# Questions tagged [visualization]

The visualization tag has no usage guidance.

47
questions

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### Why should we expect this odd behavior of negative binomial distributions?

In independent Bernoulli trials with probability $p$ of success on each trial, let $X$ be the number of failures before the $n$th success. Then
$$
\Pr(X=x) = \binom{-n}{\phantom{+}x} (-q)^x p^n \text{ ...

8
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0
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488
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### What is the nicest bijection $\textbf{R}^p \to \textbf{R}^q$ that you know?

It is well-known that bijection between $\textbf{R}^p$ and $\textbf{R}$ exist (e.g. here, though many other examples exist).
The problem with all these examples of bijections is that typically the ...

1
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0
answers

82
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### 1D representation of 2D discrete Fourier transformation [closed]

I'm not too familiar with image processing, so I need a little help:
In general, if we transform a discrete function $f$ with $n$-variables from the "spatial domain" using the Fourier ...

2
votes

0
answers

133
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### Visualisation of general 3x3 matrices, with applications to the pedagogy of linear algebra?

I've got a method for visualising non-zero $2 \times 2$ real matrices (modulo non-zero scalar factor) using the fact that:
Nonnegative determinant matrices (modulo non-zero scalar factor) are in 1-to-...

0
votes

0
answers

98
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### Possible shifts in finite elementary cellular automata

I investigated the long term behaviour of a pair of black cells ■■ on a circle of $N$ cells under the action of each of Wolfram's rules $R$. For each combination $(R,N)$ I determined the first ...

1
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0
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372
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### Astonishing affinity of Wolfram's rule 110 to the numbers 2 and 7

I investigated the evolution of a single black cell on 1-dimensional grids with periodic boundary conditions of variable sizes $N$ under Wolfram's rule 110 which is the only one for which Turing ...

0
votes

1
answer

724
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### Plot two implicit surfaces in 3D and highlight their intersection [closed]

I want to plot the two surfaces which are defined in $ \mathbb{ R }^3 \ni ( x, y, z ) $ via the equations $ 0 = y^2 - x*(x^2 + 1) $ and $ 0 = z^2 - y*(y^2 + 1) $, respectively.
Moreover, I want also ...

9
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1
answer

615
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### Aphantasiac mathematicians? [duplicate]

Over the past few years there's been a fair amount of publicity given to the phenomenon of aphantasia, the condition of being unable to form visual images in one's mind or remember what things look ...

0
votes

1
answer

72
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### Visualization PDF of distribution defined by quantiles

How can I visualise PDF of distribution defined by quantiles, that I predict with my neural network? Now I'm passing quantiles to the histogram, but I don't think it is the correct way for visualising....

11
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2
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774
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### Visualizing holomorphic differentials on a compact Riemann surface?

It is a classical result that the vector space of holomorphic differentials on a compact Riemann surface of genus $g$ has dimension $g$. I am wondering if there is a way of visualizing this wonderful ...

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0
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### Barycentric coordinates of weighted edges

Given $K_n$ with weighted edges, we can fix an edge $e_{AB}$, iterate over all non-adjacent edges $e_{CD}\in E\setminus e_{AD}$ and record how often $e_{AB}$ was in the lightest, intermediate or ...

1
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0
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### Methods for useful visualisations of complete weighted graphs

Question:
which methods for visualising complete weighted and symmetric graphs, i.e. $K_n$, are useful in the sense that they can aid in mathematical research?
The Traveling Salesman Problem may ...

1
vote

1
answer

49
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### Represent multivariate data [closed]

I am not sure if this is the best place for my question. Please delete if it is not, but I would really appreciate some suggestions.
I want to graphically represent multivariate data. I have 7 ...

28
votes

2
answers

3k
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### What (or how) are the new spaces of derived algebraic geometry?

I am a beginner in derived algebraic geometry and I am trying to develop some visual and geometrical intuition about derived schemes (and stacks), or more precisely about the new geometrical phenomena ...

16
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2
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### What's the "actual" shape of a black hole accretion disk?

