Questions tagged [triangulations]
The triangulations tag has no usage guidance.
46 questions with no upvoted or accepted answers
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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 ...
14
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420
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Minimum number of distinct triangles for tesselating the sphere
Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new ...
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3-manifolds with stacked links
Stacked spheres
A triangulation of a 2-dimensional sphere is called a stacked sphere if it is obtained inductively from the boundary of a 3-simplex by deleting a 2-face (triangle) $T$ adding a new ...
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Finite list of neighborhoods to approximate any finite simplicial complex
It is easy to see that any (locally finite) graph is homotopy-equivalent to a trivalent graph. Moreover, this can be achieved by a local construction - take neighborhoods of vertices of degree $> 3$...
- 1,842
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Can any smooth triangulation of a smooth manifold be blurred?
For the purposes of this question, let's say that a blurring
of a smooth triangulation $T$ of a smooth manifold $X$
is a smooth homotopy $h\colon [0,1] \times X \to X$ such that $h_0=\operatorname{id}...
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9
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Left adjoint functor between categories of polygons?
EDIT: Based on very helpful comments from Alec Rhea and Qiaochu Yuan I am adding some specification on objects and morphisms, hoping that this clarifies the idea behind these categories. I have also ...
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What is an intuitive explanation for a manifold to have no triangulation?
It is known that some topological manifolds, even compact and simply-connected ones, do not have admit a triangulation. One example is the E8 manifold in a dimension as low as $4$.
I am trying to ...
- 5,327
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Is there a combinatorial representation of general topological manifolds similar to triangulations?
Piece-wise linear manifolds are combinatorially represented by simplicial complexes modulo Pachner moves. However, for dimensions greater than $3$, the notions of piece-wise linear and topological ...
- 3,001
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Are triangulations of n-dimensional manifolds determined by lower-dimensional skeleta?
Suppose that $M$ is an $n$-dimensional manifold equipped with a triangulation $T$. Given $n\ge 1$, in order to recover $T$ (up to an isomorphism of simplicial complexes) one needs to know at least ...
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6
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Hamiltonicity for triangulations of the 3-sphere
A classical theorem of Whitney states that the 1-skeleton of every triangulation of the 2-sphere $\mathbb{S}^2$ has a Hamilton cycle as long as each of its 3-cycles bounds a triangle.
I'm wondering if ...
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If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ be a triangulable manifold?
If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ always be a triangulable manifold?
Let $X_d$ be a $d$-manifold which is NOT a ...
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226
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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. ...
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Triangulating piecewise-linear manifolds
Question 1: Is this the mainstream definition of a PL-manifold?
Definition. A PL-manifold is a manifold with an atlas $(\varphi_i)_{i\in I}$ in which all transition maps $\varphi_j\circ\varphi_i^{-1}$ ...
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Covering the sphere with an approximately planar grid
Consider a triangulation of a radius $R$ sphere into $n$ triangles. Must $Ω(\sqrt n)$ triangles have $Ω(1)$ relative difference from being an equilateral triangle of area $4πR^2/n$? ($Ω$ is from ...
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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 ...
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(∞,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 ...
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Retriangulating manifolds via triangulations of low local complexity
Suppose that $M$ is closed, connected PL $n$-manifold. We say that a triangulation of $M$ has local complexity at most $L$ if every zero-cell of $T$ meets at most $L$ $n$-simplices. (An alternative ...
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(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 ...
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286
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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 ...
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Volume of all Voronoi cells in n-dimensional bounded space
How can one find the volume of all Voronoi cells (bounded and unbounded) in an $n$-dimensional bounded space? For instance, consider an $N$-dimensional space (hypercube) with bounds on each dimension ...
- 131
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Minimal set of geometric moves in various equivalence classes of triangulated geometries
I would like to get to know what is the minimal set of geometric changes "aka. moves" (topology preserving modifications / Pachner moves / bistellar moves) that can transform any 3-...
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102
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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_{\...
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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, ...
- 109k
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75
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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 ...
- 3,001
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60
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Relation between symmetries of hyperbolic knot and the symmetries of a generic triangulation
For canonical ideal triangulation of a hyperbolic knot, the symmetries of the knot are the same as the symmetries of that triangulation. This is how SnapPy computes the knot symmetry group.
