# Questions tagged [triangulations]

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59 questions
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### High-Dimensional Analogs of Polygon Spaces

[Edit: I had a mistake in the numerology (took d=6,5 instead of d=5,4). Edit: I mistakenly identified my mistake, it is 6,5 but I got the indices shifted by one.] Background: Polygon spaces Given a ...
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### Homeomorphisms of the sphere mapping a geodesic triangulation to another one

Consider the standard Riemannian 2-sphere $S$, equipped with a geodesic triangulation $T$. Let $L(S,T)$ be the space of homeomorphisms of $S$ which map $T$ to a geodesic triangulation. What is the ...
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### Finite union of closed convex sets is triangulable?

I posted this question on math.stackexchange.com, but didn't get an answer. Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable, that ...
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### Can one determine the dimension of a manifold given its 1-skeleton?

This may be an easy question, but I can't think of the answer at hand. Suppose that I have a triangulated $n$-manifold $M$ (satisfying any set of conditions that you feel like). Suppose that I give ...
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### How can gauge theory techniques be useful to study when topological manifolds can be triangulated?

I was reading a review article arXiv:1310.7644 and it was explained there that in the last few years it was proven that there are topological manifolds of dimension greater than four that cannot be ...
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### What are some triangulations of Grassmannians?

A while ago I heard that there was no known triangulation of the Grassmannian of 3-planes in 6-space. To believe a statement like that, you have to be a little bit ungenerous about what you mean by "...
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### Comparing layered triangulations of 3-manifolds which fiber over the circle.

I am sorry but I am reposting this question because I wasn't logged in when I first asked it. Ian Agol has produced a method to build an ideal layered triangulation of a hyperbolic 3-manifold which ...
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### What is the number of equitriangulations of the n-cube?

I wonder if this question has been considered before and if anything is known. My search attempts have failed so far. Let's consider the n-dimesnional cube, $[0,1]^n$, and let's call a simplex with ...
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### Triangulation with simplices of same volume

Let $M$ be a Riemannian smooth compact manifold. It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ? ...
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### Proving the Gauss-Bonnet theorem for embedded surfaces using triangulations

I heard this really neat elementary proof of the "Gauss-Bonnet Theorem" : Let $S$ be a surface embedded in $\mathbb{R}^3$. Now take a triangulation of that surface, and approximate the surface with a ...
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### Hamiltonian circuit

Let us consider a disk with one labelled point on the boundary and $n$ labelled points in the interior. Let T be a triangulation of the whole disk with vertices on the labelled points such that T ...
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### Critical dimensions D for “smooth manifolds iff triangulable manifolds”

I am aware that at least for lower dimensions, "smooth manifolds iff triangulable manifolds" at least for dimensions below a certain critical dimensions D. My question is that for ...
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### Are there invariants of cell complexes similar to the Euler characteristic?

The Euler characteristic is an invariant (under homeomorphism) of manifolds that can be computed from a cellulation by (weighted) counting of different kinds of objects, namely \chi=\...
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### Efficient topological triangulations of non-convex polyhedra

Does every polyhedron in $\mathbb{R}^3$ with $n$ triangular facets have a topological triangulation with complexity $O(n)$? Suppose $P$ is a non-convex polyhedron in $\mathbb{R}^3$ with $n$ ...
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### Colorings of triangulations as a generalization of the four-color problem