[Warning: I have no expertise in general relativity, so this question might not be very rigorous]
More and more often we come across science popularization articles like this one which show beautiful ...

8
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0
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### Visualization of an algebraic stack

As the visuallization of an algebraic stack is virtually impossible I warn about this is a soft question.
I am interested in thinking visually about algebraic stacks (also higher and derived stacks, ...

7
votes

1
answer

465
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### Visualizing a Whitehead product: the attaching map $S^3\to S^2\vee S^2$

There are informative and easily accessible images and videos that illustrate the Hopf fibration $S^3\to S^2$ by describing what happens to the fibers in the unit cube $(0,1)^3\approx S^3\backslash \...

1
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0
answers

181
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### Imagining linear maps between finite fields

I can't imagine a right picture of a linear transformation $\mathbb{F}_{p} \mapsto \mathbb{F}_p$ or $\mathbb{F}_{p^2} \rightarrow \mathbb{F}_{p^2}$ etc (over the field $\mathbb{F}_p$) although they ...

11
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0
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2k
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### Visualization and new geometry in higher stacks

I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond ...

6
votes

3
answers

677
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### When is $2\varphi(n) > n$ – and how to prove it?

When coloring the squares of the Ulam spiral not only by black and white (for being prime or non-prime) but by shades of grey representing the normalized totient function $\varphi(n)/n$
and ...

5
votes

1
answer

477
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### Explaining patterns in modular multiplication graphs

Let the multiplication graph $n/m$ be the graph with $m$ points distributed evenly on a circle and a line between two points $a$, $b$ when $an \equiv b\operatorname{mod} m$.
These graphs often look ...

1
vote

1
answer

373
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### Why is $n_{n^2-1}$ the smallest graph that clearly shows the structure of multiplication by $n$?

Initially, I wanted to ask this question as a puzzle.
Consider a regular $m$-gon. Let $0$ be the lower corner and count the corners clockwise.
Let $n_m$ be the multiplication-by-$n$-graph of $...

7
votes

1
answer

1k
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### How to visualize local complete intersection morphisms?

As the question title asks for, how do others visualize local complete intersection morphisms? My experiment in asking people in real life didn't pan out, so I'm consulting the MO algebraic geometry ...

12
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2
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1k
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### How to visualize finiteness of class number?

As the question title asks for, how do others "visualize" the finiteness of class number with algebro-geometric insight? I just think of it as a result in algebraic number theory and not one in ...

3
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0
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235
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### Visualization of hidden structures in numbers

[Please allow me a note: The way desribed below allows to depict functions $f:X^2 \rightarrow Y$ completely in two dimensions (without hiding or omitting any information). This allows for depicting ...

36
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2
answers

3k
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### How to visualize Dirichlet’s unit theorem?

As the question title asks for, how do others "visualize" Dirichlet’s unit theorem? I just think of it as a result in algebraic number theory and not one in algebraic geometry. Bonus points for ...

14
votes

1
answer

2k
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### How to visualize the Frobenius endomorphism?

As the question title asks for, how do others "visualize" the Frobenius endomorphism? I asked some people in real life and they said they didn't know and that I could go and ask on MO and possibly get ...

17
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2
answers

634
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### Geometric/combinatorial depiction of algebraic identity?

I'm looking for a geometric or combinatorial depiction of the algebraic identity
$$
xyz = \frac{1}{24} \Big\{(x+y+z)^3 - (x-y+z)^3 - (x+y-z)^3 + (x-y-z)^3 \Big\}.
\label{*}\tag{$*$}
$$
Here is the ...

17
votes

1
answer

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### How to visualize a Witt vector?

As the question title asks for, how do others "visualize" Witt vectors? I just think of them as algebraic creatures. Bonus points for pictures.

1
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2
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1k
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### Understanding reduced suspension of $S^1$ [closed]

I know this is just $S^2$. To see it, I use the CW structure of $S^1$ x $S^1$ , consisting of one 0-cell, two 1-cells and a 2-cell. Then since the reduced suspension is the cartesian product ...

