Questions tagged [topology]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
64
votes
3answers
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 ...
33
votes
1answer
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 $S^2$ ...
16
votes
2answers
2k 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 ...
15
votes
1answer
727 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?
13
votes
1answer
680 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 ...
13
votes
1answer
671 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
3answers
3k 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 \...
7
votes
1answer
139 views

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

I have a list of vertices: ...
6
votes
2answers
602 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 ...
6
votes
2answers
830 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 ...
6
votes
1answer
495 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. ...
6
votes
0answers
299 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 ...
5
votes
1answer
78 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 ...
4
votes
2answers
536 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 (...
4
votes
2answers
1k views

Finding minimal distance between two surfaces

This code will display two parametric surfaces: ...
3
votes
3answers
295 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 ...
3
votes
1answer
143 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 ...
3
votes
0answers
199 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 ...
2
votes
1answer
145 views

Making 3D graphics showing sphere eversions

Were these pictures made by Mathematica or MATLAB? And how could I make ones like them?
2
votes
0answers
62 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 ...
1
vote
1answer
955 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 $x$ ...
1
vote
2answers
282 views

Check if two graph 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: ...
1
vote
1answer
959 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,...
1
vote
0answers
46 views

Persistence diagram

Is there an easy way to get a persistence diagram of given planar points? For example the following points: ...
1
vote
0answers
47 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 ...
0
votes
1answer
190 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 ...
0
votes
0answers
45 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 ?