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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16 views

Span and radius vectors

Is it possible to get any (radius) vector in 3D using only 1 fixed radius vector and all possible radius vectors? (Of course, every vector we get can be translated so that its starting point is at ...
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What are the most interesting math riddles you know?

I am talking about riddles that do not involve play on words. The riddles should be logically and mathematically correct too.
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15 views

Is there any way to visualize Ring structure using Cayley digraph?

In Nathan Carter's Visual group theory there is a nice description of how to explore group structure through Cayley digraph and how the structures reflect themselves in Cayley tables.For example ...
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1answer
21 views

Phase diagram for empirical data

The behaviour of partial or ordinary differential equations can be studied/visualized with phase diagrams. How would one plot such diagrams for empirical data, which are suspected to be governed by ...
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25 views

Visualizing derivative of a matrix-valued function of a matrix variable

Apologies if this is not at an appropriate level for this site or if it's too broad/scrambled of a question, but I was wondering how best to visualize a matrix-valued function of a matrix variable? ...
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Examples of problems/proofs which can be (surprisingly) represented in terms of graphs

I am looking for examples of problems or proofs in mathematics which have a equivalent representation in terms of graphs, which makes solving the problem easier. For example, the problem of finding ...
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98 views

Is there visual or intuitive explanation of equations of conic sections defined in traditional way?

Is there purely visual and intuitive approach for equations of conic sections using traditional definitions of ellipse (constant sum of distances from two foci), hyperbola (constant difference of ...
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53 views

How can this graph of the relationships among types of commutative rings be improved?

I made a directed graph in order to get a better understanding of the relationships between various types of commutative rings. Since I’m not very well versed in ring theory, I’m sure it ...
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2answers
58 views

minimum number of congruent rectangles

consider a 6*6 square which is dissected into 9 rectangles by lines parallel to its sides such that all the rectangles have integral sides.the question is - what is the minimum number of congruent ...
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35 views

Visualizing split extensions and central extensions

This may be a bit of a silly question, but I think it is useful to get a visual idea of the concept of group extensions. A short exact sequence $1 \to A \overset{i}{\to} B \overset{\pi}{\to} C \to 1$ ...
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How can I generalize diagram proving Mean Value Theorem to Generalized MVT, without assuming any function as a straight line?

Calculus: The Language Of Change (2005) by David W. Cohen, James M. Henle. pp. 827-829. The original colored in just blue. I annotated and added more colors. I can't recall which page presents the ...
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Why does $h = v\cos\alpha$ never enter into the Mean Value Theorem Proof?

Calculus: The Language Of Change (2005) by David W. Cohen, James M. Henle. pp. 827-829. The original colored in just blue. I annotated and added more colors. Pls see below. Why do $\color{pink}{h}, \...
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Visualization of the proof in Lee's Smooth Manifolds book that every open cover has a regular refinement

Here is the theorem Lee is proving. (It states that any open cover on a smooth manifold has a regular refinement (a refinement which is countable, locally finite, and satisfies additional ad-hoc ...
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Geometrical intuition for the action of symmetry group of a cube.

Let $G$ be the symmetry group of a cube. It has the group of rotational symmetries $H$($\cong S_4$) as a normal subgroup of index two. Now this is the kernel of some action of $G$ on a set of size ...
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58 views

Write down what metric this transformation preserves based on animation

I made this on desmos: https://www.desmos.com/calculator/u5qpd135uc I made it because I wanted to compare and contrast it with the Lorentz boost. The transformation should move a point to ...
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“Visual” Real analysis introductory text

I am looking for Real Analysis book suitable for self study which is similar to the essence of Visual Group theory by Nathan Carter, which is scrupulous and punctilious in explaining concepts via ...
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A question regarding the equivalent criterion of continuity.

We know that a function $f:X\to Y$ is continuous iff $f(\overline{ A})\subset \overline {f(A)}$ or iff $f^{-1}(B^0)\subset (f^{-1}(B))^0$ or iff $\overline {f^{-1}(B)}\subset f^{-1}(\overline {B})$....
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Reference request for studying and visualizing dual vector spaces.