5
votes

1
answer

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### Is Visualization of Data a Subject of Mathematical Research? [closed]

Please excuse my naive question, but what kind of rôle does the visualization of (especially high-dimensional) data play in mathematical research?
I know, that it plays an important rôle in the ...

20
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4
answers

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### Problems for developing mathematical visualization expertise

Einstein stated that he often explored and reasoned visually and spatially, and only after achieving understanding cast his insights into algebraic form. He could just "see" the answer. There are ...

5
votes

1
answer

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### t-Stochastic Neighbor Embedding vs Topological Data Analysis

The shortest form of this question is:
How much TDA can be done with tSNE?
Specifically, I'm referring to the application of TDA to clustering data, so, think along the lines of Ayasdi's ...

13
votes

2
answers

1k
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### Why does this Moiré pattern look this way?

I was making some gifs of Mobius transformations in Matlab, and some strange patterns began to appear. I'm not sure if a deeper knowledge of the filetype/algorithm is needed to understand this ...

5
votes

1
answer

438
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### How to visualize a category (of "combinatorial" maps)

This is a practical and very soft question, with the combinatorial database http://www.findstat.org in mind.
I have a few, around 20, families of combinatorial objects, for example Dyck paths, ...

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0
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### Open volumetric time series data set

Does anyone know where I can find a good open volumetric time series data set?
I had a look at some of Stanford's open data sets (https://graphics.stanford.edu/data/voldata/ )
But these do not seem ...

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0
answers

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### What is the state of the art of visualizing bifurcations for "difficult" dynamical systems?

This question is related to my other recent question on MO (although I am not confident that the dynamical system described in that other question is actually "difficult," in the sense that I will ...

54
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3
answers

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### The view from inside of a mirrored tetrahedron

Suppose you were standing inside a regular tetrahedron $T$ whose
internal face surfaces were perfect mirrors.
Let's assume $T$'s height is $3{\times}$ yours, so that your
eye is roughly at the ...

6
votes

1
answer

843
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### How to visualise Bollobas' 1965 theorem?

Theorem
$[n]=\{1,\ldots,n\}$. Let $\lbrace (R_i, S_i), i \in I \rbrace, R_i, S_i \subset [n]$ be such that $R_i \cap S_i = \emptyset, R_i \cap S_j \ne \emptyset (i \ne j)$. Then $$\sum_{i \in I} \frac{...

281
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47
answers

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### Examples of unexpected mathematical images

I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an ...

5
votes

2
answers

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### Visualization of the real projective plane [closed]

Consider a closed (compact and without boundary) and non-orientable 2-manifold $M$. By Whitney embedding theorem, one can embed $M$ in $\mathbb{R}^4$. $M$ cannot be embeded in $\mathbb{R}^3$ and just ...

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4
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### Visualizing functions with a number of independent variables

I need to graph real valued functions (for exposition and analysis).
The issue is: there are more independent variables so that the conventional graphing methods can't be used, and furthermore I don't ...

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4
answers

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### Picturing a Certain Torus and Klein Bottle

The other day I was explaining orientability to someone and we were walking through some of the statements about orientability on the Wikipedia page on the topic. While I was able to satisfy his ...

37
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4
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### Understanding the countable ordinals up to $\epsilon_{0}$

in a recent MO question, link, discussing the current foundations of mathematics, the author linked a video lecture by Prof. Voevodsky, which argues against the principle of $\epsilon_{0}$-induction ...

28
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12
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### Creating high quality figures of surfaces

I am not sure if this question is suitable for mo, it is more about visualization than math. Anyway, here it is:
What is the best way to visualize a 2-surface in Euclidean space with high quality?
...

19
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5
answers

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### Visualization of Riemann–Stieltjes Integrals

The Riemann–Stieltjes integral $\int_a^b f(x)\,dg(x)$ is a generalization of the Riemann integral. It is e.g. heavily used as a starting point for stochastic integration. The approximating Riemann–...

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5
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1k
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### Visual representation of mathematical research interrelationships

I remember seeing a visualization in the form of a 2d (nodal) graph of all areas of academia, with math, physics and engineering over in one section, connecting in an arc to the central area of ...