Questions tagged [visualization]

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

232 questions with no upvoted or accepted answers
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71
votes
0answers
1k views

Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles?

The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a ...
25
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0answers
580 views

Visualizing the Partition numbers (suggestions for visualization techniques)

So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so ...
22
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0answers
447 views

Geometric representation of Euler-Maclaurin Summation Formula

In Tom Apostol's expository article (here's a free link), upon seeing the figure below (or this from the Wolfram project) I was expecting more diagrams to come to continue the error decomposition of ...
19
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0answers
242 views

The famous prime race and generalizations

So I was messing around with the famous prime race that comes down to this: We make a list of primes. The list has two rows; the top row is for primes $1\mod 4$ and the bottom row for primes $3\mod 4$...
13
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0answers
288 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 $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$ ...
9
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220 views

Seeing symmetries

Preliminaries Let $[n] = \{0,\dots,n-1\}$ and $P([n])$ be the power set of $[n]$. Let the correlation between two subsets $x,y$ of $[n]$ be the number $\kappa(x,y) = 1 - \frac{2}{n}|x\triangle y|$ ...
9
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0answers
890 views

How to visualize cotangent spaces.

I was wondering how to intuitively and visually understand dual vector spaces and one-forms. So my question is (1), how to visualize cotangent spaces and (2), how to intuitively understand them? My ...
8
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0answers
200 views

Is there a way to visualize a group?

Is there a way to picture a group in ones head? I want to "see" the difference between abelian and non-abelian group. And if $f$ is a group homomorphism, is there a way to see that $\ker(f)=1\...
7
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0answers
201 views

The quadrature of the circle: comparing Archimedean and Ulam spirals

There are two closely related arrangements of the natural numbers that allow to show patterns in the distribution of some sets of numbers (multiples of 2, 4, 8, square numbers, prime numbers): the ...
6
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0answers
119 views

Visualizing rational numbers as multiplication graphs

It's an interesting fact, that there's a straight forward way to visualize rational numbers. To each rational number – given as two integers $n<m$ – there corresponds a multiplication graph $n/m$ ...
6
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0answers
223 views

Graphical multiplication tables for $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{Z}$

Inspired by Burkard Polster's beautiful video on Times Tables, Mandelbrot and the Heart of Mathematics I wondered how this graphical approach to visualize the multiplicative structure of finite rings ...
6
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0answers
184 views

Converting complex domain coloring visualizations into autostereograms: is this technique in use?

As Wikipedia says regarding the domain coloring technique for complex functions: A graph of a complex function $g : \Bbb C \to \Bbb C $ of one complex variable lives in a space with two complex ...
6
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1answer
122 views

Visual questions for 6th graders

I'm tutoring a 6th grader in math at the moment and because she never has a ton of homework I like to give her some interesting extra problems to do. It seems she really enjoyed a problem I showed ...
6
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1answer
104 views

An Auto-Generated Cartography of Mathematical Theories: Has it been done already?

While looking for a way to visualize the logical structure of mathematical theories a graph-like depiction came to my mind, where propositions are represented by vertices. An edge goes from ...
5
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1answer
247 views

Visualize Itô differentiation rule

Please help me to find an idea to visualize $$\displaystyle d{ f(t,x)} = \frac{\partial f(t,x)}{\partial t}dt + \frac{\partial f(t,x)}{\partial x}dx + \frac12 \frac{\partial^2f(t,x)}{\partial x^2} dt$$...
5
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0answers
60 views

Visualization of quadratic rings $\mathbb{Z}[\sqrt{d}]$

The extensions $\mathbb{Z}[\sqrt{d}]$ of $\mathbb{Z}$ by the root $\sqrt{d}$ of the quadratic polynomial $X^2 - d$, $d \in \mathbb{Z}$ square-free, have degree $2$ and all have the same additive ...
5
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0answers
155 views

Distributions of prime numbers

When folding the number line not into a spiral (like Ulam did) but into a zig-zag pattern (like Cantor did) there are other patterns visible in the distribution of prime numbers: [To see these ...
5
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0answers
1k views

I've animated Riemann zeta function and got a spiral. Why?

So I've obtained this peculiar visualization of Riemann zeta function: The way I do this: I treat every $nth$ animation frame as a complex plane and plot all values of $\zeta(s)$ for all $s$ such ...
5
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1answer
760 views

Intuition & Proof of rank(AB) $\le$ min{rank(A), rank(B)} (without inverses or maps) [Poole P217 3.6.59, 60]

I'm aware of analogous threads; I hope that mine is specific enough not to be esteemed one. $\mathbf{a^i}$ is a row vector. $A, B$ are matrices. Prove: $1$. $\mathbf{a^i}B$ is a linear ...
5
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0answers
336 views

Looking for proof-without-words of Bezout's identity

I'm looking for a "proof-without-words" of Bezout's identity (for integers). Does anyone know of one?
5
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0answers
586 views

Visualizing the domain of the square root

I would like to show someone the domain of the complex square root function (the 2-sheeted riemann surface). Is there a good interactive visualization software for this? I would like some sort of ...
4
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1answer
133 views

Seeking intuition for why tesselations of space by hypercubes in dimensions 8+ need not have a face-to-face pair (Keller's conjecture counterexample)

According to Keller's conjecture: In any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face. Perhaps surprisingly, this is false for every dimension greater ...
4
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0answers
51 views

What is the best approach to graphing functions from $\mathbb R^n$ to $\mathbb R^m$?

