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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.

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Visually suggestive way to present a finite group

When studying about groups, we can often grasp the structure of a small group "internally". For instance, we can "see" that $\mathbb Z/3\mathbb Z$ as a three-fold symmetric shape;more complicated ...
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47 views

Circle group $S^1$

Can someone describe the circle group $S^1$ in a easily understandable way? Are elements in $S^1$ in the form of $e^{2i\theta\pi}$? What does this look like pictorially? Doesn't necessarily need a ...
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Unit vector Identicon

Is there a way to visualize multiple n-dimensional ($n\approx300$) unit vectors with the property that large changes to the vector result in large visual changes, but small changes result in little to ...
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1answer
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If I have n posts that don't occur at the end of the fence, why are there n + 1 sections? [on hold]

https://en.wikipedia.org/wiki/Off-by-one_error#Fencepost_error: More generally, the problem can be stated as follows: If you have n posts, how many sections are there between them? ...
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How to display a 2d manifold of points?

I have created a manifold for a set of discrete (integer) points which lie in 2 dimensions. The manfiold has dimensions of $N\times N$ and I have a metric which dictates the seperation between the ...
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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 ...
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176 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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1answer
124 views

Intuition behind Well Ordering Principle and Axiom of Choice

I am learning about Axiom of Choice and Well Ordering Principle from Munkres's Topology book, but I can't quite wrap my head around it properly. I have these questions: [Munkres 0.4.3] If $A = A_1 \...
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3answers
115 views

Enigmatic patterns in Archimedean spirals

Distributing the natural numbers as circles evenly along the Archimedean spiral yields surprising patterns when changing the radius of the circles: they cover more and more of the plane, finally ...
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1answer
35 views

Adding vector fields

Consider two vector fields: $$ \vec F_1=(\sin(x),\sin(y)) $$ $$ \vec F_2=(\sin(1-x),\sin(y)), $$ where $x,y \in(0,\pi).$ Does adding the two superimposed vector fields produce a net vertical flow, ...
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Parametrizing the square spiral

Related to this question concerning number spirals I have another one, more specific. While it is rather easy to arrange the natural numbers along an Archimedean spiral by $$x(n) = \sqrt{n}\cos(2\pi\...
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How to plot ideals of rings

Im trying to better understand ideals of rings and I think being able to visualize what I'm working with would help. I want to graph them (I'm talking mostly about quadratic rings), but I don't know ...
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1answer
153 views

Visualization of groups with a normal subgroup_rev#1

Let $G$ be a group and $H \unlhd G$. In general, $H=H_Z \sqcup H_{G \setminus Z}$, where $H_Z:=H \cap Z(G)$ and $H_{G \setminus Z}:=H \cap (G \setminus Z(G))$. I'm investigating on a plausible visual ...
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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 ...
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1answer
28 views

Calculate points of a tesseract (hypercube)

I would like to know how to calculate the points of a hypercube. I am trying to use the mac app Grapher to simulate what one would look like. Does anyone know the equation I could use to generate the ...
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1answer
114 views

Proof that $n$ planes cut a solid torus into a maximum of $\frac16(n^3+3n^2+8n)$ pieces

Question: How many pieces can a solid torus be cut into with three (affine) planar cuts? A google search will quickly reveal that the answer is thirteen, as can be read about here. The picture below ...
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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 ...
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1answer
24 views

Visualization / sketch for this basic proof about subspace topology

Let $(X,d)$ be a metric space and $A\subset X$ a subset equipped with the induced metric $d_{A}$. Then the open subsets of $(A,d_{A})$ are exactly the intersections of open subsets of of $(X,d)$ ...
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Wild visualization of higher dimensions [closed]

I have a very sophisticated mental picture of higher dimensions and I really need some guidance in correcting my wild imagination. Is it ok to visualize $ \mathbb{R}^4 $ like a regular 3D space ...
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1answer
127 views

Visualization of groups with a normal subgroup

Suppose $G$ a group and $H \triangleleft G$ (proper normal subgroup). The simplest way to visualize this basic setup is that (Venn-wise) of a bubble ($H$) into a bigger one ($G$), sharing the unit and ...
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What “tools” are available within pure mathematics to visualize more advanced topics? [closed]

What "tools" exist within mathematics to visualize concepts for more advanced areas of mathematics, particularly within Analysis, Topology and Algebra ? Furthermore, can one rigorously integrate such "...
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1answer
39 views

Explicit construction and proving or disproving expander graph for this family

In combinatorics, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. In an expander graph most vertices are far apart ...
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Visual representations of groups (in their symmetric groups)_part#2

Background In this post, I have shown that a plausible visual representation of a group $K$ in $\operatorname{Sym}(K)$ can be established, where $\operatorname{Aut}(K) \setminus \lbrace \iota_{\...
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Graph intersection of two 3d loci

I'm trying to graph a 1 dimensional object that curves in 3 dimensions. The only way I've ever been able to do this is with parametrics in Mac Grapher, or by graphing two 2 dimensional surfaces in 3 ...
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2answers
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Mathematics and the art of linearizing the circle

[I edited the question and put stronger emphasis on "constant curvature" than on "naturalness".] One of the most prominent problems of ancient mathematics was the squaring of the circle: to construct ...
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2answers
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Explanation of a regular pattern only occuring for prime numbers

