Questions tagged [triangulations]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
1 answer
57 views

Annulus theorem for pseudomanifolds

Lets say I take an arbitrary closed and smooth $d$-manifolds $\mathcal{M}$. Now, it is a well-known fact that whenever I take two (sufficiently nice embedded) closed $d$-balls $B_{1}$ and $B_{2}$ in $\...
user avatar
0 votes
0 answers
23 views

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, ...
user avatar
8 votes
1 answer
348 views

Refining a triangulation

I'm reading Thurston's article "Shapes of polyhedra and triangulations of the sphere." In the introduction he claims the following: "${}^{(1)}$There are procedures to refine and modify ...
user avatar
3 votes
1 answer
94 views

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

A triangulation of a topological manifold $\mathcal{M}$ possibly with boundary is an abstract simplicial complex $\Delta$ together with a homeomorphism $\varphi:\vert\Delta\vert\to\mathcal{M}$, where $...
user avatar
  • 351
2 votes
0 answers
43 views

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 ...
user avatar
  • 2,475
1 vote
0 answers
31 views

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 ...
user avatar
1 vote
0 answers
96 views

Bordism for oriented triangulable manifolds without smooth differentiable structures

We know the bordism group for oriented smooth differentiable structures such as $\Omega_d^{SO}$ that requires the special orthogonal group structure on the tangent bundle $TM$ of manifold $M$. $$\...
user avatar
  • 9,966
12 votes
0 answers
200 views

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 ...
user avatar
  • 23.5k
5 votes
1 answer
315 views

Presentations of exotic 4-manifolds

TLDR I want to see more examples of exotic $4$-manifold (hopefully connected, simply connected, oriented, and closed). Are there known presentations of $4$-manifolds $M$ with exotic structures, ...
user avatar
  • 4,005
4 votes
1 answer
197 views

Triangulation of a simplex

I am looking for a triangulation of an $n$-dimensional simplex such that all sub-simplices are of comparable size, and are "as close as possible" to a regular simplex : the latter property ...
user avatar
  • 43
1 vote
0 answers
40 views

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 ...
user avatar
4 votes
1 answer
88 views

Ideal triangulation of hyperbolic 3-manifold with generic mapping class group

I am from physics background so I apologize in advance if my question is trivial. Kojima proves for every finite group $G$, there is a hyperbolic 3-manifold such that its mapping class group equals $G$...
user avatar
6 votes
0 answers
101 views

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 ...
user avatar
  • 955
5 votes
1 answer
333 views

Properties a triangulation must have in order to describe a manifold

I am mainly interested in the $3$-dimensional case. It is a well-known fact, following from the work of E. E. Moise and R. H. Bing in the 1950s, that every $3$-dimensional topological manifold (with ...
user avatar
0 votes
0 answers
36 views

Reference for an approximation algorithm for Delaunay triangulation in higher dimension

I understood that there is an algorithm that approximates the Delaunay triangulation (in $\mathbb R^d$, $d\geq 2$), but couldn't find any paper or reference for it. If you know such a reference, I'll ...
user avatar
8 votes
1 answer
552 views

When is a triangulation of sphere two-colorable?

Let $T$ be a triangulation of sphere. We say that $T$ is $k$-colorable if the triangles of $T$ can be assigned with $k$ colors such that any two triangles with a common edge have different colors. I ...
user avatar
  • 29.6k
1 vote
1 answer
44 views

On triangulations and "coverage" of circumcircles

Let $P$ be a convex quadrilateral defined by four vertices $a$, $b$, $c$, and $d$. Suppose that the circumcircle of $\triangle abd$ contains $c$.* Let $D(\triangle abc)$ to denote the area enclosed by ...
user avatar
1 vote
0 answers
45 views

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 ...
user avatar
  • 2,060
4 votes
0 answers
155 views

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 ...
user avatar
15 votes
1 answer
575 views

Acute triangles in "obtuse" polygons?

