Questions tagged [topology]
The topology tag has no usage guidance.
31
questions
2
votes
1answer
130 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 $$D(u,v,x,y) = sign(\det (y-u,y-v,...
2
votes
1answer
161 views
How can I create this graph in Mathematica?
I'm interested in illustrating a hyperbolic plane for a report I'm writing. Here's the metric that I'm using:
$$ds^2=dr^2+\sqrt{\lvert k\rvert}^{-1}\sinh\left(r\sqrt{\lvert k\rvert}\right)\left( d\...
0
votes
0answers
60 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{...
3
votes
1answer
57 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 ...
1
vote
0answers
65 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
48 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
1answer
156 views
5
votes
1answer
91 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
vote
2answers
371 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
1answer
162 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
1answer
154 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
0answers
72 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 ...
3
votes
0answers
261 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
0answers
47 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 ?
1
vote
1answer
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 $x$ ...
6
votes
2answers
744 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
2answers
608 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 (...
6
votes
2answers
886 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 ...
1
vote
1answer
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,...
0
votes
1answer
198 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 ...
3
votes
3answers
314 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 ...
6
votes
0answers
325 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
2answers
1k views
6
votes
1answer
516 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.
...
13
votes
1answer
697 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 ...
66
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 $S^2$
...
18
votes
2answers
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 ...
13
votes
1answer
719 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 \...
15
votes
1answer
793 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?
