Questions tagged [visualization]

For questions about visualizing mathematical concepts. This includes questions about visualization of mathematical theorems and proofs without words.

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How to visualize the graph of multivariable functions, namely functions from R^2 to R?

I'd like to get a graphical approach to Analysis in higher dimension vector spaces, such as $\mathbb{R^n}$. To make this easier, my goal is to be able to visualize the graph of functions from $\mathbb{...
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intuition behind the construction of the map for showing associativity of $\pi_1(X,x_0)$.

I am studying algebraic topology.I have started with the chapter on fundamental group.Fundamental group at $x_0$ is defined to be the set of all equivalence classes of loops based at $x_0$ together ...
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2D Ternary plot equivalent for >3 dimensions

Ternary plots can be good for visualising systems where there are three different values that always add up to a constant value (i.e. there are only two degrees of freedom). A classic example is three-...
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Help me understand higher order derivatives. [closed]

I was just wondering what a double derivative of function implies. Is it the rate of change of rate of change of the function, or something else. And if that is the case then help me understand, why <...
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Julia set of finite Blaschke product

I want to compute the Julia set of finite Blaschke product $B_3$ $$B_3 = (\frac{z+1/2}{z/2+1})^3$$ First I should analyze the map so I compute first derivative : $$ (3*(z+1/2)^2)/(z/2+1)^3-(3*(z+1/2)^...
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Intuition behind the proof that pointwise limit of measurable function is measurable.

I am self-studying measure theory.There is a very important theorem in measure theory which says that a pointwise limit of measurable functions is measurable.Now,I understand the proof but do not get ...
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If the Quotient Space contains $0+U$ as it's zero element, must we not then include it in the visualization?

In Axler's book Linear Algebra Done Right 3.ed he defines the quotient space to be $V/U=\{v+U:v\in V\}$. As an example it is stated, that if $U=\{(x,2x)\in\mathbb{R}^2:x\in\mathbb{R}\}$ then $\mathbb{...
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Converse for Cayley digraphs

"It can be shown that, conversely,..... for some group."- can you clearly show or provide good references for proof of the converse? Also, I would appreciate more of an explanation for ...
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Solving PDE in Python

Does anyone know of a "nice" library for solving PDEs in Python that will compute a functional solution, u(x_1...x_n,t). What I want is to be able to pass the PDE(eq), BCs, and/or IVPs and ...
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Transfinite Construction, an intuitive interpretation.

Theorem (Transfinite Construction). Let $W$ be a well-ordered set, and $E$ an arbitrary class. Assume: For each $x\in W$, there is a given rule $R_x$ that associates with each $\varphi\colon W(x)\to E$...
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Graphical calculus for star-autonomous categories?

1. Definiton Let $(C, \otimes, I, a, l,r)$ be a (not necessarily symmetric) monoidal category. A (planar) star-autonomous structure on the monoidal category $C$ consists of an adjoint equivalence $D \...
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How to draw a planar-embedded graph in a visually pleasing way

I have a graph with two types of vertices: "boundary" vertices have degree 1, and "interior" vertices have degree 4. I've computed a planar embedding of the graph, i.e. around each ...
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Criterion for a function to be Lebegsue measurable.

We know that a function from $\mathbb R$ to $\mathbb R$ is continuous iff the graph can be drawn without lifting the pen. I want to know if there is a similar intuitive characterisation for measurable ...
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Soft-Question: Is it possible to visualize, or make concrete, progressively more abstract mathematics? Are there mathematicians who can?

This is my first post! I'm really not good at math, and I'm trying to re-learn on Khan Academy. As I'm progressing, this question came up for me. Learning math concepts that I can visualize in my mind ...
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Let P be a tetrahedron, and let vector (V, E, F) = (4, 6, 4) represent the number of vertices, edges, and faces of P.

What are the possible (V, E, F) vectors that can be obtained by stacking pyramids over faces of P? Justify this in depth.
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Correct scaling for scatter plots — is there such a thing?

I am new to creating scatter plots and as I work my way through a Cambridge publication a guide to year 10 Australian mathematics, Advanced, I’m aware they are likely only touching upon the subject. I ...
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What is the motivation behind the definition of Lebesgue measurable set?

