Questions tagged [topology]
The topology tag has no usage guidance.
33
questions
8
votes
2
answers
240
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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:...
- 21.7k
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 ...
- 6,888
2
votes
1
answer
273
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 $$D(u,v,x,y) = sign(\det (y-u,y-v,...
2
votes
1
answer
182
views
How can I create this graph in Mathematica?
I'm interested in illustrating a hyperbolic plane for a report I'm writing. Here's the metric that I'm using:
$$ds^2=dr^2+\sqrt{\lvert k\rvert}^{-1}\sinh\left(r\sqrt{\lvert k\rvert}\right)\left( d\...
- 1,481
0
votes
0
answers
61
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{...
- 1,481
3
votes
1
answer
79
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
78
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
52
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
164
views
Finding radius when the hole is born and dead (persistence diagrams)
I have a list of vertices:
...
- 153
5
votes
1
answer
111
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 ...
- 1,505
1
vote
2
answers
529
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:
...
- 125
2
votes
1
answer
202
views
Making 3D graphics showing sphere eversions
Were these pictures made by Mathematica or MATLAB? And how could I make ones like them?
- 129
3
votes
1
answer
173
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 ...
- 35.8k
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
336
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 ...
- 4,522
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
864
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
646
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 (...
- 8,648
6
votes
2
answers
959
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 ...
- 223
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
209
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 ...
- 1,597
3
votes
3
answers
362
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
370
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
555
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.
...
- 21.7k
13
votes
1
answer
729
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 ...
- 21.7k
70
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 ...
- 2,064
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 $S^2$
...
- 1,065
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 ...
- 2,703
13
votes
1
answer
761
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 ...
- 131
7
votes
3
answers
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 \...
- 71
15
votes
1
answer
851
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?
- 2,064