I am an undergraduate student of mathematics and we have in out linear algebra course, a brief introduction of Adjoint operators and unitary operators. Now I understand that adjoint of a linear ...
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70 views

A visual proof:A projection $T$ is orthogonal if $||Tx||\leq ||x||$, $x\in V$.

Definition A linear transformation $T$ is said to be a projection if $T^2=T$. A linear transformation $T$ is said to be an orthogonal projection if for each $x\in V$,$||x-Tx||\leq ||x-w||\forall w\in ...
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How can I visualize the multivariable function z = f(x(t), y(t)) in 4 dimensions?

I am trying to visualize$\ z = f(x(t), y(t))$, and my model is that$\ x, y, z$ depend on$\ t$. The only way that I can visualize this is as a point moving in 3D space. However, wouldn't that be $\ \...
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Which matrix do we use to calculate principal components in PCA? $X^T X$ or covariance matrix of $X$?

I am reading Principal Component Analysis (PCA) from Wikipedia. Under the details section, it states that the principal components are just eigenvectors of $X^T X$ where $X$ is the data matrix. ...
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Is there any visual proof that rationals in cantor set are dense in cantor set?

I was thinking about how we can get a feel that rationals in Cantor set are dense in Cantor set. Is there any way to put this thing in a visual way? It is quite easy to think for irrationals in the ...
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58 views

Trying to visualize a polygon in a space $X$

From Rotman's Algebraic Topology: A polygon in a space $X$ is a $1$-chain $\pi = \sum\limits_{i=0}^k \sigma_i$ where $\sigma_i(e_1) = \sigma_i(e_0)$ for all $i$. By a theorem proven in the ...
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How to simply represent topological spaces satisfying separation axioms.

I am a new learner of topology and I feel confused when I am introduced to different separability axioms like $T_0,T_1,T_2$ etc.Is there any way to diagramatically represent these spaces by simple ...
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1answer
32 views

How can I visualize ${A^0}^{Y} \supset A^0$?

$(X,\tau)$ is a topological space and $Y\subset X$.How can I visualize ${A^0}^{Y} \supset A^0$,where $A\subset Y$.The proof is not very difficult but the fact is not seeming very obvious to me.Suppose ...
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Are there any “visual proofs” that have no formal written proof?

As the title says, are there any problems that have been "solved" using a visual proof but which have not been solved via a written proof?
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Does anyone know what this diagram could be about?

Does anyone know what this diagram could be about? I found it about a year ago on some blog and I tried to relocate the source of the picture but was unable to. My best guess is that maybe $R(x)$ and ...
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151 views

A question on nowhere dense sets.

Consider the $2$ definitions: A set $A$ in a topological space $(X,\tau)$ is said to be a nowhere dense set if it is not dense in any nonempty open set. A Set $A$ in a topological space $(X,\tau)$ ...
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Using rotational sweeping, define by parametric functions x(u,v), y(u,v), z(u,v), u,v Î[0,1] the surface displayed in Figure Q1.

I am having difficulty visualizing. Can someone help me? The figure can be seen by clicking on the underlined Figure 1 Figure 1
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35 views

Can we interpolate Pascal's triangle?

Answer to the title: of course we can interpolate Pascal's triangle with $z(x,y)= {x \choose y}$, but my laptop cannot handle the graph of $z(x,y)$, and I cannot find one online. I am looking for ...
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Visualization in a metric space?

Metric space is the first phase of abstract analysis,even in real analysis we do everything with proper rigor and do not depend on pictures only,although pictures are necessary to understand a rough ...
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53 views

How to visualize $(A\cap B)^\mathrm{o}=A^\mathrm{o} \cap B^\mathrm{o}$?

We all know that $(A\cap B)^\mathrm{o}=A^\mathrm{o} \cap B^\mathrm{o}$, where $A,B \subset X$ which is a metric space. The proof is not also difficult, but actually I cannot visualize or feel ...
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How to visualize the quotient process using Cayley diagram?

The book visual group theory by Nathan Carter does a great job explaining the Direct product with the help of multiplication table. But it does not discuss the quotient process in terms of the ...
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What is a good software to use to generate a visualization of a graph G=(V,E)?