We study the definition of Lebesgue measurable set to be the following: Let $A\subset \mathbb R$ be called Lebesgue measurable if $\exists$ a Borel set $B\subset A$ such that $|A-B|=0$,where $|.|$ ...
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Visualise in $\mathbb{R}$: inverse of open set is open $\implies f$ continuous

Theorem: Let $(S,d_S), (T,d_T)$ be metric spaces and $f: S\to T$. Then $f$ is continuous (1) iff every open set $D\subset T$ has an open inverse image $f^{-1}(D)$ (2). The theorem in itself is very ...
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How are Mandelbrot Animations Made?

I have always been interested in learning about how computers are able to animate the "Mandelbrot Set" (https://en.wikipedia.org/wiki/Mandelbrot_set). I tried to learn about how pictures of ...
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Is there a handy mnemonic or visual aid for all the basic set theory function relations, such as $f(A\cap B)\subseteq f(A)\cap f(B)$?

Glancing at Appendix A of John Lee's Introduction to Topological Manifolds, I noticed the following list of rules: I'm familiar with rules like these and have no trouble proving them—that's not my ...
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0.5 times 0.5 equals 0.25, but how does this work with repeated addition?

So I'm trying to brush up on my math as an old adult, and I'm currently working through the very basics of math again. I'm trying to truly understand and visualize the various operations I'm engaging ...
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2 votes
4 answers
287 views

I don't fully understand why Pythagorean theorem works with velocity vectors.

I get why it works with displacement because that's what the theorem was originally meant for, lengths.... I find it harder to wrap my head around it when its velocity. If anyone has a good ...
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visualizing conformally compactified Minkowski immersed in $\mathbf L^3$ [closed]

Boy's surface is an immersion of the real projective plane in $\Bbb R^3.$ And the real projective plane is a compactification of $\Bbb R^2$. I've seen images of Boy's surface on the web. I'm ...
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Is there a relationship between $\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n} = 1$ and $\int_{1}^{\infty} \frac{1}{x^2} \,dx = 1$?

A classic example of an infinite series that converges is: ${\displaystyle {\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n}=1.}$...
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Why does the solution of a problem with an equality constraint would remain at least a local solution if that constraint were removed?

To perform constrained maximization, we can construct the generalized Lagrange function of $-f(x)$, which leads to this optimization problem: $$ \min _{x} \max _{\lambda} \max _{\alpha, \alpha \geq 0}-...
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Visualizing a 4D object in 3D space

out of curiosity i am trying to visualize a 4D object in 3D space in Blender (3D modeling software). Using Python i have access to all the generated 4D vertices, 4D Edges and 4D polygons of the 4D ...
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5 votes
1 answer
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Patterns in modular multiplication and frequency tables

I moved this question from MathOverflow since it seemed not appropriate there. Consider the multiplication table $f(n,m) = (n\cdot m) \text{ mod } N$ for $n,m \leq N/2$. For $N=30$ it looks like this:...
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Geometrical interpretation of feasible regions of three different convex optimization formulations

I am trying to understand and compare the feasible regions of three convex optimization formulations "geometrically". I understand the feasible region would depend on the parameters. Let me ...
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Looking at this graph would you say there is no correlation between these two variables?

I am trying to figure out if there is a correlation between Military Enlistment and the number of U.S. War Films Released between 1980-2021? This is a graph I made in Tableau for data for the ...
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Describing the action of this Matrix?

Let $A = \begin{bmatrix} 1 & 1 \\ 0 &-1 \end{bmatrix}$ Then $A \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} $ and $A \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \...
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How to get equations of fluid flow velocity with vorticies

I am looking for the equation of the velocity field $\textbf{u} = u(x,y)\hat{i} + v(x,y)\hat{j}$ of a two-dimensional steady flow, which streamlines look similarly to this image: example image of the ...
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1 answer
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Can we visualize the action of $L_A$ easily?

(1) Let $A$ be an $n\times n$ real symmetric matrix. Then, $P^{-1}AP$ is a diagonal matrix for some orthogonal matrix $P$. Let $L_A:\mathbb{R}^n\ni x\mapsto Ax\in\mathbb{R}^n$. In this case, we can ...
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1 vote
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Is there a program or website for visualization of 2d and 3d matrix transformations?

Is there a program or website for visualization of 2d and 3d matrix transformations? Hello everyone !!! How to look at the transformation (when multiplying two matrices or a matrix by a vector) of ...
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2 answers
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How can you visualize Independence with Venn Diagrams?

