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Questions tagged [visualization]

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

23
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535 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 ...
16
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0answers
196 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 $3\mod 4$...
15
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0answers
269 views

Geometric representation of Euler-Maclaurin Summation Formula

When reading Tom Apostol's expository article (or the free link), I was expecting more diagrams to come that follow the figure below (or this from the Wolfram project). It was a disappointment not ...
11
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0answers
497 views

Visualizing Hopf fibration $S^3\to S^2$ as a based map $S^1\to \mathrm{Map}(S^2,S^2)$

A fiber bundle $F\to E\to B$ may be interpreted as $E$ being a bunch of $F$s arranged in the shape of a $B$. For instance, a Mobius band $M$ is a bunch of line segments $[0,1]$ arranged in the shape ...
10
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0answers
604 views

Visual intuition for direct sum vs. tensor product of vector spaces

I completely understand the formal mathematical distinction between the direct sum and the tensor product of two vector spaces. I also understand that the direct sum has a nice visual interpretation (...
9
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0answers
215 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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637 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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188 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 $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$ ...
8
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0answers
192 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
69 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 ...
6
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88 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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134 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
143 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 ...
5
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57 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
113 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
699 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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0answers
178 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 ...
5
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0answers
309 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
532 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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0answers
93 views

Lagrange spectrum in diophantine approximation theory

Context. Hurwitz' theorem states that for every irrational $\xi$, there is infinitely many rationals $p/q$ such that $$\left\vert \xi-\frac pq\right\vert<\frac 1{q^2\sqrt 5}.$$ The number $\sqrt ...
4
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55 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
103 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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603 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
422 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} \...
4
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0answers
330 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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0answers
46 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
103 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
38 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
49 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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0answers
188 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
68 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 ...
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0answers
525 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
250 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
364 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
680 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
168 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 ...
3
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0answers
152 views

Illustrations of a line and a curve intersecting for complex field

Are there nice illustrations on the Net of say $y=a·x+b$ and $y=x^2$ intersecting where x and y are complex? I'm thinking of the amplitude of y being depicted as height above the complex plane with ...
2
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0answers
73 views

A summation of the multiplication of reflected Mobius functions and their behavior for different values of $k$

$$ \Psi_k(N)=\sum_{n=1}^{N} \mu(n)\mu(k-n)$$ where $\mu(n)$ is the Mobius function. This function is interesting to me because for the case of $k=N$ it has the symmetric property of being odd with ...
2
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0answers
44 views

Is this a new visual for finding the genralized Sum of Positive Integer Powers formulas?

In 2012-2013, I (with experimentation and hard work) founded this image to iteratively compute the formulas for the sum of powers. In 2015, I uploaded this three part video series on it. (In the ...
2
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0answers
43 views

Animate this Moire pattern. What mathematical tools could be used to analyze this moving pattern?

For a mathematical art project, I want to animate the following pattern I made on desmos. It seems to be a Moire pattern. However I cannot make the pattern move smoothly and continuously because ...
2
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0answers
34 views

Conic sections in addition and multiplication graphs of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

Compare two kinds of addition and multiplication graphs of the cyclic groups $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$: group tables (with colored circles on a rectangular grid) line graphs (with ...
2
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0answers
36 views

Any good visualization tools to see certain matrix groups?

I've been studying differential geometry and algebraic topology for a bit, and something that keeps coming up are the manifolds $GL(n,\mathbb R),$ $SL(n,\mathbb R)$, $O(n),$ $SO(n),$ etc. I'm ...
2
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0answers
55 views

Another color scheme for 3D visualizations of complex functions

I am looking for visualizations of complex functions $f(z) = r(z)e^{i\varphi(z)}$ which plot the magnitude $r(z)$ as height and display the argument $\varphi(z)$ on this graph. On the cover of ...
2
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0answers
49 views

Projecting 6D cartesian coordinates to lower dimension

I've got a noisy set of points that lie on/near a unit 6-sphere. I want to visualise their proximity to the sphere in some way. The only way I can come up with now is a histogram of the norm of the ...
2
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0answers
35 views

Drawing defects in the plane surrounding lattice squares

Localized curvature of a surface that is otherwise flat is sometimes called a curvature defect. If we put a cube on a table and cover it with a table cloth and wrap it with three rubber bands, the top ...
2
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0answers
68 views

Parametrizations for smooth tori of genus $n > 1$ in $R^3$

I am looking for parametrizations of smooth 2-tori in $R^3$ with multiple holes (genus > 1). For instance, Wikipedia has pictures of smooth-looking 2-holed and 3-holed tori. I was wondering what ...
2
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0answers
23 views

Patterns emerging from striped yarns in knitting

There's an interactive tool on this page that shows what happens when you knit a symmetrical striped yarn back-and-forth at different stitch counts (you can drag the stitch counter and it changes the ...
2
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0answers
134 views

Visually solving the integral of $\sin(x)$ from $0$ to $\frac{\pi}{2}$

I just tried to visually solve the integral of $\sin(x)$ from $0$ to $\frac{\pi}{2}$ by considering the unit circle. I tried to sketch my idea. The link is below: https://drive.google.com/file/d/0B-...
2
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0answers
153 views

Hopf map visualization (animation request)

Let $\phi:D^3\to S^2$ be the composition $D^3\to S^3\to S^2$, the first map being the quotient by the boundary and the second map being the Hopf map. Then: $$f_t:x\mapsto(1-t)x+t\phi(x)$$ is a ...
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67 views

Visualizing the minimization of $\|Ax-b\|_2^2$?

Consider the least-squares problem $$\text{minimize} \quad \|Ax-b\|_2^2$$ where $A$ is an $m \times n$ matrix and $b$ is an $m$-vector. How does it looks like geometrically? Could someone draw a ...