Let $P$ be a convex polygon. Suppose every interior angle of $P$ is obtuse. Is it always the case that there exist three vertices $p, q, r$ of $P$ such that $\triangle pqr$ is acute? I conjecture ...
user avatar
6 votes
0 answers
161 views

Can $X_4 \times S^1$, $X_4 \times I^1$, or $X_4 \times \mathbb{R}^1$ be a triangulable, PL or DIFF manifold, if $X_4$ is a non-triangulable manifold? [duplicate]

Question: If $X_4$ is a non-triangulable topological (TOP) manifold, can $X_4 \times S^1$, $X_4 \times I^1$, or $X_4 \times \mathbb{R}^1$ be a triangulable manifold? can $X_4 \times S^1$, $X_4 \...
user avatar
  • 9,966
6 votes
0 answers
190 views

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 ...
user avatar
  • 9,966
3 votes
1 answer
74 views

Surfaces generated by minimum-weight triangulations

The minimum-weight triangulation of simple polygons can be efficiently calculated in $O(n^3)$ time by dynamic programming, while it is $\text{NP-hard}$ for pointsets in general. Looking at the ...
user avatar
3 votes
0 answers
84 views

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 ...
user avatar
  • 18.5k
2 votes
0 answers
22 views

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{...
user avatar
14 votes
0 answers
302 views

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 ...
user avatar
  • 1,820
8 votes
0 answers
147 views

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 ...
user avatar
  • 2,475
9 votes
1 answer
269 views

Local behavior of smooth triangulations

If $M$ is a smooth $n$- manifold, a smooth triangulation is defined to be a homeomorphism from a simplicial complex $K$ to $M$ whose restriction to each simplex is a smooth embedding. It's a well-...
user avatar
11 votes
3 answers
704 views

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

There are exotic versions of $RP^4$, constructed by Cappell-Shaneson, which are homeomorphic but not diffeomorphic to the standard $RP^4$. One way to distinguish them is via the $\eta$ invariant of $...
user avatar
  • 565
0 votes
0 answers
30 views

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 ...
user avatar
7 votes
0 answers
171 views

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 ...
user avatar
  • 7,328
1 vote
0 answers
98 views

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 ...
user avatar
  • 11
3 votes
1 answer
69 views

Uniqueness constraints for Delaunay triangulation

Commonly the assumption that is made on point sets that shall be Delaunay-triangulated is that no three are collinear and no four are cocircular. Those assumptions are however too restrictive: if ...
user avatar
10 votes
2 answers
691 views

On Gromov's proof of the systolic inequality $\operatorname{Sys}_1(M)\leq 6\operatorname{FillRad}(M)$

In the page 10 of the paper "Filling Riemannian manifolds" by Gromov (ProjetEuclid link), the author proves the following inequality (1.2) relating the systole and the filling radius of manifolds. $$\...
user avatar
  • 439
2 votes
0 answers
43 views

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 ...
user avatar
  • 2,475
4 votes
0 answers
133 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 ...
user avatar
4 votes
1 answer
818 views

On Thurston's triangulations of sphere

I have two questions from Thurston's paper [1]. In the paper [1], Thurston talks about classifying certain classes of triangulations of the sphere. Here a triangulation of a sphere a Topological ...
user avatar
  • 239
2 votes
0 answers
184 views

Category of Manifolds and Maps: TOP $\supseteq$ TRI $\supseteq$ PL $\supseteq$ DIFF? [closed]

Please let me denote the following (TOP) topological manifolds https://en.wikipedia.org/wiki/Topological_manifold (PDIFF), for piecewise differentiable https://en.wikipedia.org/wiki/PDIFF (PL) ...
user avatar
6 votes
1 answer
383 views

Are triangulations of compact manifolds PL homeomorphic?

I have frequently come across the statement "Any two triangulations of a compact n-manifold are related by bistellar moves" attributed to Pachner via Lickorish's paper 'Simplicial moves on complexes ...
user avatar
8 votes
0 answers
153 views

Is every compact smooth Riemannian manifold bilipschitz equivalent to a finite simplicial complex?

Let $M$ be a compact smooth Riemannian manifold. Then it admits a triangulation, i.e. a finite simplicial complex $K$ which is homeomorphic to $M$. Any such simplicial complex carries a natural metric ...
user avatar
  • 81
12 votes
0 answers
128 views

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 $> 3$...
user avatar
1 vote
1 answer
167 views

A totally geodesic triangulation

Let $M$ be a compact orientable $n$ dimensional Riemannian manifold. Is there a triangulation of $M$ such that every $k$ dimensional face of each simplex is a totally geodesic submanifold, $\forall k ...
user avatar
16 votes
1 answer
825 views

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 ...
user avatar
  • 6,082
8 votes
1 answer
686 views

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 For ...
user avatar
  • 9,966
4 votes
2 answers
195 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 ...
user avatar
  • 141
2 votes
0 answers
82 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 ...
user avatar
  • 2,921
2 votes
0 answers
90 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_{\...
user avatar
  • 2,611
3 votes
2 answers
262 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 ...
user avatar
5 votes
0 answers
219 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 ...
user avatar
  • 8,479
9 votes
3 answers
356 views

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 \begin{equation} \chi=\...
user avatar
  • 2,475