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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3answers
71 views

How to visualize this modular arithmetic problem?

In a certain medical group, Dr. Schwartz schedules appointments to begin 30 minutes apart, Dr. Ramirez schedules appointments to begin 25 minutes apart, and Dr. Wu schedules appointments to begin 50 ...
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1answer
144 views

Visualisation of ordinal vs cardinal arithmetic

I came around a nice article talking about large countable ordinals with examples such as: Let's have a book with $\omega$ pages (each 1/2 the thickness of previous so we can fit them in a book of ...
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3answers
66 views

How to think about $|a| \leq b$

Let $a,b \in \mathbb{R}$. $a \leq |b|$ is equivalent to the expression $-b \leq a \leq b$. Easy, geometrical, elegant, intuitive. But what about $|a| \leq b$ Suppose $b \geq 0$, then $|a| \leq ...
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1answer
99 views

Visual intuition for $\frac{1}{b - a} \int_{a}^{b} u(t) \ \text{d}t \in \overline{\text{co}}(u([a,b]))$

Let $u: [a,b] \to X$ to be a continuous function and $X$ a Banach space. Then $$ \frac{1}{b - a} \int_{a}^{b} u(t) \ \text{d}t \in \overline{\text{co}}(u([a,b])) $$ holds, where $\overline{\...
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1answer
132 views

Visualization of an algebraic stack

I am interested in thinking visually about algebraic stacks (also higher and derived stacks, but let´s start from the beginning). As the visuallization of an algebraic stack is virtually impossible I ...
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0answers
35 views

Kerr metric in SageMath causal visualization

I am studying the Kerr Metric via the SageMath manifolds module and would like to visualize the casual surfaces generated by light cones. Like the circles in this picture: Can anybody provide clues ...
5
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3answers
187 views

How can you mathematically define a “wobbly” function?

There are a lot of functions that look wobbly. For example $x^4 + x^3$ looks a little wobbly when it gets near the x axis. The function $\sin(x)$ is extremely wobbly. The function $\sin(x) + x$ is ...
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0answers
43 views

I attempted to visualize dot product of complex vectors. What do you advice?

I am an electronics undergraduate student currently learning wavelets. In the book A First Course in Wavelets with Fourier Analysis authors first introduce complex vectors and their dot products. Then ...
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0answers
35 views

Visual Intuition: Gaussians closed under addition

I'm trying to develop some intuition for the fact that the family of Gaussian distributions is closed under addition. I.e. if $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$, then $Y = \sum_iX_i$ is also ...
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0answers
129 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 ...
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1answer
77 views

Closed expression and physical interpretation of the median

Opposed to the arithmetic mean there is no immediate closed expression for the median of a distribution $n(x)$ of a variable $x\in\mathbb{N}$ over a population of $N$ items, at least not when ...
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2answers
25 views

Area of revolution of a square

A square of side length 1 is rotated 360 degrees about one of its vertices. What is the area of the region the square covers while rotating? I don't know how to visualize this as a geometric shape ...
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2answers
93 views

Asymptotic expansion of $Li^{-1}$ and zeros of $F(s)$ and $G(s)$

If you downvote please leave some constructive feedback. I would like to compare and visualize/gain insight about the zeros of two functions, $F(s)$ and $G(s).$ $\pi(m)$ is the prime counting ...
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0answers
79 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 ...
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1answer
239 views

How to see if a subgroup is normal from Cayley graph

Let be a Cayley diagram of group $G$. Let $H$ be the orbit of element p. Is $H$ a normal subgroup of $G$? Is there a simple way to check that because going by definition seems complicated. I tried ...
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0answers
69 views

Visualising connectedness in topology

Recently started a topology course, I am struggling to visualise connected topological spaces in topology, I am understanding the definition when it comes to clopen subsets and disjointness, ie $[0,1) ...
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0answers
45 views

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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0answers
128 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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1answer
37 views

If I have n posts that don't occur at the end of the fence, why are there n + 1 sections? [closed]

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? The correct ...
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0answers
51 views

Educational gif-animations on mathematical analysis [closed]

I drew some simplified gif-animations by math. Do you think, they will help in the study of mathematics to people, who have difficulties with this?
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0answers
66 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 ...
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0answers
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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1answer
165 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
171 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 ...
1
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1answer
268 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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2answers
549 views

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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0answers
33 views

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 ...
2
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1answer
169 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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0answers
186 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 ...
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1answer
45 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 ...
8
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1answer
393 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 ...
2
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0answers
56 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 ...
0
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1answer
32 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 $(X,d)$ ...
2
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1answer
174 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 ...
6
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0answers
93 views

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 "...
0
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1answer
74 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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0answers
22 views

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 ...
12
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2answers
1k views

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 ...
12
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2answers
1k views

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
233 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
51 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 ...
7
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2answers
263 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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0answers
133 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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3answers
241 views

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 ...
1
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3answers
198 views

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.: $(...
11
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1answer
266 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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0answers
125 views

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 ...
0
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1answer
73 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 $\mathbf{w}$, $\mathbf{x}$ ...
2
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1answer
89 views

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....
4
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0answers
388 views

Visual representations of groups (in their symmetric groups) [closed]

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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