I'm looking to 'visualize' higher dimensional functions in the best way possible (using plotting software,i.e., we are given a range of values in the domain to plot to the image of a function). ...
4
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0answers
128 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 ...
4
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0answers
74 views

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, ...
4
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0answers
66 views

Graph layout that reflects graph symmetries

I am looking for practical computational methods to lay out graphs in such a way that the geometry of the drawing reflects some of the symmetries of the graph. Here are two example drawings of two ...
4
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0answers
138 views

Iterative construction of the real projective space

I can visualize the construction of $\mathrm{RP}^2$ from a disc $B^2$ whose boundary $S^1$ is subjected to the antipodal identification. This can be obtained by glueing the edge of a Möbius strip $M$, ...
4
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1answer
1k views

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,...
4
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0answers
623 views

Is this a legit way to visualize complex functions?

I am doing laplace transform in a class and I hate how there seems to be no graphical support when things are transformed to laplace domain i.e. nobody cares what they look like in laplace domain But ...
4
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0answers
553 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} \frac{...
4
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0answers
345 views

Visualization of immersed submanifold

I am trying to visualize the difference between immersed submanifold and embedded submanifold. At first, I thought that, for example, if I can embed manifold $M$ in $\mathbb{R}^4$ and if my friend can ...
3
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1answer
74 views

Geometric Intuition of the Dot Product

First of all, sorry for my poor English and thanks for your time. I’m having problems to understand the intuition behind the dot product. I know how to calculate the dot product with the algebraically ...
3
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0answers
33 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? ...
3
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0answers
61 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 ...
3
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0answers
128 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 ...
3
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0answers
34 views

How to visualize and explain complicated 3-dimensional structures?

I'm studying this example of nonshellable but constructible 3-ball on 10 vertices and 21 facets. Other than just staring at the pictures and hoping that they would eventually make sense, is there any ...
3
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0answers
148 views

What's the geometric interpretation of the square root of a matrix?

Question: If I have a matrix $A$, I know that its square root is a matrix that has the same eigenvectors as $A$ but its eigenvalues are the square roots of the eigenvalues of $A$. What does this ...
3
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0answers
235 views

Visualizing Cauchy's integral theorem (and complex integration in general)

(I edited the question due to a hint from Giuseppe Negro who pointed out that I forgot about $dz$.) Consider Cauchy's integral theorem, i.e. $$\oint_\gamma f(z)dz = 0 $$ for holomorphic functions $...
3
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0answers
46 views

Where can I find 'Emily', the matrix visualization tool?

While searching for a tool to visualize sparse matrices I discovered this paper about 'emily', a piece of software which has everything I need. However, I cannot find a place to download or purchase ...
3
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0answers
113 views

Symmetries in the equations and graphs of complex-valued polynomials

I tried to visualize the complex roots of a polynomial with real coefficients $a_i \in \mathbb{R}$: $$f(z) = z^n + a_{n-1}z^{n-1} + \dots + a_1z + a_0$$ following some obvious thoughts: For any ...
3
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1answer
91 views

Is there a relation between isometric and orthographic measurements?

This image shows a couple of different isometric projections. In the black shows the figure's "true" dimensions in an orthographic projection while the red shows the dimensions in an isometric ...
3
votes
1answer
123 views

Emil Artin on visualization of matrices

Someone called my attention to the fact that Emil Artin made very important remarks on the visual representation of matrices in some of his books. Could anyone tell me which precise book that is? ...
3
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0answers
393 views

Hyperplane in a complex vector space

This is my first question on MSE, I'm sorry if there already exists similar questions, I couldn't manage to find it. My friend, who studies Physics, asked me about the meaning of "functional" so I ...
3
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0answers
204 views

Visualising surface integrals

For a current problem I am working on, I have run into angular surface integrals, i.e. the differential solid angle $\text{d}\Omega$. Specifically the surface integrals are defined by \begin{equation}...
3
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0answers
83 views

Visualization of the fact that the integers defining lens spaces must be coprime

This is related to this question I asked: Visualization of Lens Spaces and is also related to this question by @Earthliŋ: Why are the integers appearing in lens spaces coprime? I understand the ...
3
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0answers
602 views

Visualize normal subgroup, normalizer, cosets.

A few important aspects of the relationship $H \lhd N_G(H) \le G$ are highlighted in Figure 7.31. First, the size of $N_G(H)$ is some multiple of |H|, and the size of G is some multiple of $N_G(H)$, ...
3
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0answers
281 views

Complex analysis visualization (Cauchy Theorem, Residue Theorem)?

I usually think of complex functions on the complex plane like vector fields. So basically what I have problems with is visualizing firstly Holomorphic functions. I have also read and successfully ...
3
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0answers
418 views

Power-set in Hypercube: historical background of indexing each term like Hasse Diagram?

My instructor wants references about the indexation over the hypercube, related question here, he does not believe that I was the first who used it -- [update] thanks to a comment, the name is Hasse ...
3
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0answers
743 views

Imagining four or higher dimensions and the difference to imagining three dimensions

I’m very interested in how people envision four or higher dimensions. And I’m especially interested in how geometers and topologists who actually work in four dimensions do. Now I know of the video ...
3
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0answers
170 views

Visualizing and manipulating 4-dimensional data with 3D technology

It is possible to visualize 3 dimensional data (like a scatter plot) by projecting it on a 2 dimensional screen in a way that allows to interact with it in an intuitive way. Is it possible to ...

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