Questions tagged [topology]

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3 votes
1 answer
175 views

Generating a non-convex polyhedron from polygons and normals

I am working on developing "directional offset" module, which requires pretty tricky mesh generation: offset values vary in different directions. Suppose I have list of points and polygons, ...
  • 303
6 votes
0 answers
128 views

Animating a Sphere Eversion (Turning a sphere inside out)

I'm interested in animating a sphere eversion like the one shown in the following gif. The full video is here: https://www.youtube.com/watch?v=OI-To1eUtuU Normally, I would offer my own code as a ...
  • 4,825
8 votes
2 answers
244 views

MorphologicalEulerNumber misbehaving in 3D?

Context I would like to compute the MorphologicalEulerNumber of 3D GaussianRandomField as a function of height above a given threshold. Attempt I proceed as follows:...
  • 22k
7 votes
1 answer
170 views

Running Mathematica in 'SpouseMode'

In this previous question I looked into a deprecated capability of Mathematica from Version 2. Mathematica command that allows it to read my intentions Thanks to all for your answers. I note that ...
  • 6,978
2 votes
1 answer
297 views

How can I reduce the error estimate of numerical integration in Mathematica?

Before introducing the integral I want to go through some definitions. Define triangles $T_1$ and $T_2$ by the points $\{a,b,c\}$ and $\{d,e,f\}$ respectively. Define $$D(u,v,x,y) = sign(\det (y-u,y-v,...
2 votes
1 answer
196 views

How can I create this graph in Mathematica?

I'm interested in illustrating a hyperbolic plane for a report I'm writing. Here's the metric that I'm using: $$ds^2=dr^2+\sqrt{\lvert k\rvert}^{-1}\sinh\left(r\sqrt{\lvert k\rvert}\right)\left( d\...
  • 1,521
0 votes
0 answers
62 views

How can Mathematica be used to compute distance from a metric?

I have a metric: $$g_{\mu\nu}=\begin{bmatrix}-c & 0 & 0 & 0\\0 & a(t) & 0 & 0\\0 & 0 & a(t)\space r & 0\\0 & 0 & 0 & a(t)\space r\space \sinθ\end{...
  • 1,521
3 votes
1 answer
89 views

Euler Characteristic of MengerMesh

When I ask Mathematica to give me the Euler characteristic of MengerMesh[1, 3], it returns 16. It seems to me the 1-step Menger sponge has genus 5 and should ...
  • 101
1 vote
0 answers
81 views

Persistence diagram

Is there an easy way to get a persistence diagram of given planar points? For example the following points: ...
  • 933
1 vote
0 answers
53 views

Plot on fundamental square of a torus [duplicate]

I have line on the 2d plane, given by say f(t). I want to plot it in the torus by plotting Mod[f(t),2\[Pi]]. However, there are some sharp horizontal and vertical ...
7 votes
1 answer
167 views

Finding radius when the hole is born and dead (persistence diagrams)

I have a list of vertices: ...
6 votes
1 answer
118 views

Interfacing with an external computational homology program called CHomP

I have an external program called CHomP. I am trying to send a command to CHomP via RunThrough, and read the output string. It ...
  • 1,599
1 vote
2 answers
555 views

Check if two graphs have same shape

How can I check if two graphs have same shape or not? By shape I mean the two graphs are equivalent under a set of "name" replacements, like: ...
2 votes
1 answer
212 views

Making 3D graphics showing sphere eversions

Were these pictures made by Mathematica or MATLAB? And how could I make ones like them?
3 votes
1 answer
179 views

Computing the topological genus from a parametric function

Recall that the topological genus of a surface (or Euler characteristic) is (in essence) the number of its "holes." Thus the genus of a baseball is 0 while the genus of a donut or handled coffee cup ...
2 votes
0 answers
84 views

Built-in functions for Homotopy group

Does Mathematica have built-in functions for Homotopy group, for example for computing $$\pi_n(S^m)=?$$ or more generally $n$-th homotopy group, $$\pi_n(X)=?$$ for some $X$. (I search, but I could ...
  • 823
3 votes
0 answers
354 views

Help to finish a Sperner lemma application

This is the code for the celebrated Sperner's Lemma in two dimensions --- which is equivalent to Brouwer Fixed Point theorem. Incidentally, it's my first program without any help --- until I ask this ...
0 votes
0 answers
50 views

Compactness and Connectedness of sets

Can we check Compactness/Connectedness of sets in $R^{2}$ using some mathematica code ? If Yes , Can anyone tell how ?
  • 123
2 votes
1 answer
2k views

