes of triangulations of the sphere. Here a triangulation of a sphere a Topological ...