# Questions tagged [visualization]

The visualization tag has no usage guidance.

**5**

votes

**3**answers

544 views

### 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 ...

**6**

votes

**1**answer

119 views

### 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 ...

**2**

votes

**1**answer

320 views

### 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 $...

**6**

votes

**1**answer

232 views

### 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 ...

**10**

votes

**2**answers

637 views

### 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**

votes

**0**answers

188 views

### 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 ...

**28**

votes

**2**answers

2k views

### 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 ...

**11**

votes

**1**answer

854 views

### 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 ...

**16**

votes

**2**answers

430 views

### 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 ...

**14**

votes

**1**answer

568 views

### 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**

vote

**2**answers

187 views

### 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 ...

**4**

votes

**1**answer

202 views

### 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 ...

**19**

votes

**4**answers

689 views

### 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 ...

**2**

votes

**1**answer

696 views

### 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 ...

**10**

votes

**2**answers

522 views

### 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

289 views

### 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, ...

**1**

vote

**0**answers

45 views

### 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 ...

**0**

votes

**0**answers

202 views

### 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 ...

**49**

votes

**3**answers

3k views

### 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 ...

**5**

votes

**1**answer

524 views

### 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} \...

**218**

votes

**37**answers

92k views

### 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

2k views

### 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 ...

**7**

votes

**4**answers

2k views

### 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 ...

**4**

votes

**4**answers

686 views

### 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 ...

**33**

votes

**5**answers

3k views

### 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 ...

**21**

votes

**11**answers

3k views

### 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?
...

**15**

votes

**4**answers

12k views

### 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–...

**10**

votes

**5**answers

982 views

### 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 ...