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0
votes
0answers
27 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
121 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
113 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 ...
3
votes
2answers
345 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 ...
5
votes
2answers
239 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
309 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 ...
0
votes
1answer
138 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
236 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 ...
4
votes
0answers
142 views

Are there any build-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 build-in or third-party packages for general topology or algebraic topology in Mathematica? Through googling, ...
4
votes
2answers
622 views

Finding minimal distance between two surfaces

This code will display two parametric surfaces: ...
6
votes
1answer
364 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
503 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 ...
47
votes
4answers
2k 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 ...
29
votes
1answer
779 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 $S^2$ ...
11
votes
2answers
1k 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
492 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 ...
6
votes
3answers
1k 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 ...
12
votes
1answer
518 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 ...