Questions tagged [visualization]
For questions about visualizing mathematical concepts. This includes questions about visualization of mathematical theorems and proofs without words.
1
vote
0answers
26 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 ...
1
vote
0answers
24 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 ...
3
votes
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 ...
2
votes
1answer
66 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 ...
0
votes
2answers
20 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 ...
4
votes
2answers
83 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 ...
2
votes
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
votes
2answers
101 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 ...
0
votes
0answers
55 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) ...
1
vote
0answers
42 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
votes
0answers
51 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 ...
0
votes
0answers
6 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
votes
1answer
26 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?
...
2
votes
0answers
37 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?
0
votes
0answers
16 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
votes
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 ...
8
votes
0answers
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 $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$
...
0
votes
1answer
133 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
votes
3answers
122 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
votes
2answers
83 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\...
1
vote
0answers
26 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
votes
1answer
155 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
votes
0answers
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 ...
0
votes
1answer
33 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
votes
1answer
124 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
votes
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 ...
0
votes
1answer
25 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
vote
0answers
75 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 ...
2
votes
1answer
135 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
votes
0answers
92 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
votes
1answer
41 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
vote
0answers
88 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_{\...
1
vote
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
votes
2answers
926 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
votes
2answers
840 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
218 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
33 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
190 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
votes
0answers
86 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
votes
3answers
192 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
vote
3answers
52 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
180 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$...
0
votes
0answers
43 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
votes
1answer
30 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
votes
1answer
61 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
votes
0answers
311 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 \...
21
votes
2answers
396 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
101 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
77 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/...