Consider multiplication group tables modulo $n$ with entries $k_{ij} = (i\cdot j)\ \%\ n$ visualized according to these principles: Colors are assigned to numbers $0 \leq k \leq n$ from $\color{...
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1answer
217 views

Strangely but closely related parametrized curves

Compare the following two parametrized curves for $k \in \mathbb{N}^+$: $$x_r(t) = \cos(t)(1 + r\sin(kt))$$ $$y_r(t) = \sin(t)(1 + r\sin(kt))$$ with $0 \leq t < 2\pi$ and $0 \leq r \leq 1$ (being ...
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0answers
31 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 ...
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2answers
187 views

Compute fundamental group _visually_ by the polygonal representation of the space

Some time ago, before I learnt about covering spaces and Seifert-Van Kampen theorem, I tried to compute visually the fundamental group of some spaces. For example I figure out by myself that the ...
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82 views

$\operatorname{Spec}(\mathbb{Z}[x])$ as a ''classical'' topological space?

I've read recently the definition of the prime spectrum associated to a ring and how it can be made into a topological space. I've read that $\operatorname{Spec}(\mathbb{Z}[x])$ is really important ...
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Visualized group tables for $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

Let me share a kind of visualization of group tables which is especially well suited for cyclic groups like $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$. In these groups you can easily give colors to ...
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3answers
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Visualization of binomial coefficients to the 4th power

I have been reading about binomial coefficients in Wikipedia. Where there is a visualization of binomial expansion up to the 4th power: I do not understand the sequence for the 4th dimension, i.e.: $(...
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1answer
176 views

Visualizing quadratic residues and their structure

[I corrected the pictures and deleted one question due to user i707107's valuable hint concerning cycles.] Visualizing the functions $\mu_{n\%m}(k) = kn\ \%\ m$ (with $a\ \%\ b$ meaning $a$ modulo $b$...
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visualising parametric equations

Whenever I'm doing a question on curves, and I am given the equation without the graph I can reasonably visualise in my head what that graph would look like, and for the more complicated equations, I ...
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1answer
27 views

Library for visualizing computation graph

Does anyone know a tool or library (preferable JavaScript) that can visualize an equation as a computational graph, such that for example the sigmoid function with inputs $\mathbf{w}$, $\mathbf{x}$ ...
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1answer
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Doubt in understanding theorem 4.1 Real analysis Stein and Shakarachi

I came across the following theorem. I understood that $F_k(x)\to f(x)$ pointwise as $k\to \infty$, but I do not understand how $E_{l,j} $ is defined over a range of $F_k$. I am not able to visualise....
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307 views

Visual representations of groups (in their symmetric groups)

Given a group $G$, left and right multiplications establish the subgroups $\Theta:=\lbrace \theta_a \mid a \in G \rbrace \le \operatorname{Sym}(G)$ and $\Gamma:=\lbrace \gamma_a \mid a\in G \rbrace \...
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2answers
389 views

Polar plots of $\sin(kx)$

The plots of $\sin(kx)$ over the real line are somehow boring and look essentially all the same: For larger $k$ you cannot easily tell which $k$ it is (not only due to Moiré effects): But when ...
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1answer
100 views

The slope of $nx\ \%\ m$

(There is a follow-up question at MO.) Let $x\ \%\ m$ be the residue of $x$ modulo $m$, i.e. $$x \equiv x\ \%\ m\pmod{m}$$ Let $\mu^n_m(x)$ denote multiplication by $n$ modulo $m$, i.e. $$\mu^n_m(...
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1answer
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Html5 Math Applets. Interactive free online

I was an aficionado at collecting links from websites with math java applets that allowed interaction to learn mathematical concepts visually and interactively. My favorite was http://www.ies-math.com/...
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Proof without words of $\oint zdz = 0$ and $\oint dz/z = 2\pi i$

I found this visual "proof" of $\oint zdz = 0$ and $\oint dz/z = 2\pi i$ quite compelling and first want to share it with you. But I have a real question, too, which I will ask at the end of this post,...
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0answers
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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 ...
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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 $...
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What kind of matrices can be visualized by graphing the column vectors?

In the 3Blue1Brown video “Linear transformations and matrices” linear transformations are visualized by overlaying gridlines which have a position determined by the values of the transformed basis ...
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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 ...
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1answer
142 views

Patterns in division graphs modulo $n$

(I made an edit due to hints from Alex Ravsky. Thanks to him.) General division graphs with nodes $1,2,\dots N$ and an edge between $n$ and $m$ when $n$ divides $m$ or $m$ divides $n$ are sparse and ...
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2answers
243 views

An enigmatic pattern in division graphs

Draw the numbers $1,2,\dots,N$ on a circle and draw a line from $n$ to $m>n$ when $n$ divides $m$: For larger $N$ some kind of stable structure emerges which remains perfectly in place for ever ...
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Almost- and non-primes in the Ulam spiral

There are lots of explanations of the preferred diagonal distribution of prime numbers in the Ulam spiral: I wonder if these explanations can also explain the observation that when highlighting also ...
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0answers
112 views

3D-plots of complex functions

Commonly it's believed that one cannot fully visualize a complex function $f:\mathbb{C}\rightarrow \mathbb{C}$ because the full plot would have to be 4-dimensional. But is this true? And why would the ...
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1answer
41 views

Venn Diagram tool to help with learning set theory

I want to use Venn Diagrams to help me to develop confidence and skill in set theory, including various proofs. I found this tool online: https://www.wolframalpha.com/widgets/view.jsp?id=...