Imagine two events $A$ and $B$ that are not mutually exclusive, such that $P(A) = 0.3$ and $P(B)=0.4.$ Consider the Venn diagram of the two overlapping sets, and visualize moving them closer together ...
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Plotting Spherical Harmonics - Phase Coloring?

Looking at the wikipedia table of spherical harmonics, I'm trying to figure out how they are computing phase and using it set the hue on the plots. I can't figure this out for either the real plots (...
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5 votes
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What is the Cayley graph for alternating group A6?

According to ATLAS of Group Representations, the alternating group $A_6$ is a group of order 360 which has presentation $$ \langle a,b \mid a^2 = b^4 = (ab)^5 = (ab^2)^5 = 1 \rangle. $$ If we draw its ...
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Drawing a line from n-dimension in 3D by preserving angles!!

I am brainstorming on something. There might be a standard method for my problem, but I always like to rethink stuff. Here is the problem: I am working on a program mapping my data into a line in a ...
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2 votes
1 answer
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How do two tori glued along the identity of $S^1 \times S^1$ look like?

I have got trouble imagining the outcome of gluing two solid tori $D^2 \times S^1 \sqcup_{\text{id}}S^1 \times D^2$ along the identity of $S^1 \times S^1$. If I understand correctly, the tori are ...
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diagram of inner product - generalization?

Consider a sort of graphical language of linear algebra...tensor network diagrams. So consider the space of $n$ non-intersecting paths connecting a node $A$ to a node $B.$ This "picture" is ...
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Meaning of opposite sign in Fourier's exponents

I'm currently exploring the meaning, intuition and differences of the various types of Fourier transforms. I can't say everything is ok, but I think I got most of them (hopefully, Dunning Kruger is ...
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Visually intriguing unsolved problems which are easy to explain

I have come across a list of visual proofs which are wrong (Visually deceptive "proofs" which are mathematically wrong) visual proofs which are not wrong (Proofs without words) visually ...
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3 votes
4 answers
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How can we visualize $a^2+b^2-c^2$ for a triangle of sides $a$, $b$, $c$?

Let $a$, $b$, and $c$ be three lengths of sides of a triangle, that is, $a+b>c$. How can we visualize the value $a^2+b^2-c^2$ as length of some segment or area, ... constructed from the triangle $...
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3 votes
0 answers
130 views

Visualizing $\det (\exp M+ \exp N)$

I am interested in the follow expression $$ \det (\exp M+ \exp N) $$ where $M$ and $N$ are $2\times 2$ real matrices (but also in the general case where the matrices are $n\times n$). which represents ...
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3 answers
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Does the graph of a function $f(x)$ versus its derivative $f'(x)$ have a special purpose?

I can't find any insights online on how useful the graph of function $f(x)$ (on the $y$ axis) versus it's derivative $f'(x)$? (on the $x$ axis) does it provide some useful informations if any? For ...
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1 vote
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visualizing solution of a Lagrange differential equation

I want to visualize the solution of a Lagrange differential equation $y=2xy'-3(y')^2$ (Example 1 in here). Analytic solutions are fine: a singular solution $y=0$, and a parametric family of solutions. ...
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1 answer
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Why isn't there an app that allows you to enter in all the rules of a given formal system so that the app supports all formal systems of math?

I'm jumping around between articles about ETCS to Simple TT to Calculus of Constructions wondering what my app should focus on. I'm wondering, why there isn't yet a software app in which you can ...
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building intuition of trajectories from visualization of phase portrait

lets say we have a system of ordinary differential equations and want to how the trajectories starting from all possible initial states evolve, without theoretically proving any properties (e.g. ...
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16 views

Open Source Tools for Graph Analytics

What are the best open-source (no code, GUI based) tools for visualizing and analyzing Network Data. I am currently using NodeXL and Giphy also looks interesting. Any other options?
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Visualization of simple topological spaces

I am not so good with visualizing some spaces and would appreciate if you help me. Are my drawings of the topological spaces correct? Why does $\mathbb{R^2} X S^1$ look like torus without boundary? (...
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How to visualize multiplication, in the Odds form of Bayes's Theorem, as pie charts?

Here I'm asking solely about the circle pictograms. Please eschew numbers as much as possible. Please explain using solely the circle pictograms. Undeniably, I'm NOT asking about how to multiply ...
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