Are there any good softwares out there where I can input a list of edges and have it generate a graph visualization for me?
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Visualizing Laplace Transform

I watched Linear algebra series on 3Blue1Brown channel, This blow my mind, I loved it and now I can visualize Linear transformation works. But what about Laplace transform, it is also a linear ...
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52 views

Intuitive idea behind orbit stabilizer theorem.

Recently I studied the orbit stabilizer theorem which is as follows: Suppose $G$ is a group acting on $X$ (i.e.$X$ is a $G$-set). Let $x\in X$,then define, $\operatorname {orb}(x):=\{g.x:g\in G\}$ ...
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Visualization of a “Not so intuitive” problem of linear algebra.

I recently encountered a problem in Hoffmann-Kunze linear algebra: If $(.,.)$ is the standard inner product on $\mathbb C^2$ then show that $(Tv,v)=0 \forall v\in \mathbb C^2 \implies T=0$, I think ...
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23 views

Visualization of standard metric on Frechet's sequence space.

Suppose $X$ is the set of all sequences in $\mathbb R$.Define a metric called Frechet metric on $X$ by $d(x,y)=\sum_{n=1}^{\infty} \frac{1}{2^n} \frac{|x_n-y_n|}{1+|x_n-y_n|}$,I want to investigate ...
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36 views

Using Virtual Reality to visualize mathematical concepts

I hope this is an appropriate place to ask this question. I have some experience in 3D graphics programming, and I'm interested in creating mathematical models in virtual reality. Especially topology ...
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56 views

Understanding third isomorphism theorem with simple pictures.

Recently I encountered the third isomorphism theorem in groups which says that if $G$ is a group and $H_1$ and $H_2$ be normal subgroups of $G$ such that $H_1 \subset H_2$,then show that $(G/H_1)/(H_2/...
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arbitrary transformation of data between linear and log-like scales

[I'm not very sure on how to formulate this question (or if it belongs here btw), so apologies in advance and I'd be happy to get feedback on other sites this question may fit better] Suppose that I ...
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is there something like the stem-and-leaf plot for timeseries?

When wanting to quickly take a look at the distribution of a sequence of values, the stem-and-leaf plot is an incredibly simple, yet powerful tool. It takes a few minutes to teach the computer to draw ...
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86 views

Proof of Cauchy Schwarz inequality for an inner product space-A visual approach.

Recently I studied Cauchy Schwarz inequality within inner product space.I found a proof in Hoffmann Kunze linear algebra book which constructs a vector $\gamma=\beta-\frac{<\beta,\alpha>}{||\...
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34 views

Visualizing the transformation from surface to contour plot in Windows

According to my understanding, this transformation from a surface in $\mathbb{R}^3$ to contour lines in $\mathbb{R}^3$ to a contour plot in $\mathbb{R}^2$ was made in Grapher, a program exclusive to ...
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129 views

Are pictures legitimate as a proof in mathematics?

While I'm studying Topology (teaching it myself with videos and books) I've seen some 'proofs' with pictorial approach and solution, I haven't seen it before. So is it legitimate?
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geometric intuition behind a homeomorphism

I've read that geometrically speaking, a homeomorphism from $M$ to $N$ is a bijection that can bend, twist, stretch and wrinkle the space $M$ to make it coincide with $N$ but it cannot rip, ...
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Given Cayley Diagram of several groups,how to construct one for their product and quotient?

I know how to represent a group as a Cayley diagram.But problem arises when I deal with quotient or product of $2$ or more groups.I want to know the process of obtaining the Cayley diagram for product ...
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75 views

A way to directly see $a^2+b^2=f^2$ for hyperbolae?

with focal length $f$ and $a,b$ from the standard equation (up-down opening, origin centered): $$ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 $$ Using the definition that a hyperbola is the points whose ...
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101 views

Are there projects that visualize how proofs relate to each other (similar to what the Paperscape Project does for publications)?

The Mathematics Genealogy Project lists mentoring relationships between mathematicians, the Paperscape Project visualizes which publications are "close" to each other (by analyzing citations and ...
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How to visualize on Venn diagram P(A ∩ B), P(A | B) when A, B is independent? [duplicate]

The equation in question is: P(A, B) = P(A | B) * P(B), and P(A,B) = P(A) * P(B). You can see in the image the portion of A ∩ B. Venn diagram of A intersect B In A independent from B case, you can ...

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