Plotting open balls for the given metric spaces [closed]

We're given a metric space (R,d) defined as follows: $$d(x,y) = |x-y|$$ We need to draw a open ball for this metric space with centre and radius of our choice. Open ball definition: For a fixed $x$ ...
  • 123
6 votes
2 answers
882 views

Small World network on a square grid

My aim is to generate a 2D Small World network on a square grid. i.e. 20x20. With a probability of 5%, one node rewires from an adjacent node to a random node of the grid, allowing some long distance ...
4 votes
2 answers
657 views

Visualization of Homotopy equivalence between "heart curve" and zero

It is well-known that a "heart" is topologically equivalent to a "zero". where $$(x^2+y^2-1)^3=x^2y^3$$ is the heart equation; and $$\frac{x^2}{2}+\frac{y^2}{3}=1$$ is the equation of the zero shape (...
  • 8,668
6 votes
2 answers
983 views

ParametricPlot3D of Boy's surface

I'm trying to visualize Boy's surface using Bryant's parametrization, as per the MathWorld article. However, I'm not sure I understand the parametrization, and I don't know how to implement it when ...
2 votes
1 answer
1k views

Package for symbolic computation of christoffel symbols and parallel transports in Riemannian geometry, given the metric

I've no knowledge in mathematica (but I do in matlab), but I'd really appreciate if someone could mention what is/are the best and easy to learn mathematica package(s) for symbolic and numerical (both,...
  • 121
0 votes
1 answer
212 views

Stitch edge of disk to edge of square in 3D space (with deformations) [closed]

Topologically each point on the edge of a square can be mapped uniquelly (including the corners?) to a point on the edge of a circle. It seems it might be possible to deform in 3D space the square ...
  • 1,607
3 votes
3 answers
367 views

I have a 2D space and I want to make it into a torus to replicate a paper

Suppose we have a 2D grid, divided by cells, and that we assign people to each cell. Each person has 4 neighbors, one in the cell above, another in the cell below, and neighbors in the cells to the ...
  • 119
6 votes
0 answers
382 views

Are there any built-in or third-party packages for general topology or algebraic topology in Mathematica?

I am learning general topology (wiki) and algebraic topology (wiki). Are there any built-in or third-party packages for general topology or algebraic topology in Mathematica? Through googling, I ...
  • 808
5 votes
2 answers
1k views

Finding minimal distance between two surfaces

This code will display two parametric surfaces: ...
  • 411
6 votes
1 answer
573 views

Connect neighbouring points as list of segments in 2 D

Context I am interested in connecting neighboring points in 2/3D as list of segments. I am guessing this is something within the reach of graph theory, which is well implemented in mathematica. ...
  • 22k
13 votes
1 answer
738 views

Morphological Filtering in 3D to produce skeletons

Context As a follow up of this question and that answer, I would like to identify the special lines separating 3D watersheds. These are useful in the context of astronomy to identify the ...
  • 22k
71 votes
3 answers
3k views

Homotopy Visualization

I noticed that both the lower cased 'i' and the Apple logo  are topologically equivalent to the disjoint union of two closed discs. I'd like to animate a homotopy from the left to the right, can ...
  • 2,084
33 votes
1 answer
1k views

Proving the hairy ball theorem using xAct

I would like to formally prove the hairy ball theorem in Mathematica, initially just for $S^2$, and then see about generalizing. An approach I thought about to use the xAct package to define $S^2$ ...
18 votes
2 answers
3k views

Morphing a "sheet of paper" into a torus

How can I visualize the standard topological "rubber-sheet" construction of a torus, that is, morphing a square into a torus? How can I start or are there any examples in the Mathematica ...
  • 2,723
13 votes
1 answer
777 views

Has anyone implemented cohomology for complex manifolds?

I'm investigating the cohomology of complex manifolds, and need to construct chain complexes, Mayer-Vietoris sequences etc. I use Mathematica for numerically tracing manifolds with geodesics, but ...
7 votes
3 answers
4k views

Plotting the open ball for the post office metric space

The post office metric space, $P$ has the distance function defined as follows: $$ d_P (\mathbf{x},\mathbf{y}) := \begin{cases} 0 & \mathbf{x} = \mathbf{y}\\ \Vert \mathbf{x}\Vert_2+\Vert \...
  • 71
15 votes
1 answer
873 views

Generating a topological space diagram for an n-element set

Over on StackOverflow I asked a similar question for the n=3 case, but the answer given doesn't easily generalize. How can I make a diagram such as this: But for a general n-element space instead?
  • 2,084