Is there ...
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23
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What is known about $\operatorname{card}_E(\mathrm{MST}\cap\mathrm{MWT})$?
It is a wellknown fact of computational geometry that the edges of Minimum-weight Spanning Tree are also found in the Delaunay Triangulation of a planar pointset $\mathcal{P}$, i.e. $\operatorname{...
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51
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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 ...
- 3,001
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87
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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 ...
- 3,079
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Does any smooth oriented closed orbifold have a fundamental class
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 ...
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Existence of triangulation of Lipschitz domains
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 ...
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Homotoping a 2-plane field on a closed orientable 3-manifold to a contact structure
I am a beginner in Contact Geometry.
To prove that the inclusion $\text{Cont}(M)\hookrightarrow \text{Dist}(M)$ induces surjection at $\pi_0$ level, the closest I got was based on Ko Honda's notes.
...
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Calculating an optimal scaling factor for Delaunay triangulations
consider a finite set $\mathcal{P}(x,y)=\lbrace(x_1,y_1),\dots,\,(x_n,y_n)\rbrace$ of points in the Euclidean plane and let $\mathrm{DT}(x,y)$ be the Delaunay triangulation of $\mathcal{P}(x,y)$
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Number of polyhedral covers of a triangulation of $S^2$
For a given triangulation (combinatorial Type I. or Type II.) of a $2$-sphere, what is the number of unique polygonal covers with $n$ polygons where ($n$ goes from $2$ to $N$)?
Under polygonal cover, ...
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References/applications/context for certain polytopes
First, let's consider an almost trivial notion. With any subspace $V\subset \mathbb R^n$ we associate a convex polytope $P(V)\subset V^*$ as follows. Each of the $n$ coordinates in $\mathbb R^n$ is a ...
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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&...
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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,...
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0
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How to do an elevated 2D Delaunay triangulation?
This is what I call an elevated Delaunay triangulation:
This is also called a 2.5D Delaunay triangulation.
To do it, I simply perform an ordinary 2D Delaunay triangulation with the (x,y)-coordinates, ...
- 2,560
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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 ...
- 2,560
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refining a coherent triangulation
I am relatively new to this topic, so this question may be easy/naive to some experts. Here goes..
I have a finite set of points $S\subset\mathbb R^2$ (you may increase the dimension of the ambient ...
- 2,275
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Regular triangulation of hypercube
I have started studying regular subdivisions of the $n$-cube, and came across the following post: Regularity of Delaunay triangulation of a hypercube.
My question is whether the "standard ...
- 11
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Interior and boundary vertices of weighted graphs
Xu He's article Rigidity of Infinite Disk Patterns and I have a problem with a statement he makes on page 7.
He considers weighted embedded planar graphs $G=(V, E)$ with weight function $\Theta: E \...
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Finding Equal-volume Triangulations in Homogenic Coordinates
Given the $n$-dimensional triangulation $\mathbb{T}^n$ of a finite set of points $\{p_1,\ ...\ ,\ p_{n+k}\} \subset \mathbb{R}^n$, is it always possible to find $n+k$ positive real weights $\{\omega_1,...
- 13.2k
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dissections and vertices of non-convex polytopes
Let us call a finite union $P$ of $n$-dimensional compact convex polytopes in $\mathbb{R}^n$ a non-convex polytope. Recall that a dissection of $P$ is a finite collection $T$ of $n$-dimensional ...
- 14k
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Number of tetrahedra inside a sphere with boundary A
I understand, that there are some combinatorial problems which are not yet solved regarding gluing triangulations in 3D. At least last time I checked, it was not yet known exactly how many ...
- 183
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Naming convention for different type of triangulations
When studying random geometries and related mathematical/physical stuff conflicting naming convention pops up regarding the naming of the different ensemble types of triangulations (in general ...
- 183
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Restrictions on crossing edges in Delaunay triangulations
what can be said about crossing edges in Delaunay triangulations, i.e. about pairs of edges that constitute to the heaviest perfect matching int the $K_4$ induced by the quadruplet of adjacent ...
- 13.2k