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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What are good graphs to compare two sets of data of network delays?

I have two sets of numbers, each set contains 100 numbers, which are delays that I have measured from a network. I want to visualize the comparison between these two sets, instead of just provide two &...
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3answers
65 views

Visualization for Euclidean Algorithm [duplicate]

I want to really understand the Euclidean Algorithm. A key component in the algorithm is fact that common divisors of two integers are common divisors of their difference. I can see from the ...
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1answer
156 views

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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73 views

Is it a bad idea to use visualizations in studying linear algebra? [closed]

many concepts in linear algebra are explained using visualizations. However, many practical applications of linear algebra use spaces whose dimension is larger than 3 which you cannot visualize. When ...
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2answers
60 views

There are 15 aliens: 8 Plutonians and 7 Venusians. How many ways can I form a group of 4 with at least 1 Venusian?

I understand that this answer is $1295$. I can get this by either of: $${{7}\choose{1}}{{8}\choose{3}}+ {{7}\choose{2}}{{8}\choose{2}}+{{7}\choose{3}}{{8}\choose{1}}+{{7}\choose{4}}{{8}\choose{0}}$$ ...
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1answer
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Drawing an approximation to a circle in isometric projection

A circle viewed from from the side is an ellipse. A common approximation can be found on the web (eg do a google image search for isometric circle). This produces something like (with arc centers T,U,...
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Snake lemma, pictorially on simplicial complexes

I'm trying to understand what the snake lemma 'computes' on small examples. Consider this: It seems to me that given two 'defective' / 'incomplete' simplicial complexes that are related through a ...
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1answer
798 views

Characterize normal subgroups - Find all subgroups of $S_3$ conjugate to $\{id, (1,3) \}$ - Fraleigh p. 143 14.29

(27.) A subgroup H is conjugate to a subgroup K of a group G (viz. p. 141 $K \le G$ is a conjugate subgroup of $H$), if $i_g[H] = gHg^{-1} =K$ for some $g \in G$. Show that conjugacy is an ...
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19 views

Intuition for curved Lorentzian manifolds through Euclidean embedding

How can one find embeddings of parts of curved Lorentzian manifolds into low dimensions ($\leq 3$) for visualization and intuition building? For example, one can embed the space $H_2$ into $\mathbb R^{...
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3answers
53 views

What is the derivative of the function $f(x)=ix?$ Is it $i$?

Why is this? How is $i$ the slope of the function? Where is it the slope? I understand taking the derivative with the power rule in, for example, the parabola $x^2$ becoming $2x$ and seeing where that ...
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1answer
14 views

Function that produce a spike in 3D

I am looking for a function that would produce a spike, of any size, in the z-direction at a given x and y coordinate. A little bit like the https://en.wikipedia.org/wiki/Dirac_delta_function but that ...
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2answers
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Representing region with integrals, but with $ y $ variable instead of $ x $.

I'm having trouble getting this through my head. I have an exercise in which I'm told that $f(x) = \sqrt x $ and $ g(x) = 2-x $. Using definite integrals, I'm supposed to find the area of the region ...
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1answer
53 views

If $f:\mathbb{C}\to\mathbb{C}$ is a continuous function that is analytic off $[−1, 1]$, then $f$ is entire

I'm trying to solve Show that if $f:\mathbb{C}\to\mathbb{C}$ is a continuous function such that f is analytic off $[−1, 1]$ then $f$ is an entire function. This is a solution of Andreas Kleefeld: ...
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Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are ...
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2answers
38 views

Why can the average/midpoint of two numbers be described as the sum of the numbers divided by two?

Say I have two numbers, A and B. The "average" or the midpoint of the two numbers is given by $$\frac{A+B}{2}$$ My question is, why does this formula work? Intuitively, I can derive the equation as ...
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138 views

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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1answer
137 views

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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24 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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1answer
54 views

What are the most interesting math riddles you know? [closed]

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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0answers
19 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
23 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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123 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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0answers
30 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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2answers
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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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26answers
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Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction

I recently proved that $$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$ using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation ...
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58 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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4answers
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Visual research problems in geometry

I am considering doing research in mathematics to be my career (and my life) someday. I'm a visually oriented person in general; for example, I prefer chess over cards because when I play chess, I ...
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1answer
116 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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2answers
60 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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1answer
37 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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1answer
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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283 views

Figures and Numbers: Relating properties of geometric shapes and their Fourier series

Consider two types of parametrized curves $\gamma:[0,2\pi]\rightarrow \mathbb{R}^2$ open curves $\gamma_\sim(t) = (t,a(t) + b(t))$ closed curves $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$ ...
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2answers
56 views

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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“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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64 views

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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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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81 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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1answer
16 views

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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14 views

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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1answer
33 views

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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3answers
2k views

Textbooks for visual learners

Perhaps this question has already been asked (if so, please let me know) but I am looking for books that appeal to visual learners. I discovered that I am able to understand concepts much quicker ...
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1answer
63 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 book, all ...
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216 views

Stellating the Octahedron

I have a few related questions and I'd be happy to get some help with any one of them. Is the stellation of a polyhedron generally a 'messy' affair that involves trimming away portions of the ...
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18 views

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
33 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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42 views

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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1answer
247 views

Visualize Itô differentiation rule

Please help me to find an idea to visualize $$\displaystyle d{ f(t,x)} = \frac{\partial f(t,x)}{\partial t}dt + \frac{\partial f(t,x)}{\partial x}dx + \frac12 \frac{\partial^2f(t,x)}{\partial x^2} dt$$...
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2answers
153 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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27 views

Visualization of imbedding of compact manifold

This is Theorem 36.2 from Munkres' Topology. If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^N$ for some positive integer $N$. Proof. Cover $X$ by finitely many open sets $\{...
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127 views

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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