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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0answers
28 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 ...
0
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0answers
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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0answers
5 views
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 ...
-1
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1answer
23 views
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?
...
0
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0answers
14 views
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 ...
2
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0answers
41 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 ...
8
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0answers
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 $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$
...
1
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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 \...
4
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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 ...
1
vote
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, ...
6
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2answers
71 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
24 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
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 ...
6
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0answers
85 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 ...
0
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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 ...
5
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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 ...
2
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0answers
41 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
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)$ ...
1
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0answers
74 views
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 ...
1
vote
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 ...
6
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0answers
91 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
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 ...
1
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0answers
81 views
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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0answers
13 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 ...
11
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2answers
923 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 ...
11
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2answers
827 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{...
19
votes
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 ...
2
votes
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 ...
7
votes
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 ...
0
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0answers
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 ...
17
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3answers
190 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
42 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.: $(...
12
votes
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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0answers
33 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
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}$ ...
2
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1answer
59 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....
3
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0answers
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 \...
21
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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 ...
2
votes
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(...
1
vote
1answer
74 views
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/...
31
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0answers
745 views
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,...
2
votes
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 ...
3
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0answers
96 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 $...
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0answers
27 views
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 ...
2
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0answers
53 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 ...
7
votes
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 ...
24
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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 ...
1
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0answers
46 views
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 ...
1
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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 ...
1
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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=...