My question can be viewed as a generalization of the four-color problem. Instead of a planar graph, consider a triangulation of a d-sphere. One wants to color vertices with N colors so that no two ...
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Let $P$ be a set of three noncollinear points in $\mathbb{R}^2$. Iteratively form the Delaunay triangulation $\cal T$ of $P$, and then augment $P$ by the circumcircle centers of all triangles in $\... 0answers 155 views ### Are there non-cuspy triangulations of smooth manifolds? In (as it turned out my misunderstanding of) the literature, a "smooth triangulation" seems to mean: a homeomorphism from a simplicial complex, such that on each simplex the map can be extended to a ... 6answers 423 views ### Characterizing Convex Configurations of Quadrupels of Coplanar Points via Linear (In-)equalities between Distance Sums or Differences Given 4 points$A$,$B$,$C$and$D$in general position in the euclidean plane, is it possible to determine from the 6 distances$AB$,$BC$,$CD$,$AD$,$AC$and,$BD$alone, whether every point is a ... 1answer 200 views ### Do random triangulation edge-flips maintain randomness? Let$S$be a fixed set of$n$points in the plane in general position. Let$T$be a triangulation of$S$, (somehow) selected uniformly at random from all triangulations of$S$. (There are an ... 2answers 85 views ### Complexity of Random Delaunay Triangulation in 3D My question: Is the number of cells in a three-dimensional Poisson-Delaunay triangulation with$n$vertices$\mathcal O(n)$with probability one? which is equivalent to the question Is the ... 1answer 283 views ### What properties does generalized Delaunay triangulation have? Suppose that instead of the usual circle, we pick some other convex set D and make the Delaunay triangulation of a finite planar point set with respect to this set, i.e. connect two points if there is ... 1answer 140 views ### Geometric realization of an abstract triangulation of the plane Can every abstract simplicial complex whose geometric realization is homeomorphic to$\mathbb{R}^2$be realized by a rectilinear triangulation of the Euclidean plane? Alternatively put, can a curvy (... 1answer 155 views ### The number of simplicial and general$d$-polytopes with$d+3$labelled vertices Micha Perles used Gale diagrams to compute the number of simplicial$d$-polytopes with$d+3$vertices and of general$d$-polytopes with$d+3$vertices. The computation can be found in Chapter 6.3 of ... 1answer 612 views ### Simplex in convex polytope, pulling triangulation Let$P$be a convex$d$-dimensional polytope. I have two questions, related to triangulations of$P$. Question 1: Let$p$be in the interior of$P$. Can I always find a triangulation of$P$, such ... 0answers 151 views ### Most regular way to triangulate$\mathbb{R}^3$? By "regular", I'm going by a property of Delaunay Triangulation, which is to maximize the minimum angle. In$\mathbb{R}^2$, tiling with equilateral triangles gives you a minimum angle of 60 degrees. ... 0answers 219 views ### (∞,n)-category of triangulated cobordisms What is an accepted definition of a (∞,n)-category of triangulated cobordisms? Is there one that has a forgetful functor to (Rezk - Hopkins -) Lurie's smooth cobordisms? Does it shed light on how ... 0answers 203 views ### Mean number of$n$-simplices per$(n-2)$-simplex in a triangulated$n$-manifold Work by Tamura (extending results by Luo and Stong) shows the following. Theorem: For any closed 3-manifold$M$and any rational number$4.5 < r < 6$there is a triangulation$T$of$M$for ... 2answers 215 views ### Minimum weight triangulation of lattice points in a circle Let$r$be a natural number, and consider the$\mathbb{Z}^2$lattice points$S$inside or on the circle$C$of radius$r$centered on the origin. Let$P$be the convex hull of$S$; so$P$is inscribed ... 1answer 2k views ### practical algorithm for constrained triangulation in two dimensions? I'm looking for an algorithm that is easy to implement in practice (resulting in small amount of code), preferably incremental. As far as I know, the biggest problem with incremental constrained ... 1answer 273 views ### Dehn-Sommerville relations for$\Delta$-complexes Let$M$be a closed, triangulated manifold of dimension$m$and$K(M)$be its triangulation. Let$f_i$denote the number of$i$-simplices of$K(M)$. As proved by Klee the face numbers satisfy the ... 0answers 205 views ### regular triangulations of the product of two simplices Is description of all regular triangluations of$\Delta^n\times \Delta^k$known? (Regular triangulations are those which correspond to vertices of Gelfand--Kapranov--Zelevinsky secondary polytope, or, ... 2answers 613 views ### Euclidean triangulation of the plane with degree 7 at each vertex. Hyperbolic plane has a beautiful triangulation by congruent hyperbolic triangles where all the vertices of the triangulation have degree 7, this is of course not possible in the euclidean plane, even ... 1answer 130 views ### Triangulations of translation surfaces whose edges are shorter than the diameter Let$S$be a compact translation surface (i.e. a surface endowed with a singular flat metric such that singular points are locally isometric to a cone of angle an integer multiple of$2\pi$, and that ... 1answer 101 views ### mean length of the non-crossing graphs on n points My original question is rather vague so I'll start with a precise example and then indicate possible generalisations. Given a n-tuple$x=(x_1,\dots,x_n)$in, say, a square with side-length$1$in the ... 0answers 73 views ### Existence of a “generic enough” lattice point interior to a lattice triangle Let$T$be a lattice triangle in$\Bbb R^2$(i.e. the convex hull of three noncolinear points in$\Bbb Z^2$), and assume it has at least one interior lattice point. Is it always possible to find a ... 0answers 79 views ### Find a certain triangulation subordinate to a given covering of a manifold Let$\{U_\alpha\}$be a covering of a smooth manifold$M$. Replacing it by a refined covering if necessary, we may assume some good properties of it, like, (1) any intersection$\cap_{i=1}^k U_{\...
This thread:triangulation of orbifolds has shown that any smooth closed orbifold has a triangulation. My further question is: if the difference of any two triangulations $P$ and $Q$ is a boundary of a ...
Consider a bounded Lipschitz domain $\Omega \subset \mathbb R^n$. Q1: Can its closure $\overline\Omega$ be triangulated? Q2: If yes, can the triangulation be chosen as finite? Furthermore, how ...