Questions tagged [visualization]
For questions about visualizing mathematical concepts. This includes questions about visualization of mathematical theorems and proofs without words.
10
votes
0answers
69 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
30 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.: $(...
0
votes
0answers
41 views
Permutations of quadratic residues modulo $p$
When bending the square $[0,n[\times[0,n[$ to a torus, the quadratic residues $k^2\ \%\ n$ — with $0 < k < n$ and $a\ \%\ b$ meaning $a$ modulo $b$ — lie on a parabola $P_1$ with ...
1
vote
0answers
18 views
Comparing four ways to visualize complex functions
Among many others there are these four ways to visualize a complex functions $f(z)$:
Choose
a straight line going through the origin
a circle around the origin
For each $z$ on this curve draw a ...
12
votes
1answer
165 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
24 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
18 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
54 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
262 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
votes
2answers
366 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
98 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.
$$\mu^n_m(...
1
vote
1answer
68 views
Html5 Math Applets. Interactive free online
I was an aficionado at collecting links from websites with math java applets that allowed interaction to learn mathematical concepts visually and interactively. My favorite was http://www.ies-math.com/...
30
votes
0answers
652 views
Proof without words of $\oint zdz = 0$ and $\oint dz/z = 2\pi i$
I found this visual "proof" of $\oint zdz = 0$ and $\oint dz/z = 2\pi i$ quite compelling and first want to share it with you. But I have a real question, too, which I will ask at the end of this post,...
2
votes
0answers
34 views
Any good visualization tools to see certain matrix groups?
I've been studying differential geometry and algebraic topology for a bit, and something that keeps coming up are the manifolds $GL(n,\mathbb R),$ $SL(n,\mathbb R)$, $O(n),$ $SO(n),$ etc.
I'm ...
4
votes
0answers
91 views
Visualizing Cauchy's integral theorem (and complex integration in general)
(I edited the question due to a hint from Giuseppe Negro who pointed out that I forgot about $dz$.)
Consider Cauchy's integral theorem, i.e.
$$\oint_\gamma f(z)dz = 0 $$
for holomorphic functions $...
0
votes
0answers
26 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
votes
0answers
50 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 ...
6
votes
1answer
137 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 ...
23
votes
2answers
238 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
vote
0answers
40 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
vote
0answers
110 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
vote
1answer
31 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=...
12
votes
1answer
118 views
My visual interpretation of $1+2+3+ \dots +n$
To be frank, I didn't learn any sort of proof for this (visual or non-visual), so I came up with this proof through trial and error.
Moreover, I haven't checked my proof online yet, therefore I am ...
3
votes
4answers
90 views
Visual example of an infinite planar graph with degree sequence $(4^4,6^\infty)$
After reading some graph theory and talking with experts, I was intrigued. I would like to construct and visualise an infinite planar graph with degree sequence:
$$D=(4^4,6^\infty)$$
where the ...
0
votes
1answer
43 views
Visualizing double points.
I was trying to visualize by drawing a curve / figure to get a double point on a curve. As per the Wolfram article, a double point is a point traced out twice as a closed curve is traversed.
Any ...
0
votes
0answers
46 views
Game Theory: Finding the nash Equilibirums of 2x3 bimatrix by graphically representing the best replies
I am having troubles understanding the graphical method for solving a 2 x 3 bimatrix and I turn to all of you for help:
To give an overview of my doubt, consider a 2 x 3 bimatrix game, (A,B)
...
6
votes
1answer
62 views
Geometric understanding of subtracting lambda from diagonals
Given the definition of eigenvalues/eigenvectors:
$Av = \lambda v $
you could rearrange the terms to be:
$(A - \lambda I)v = 0$
Geometrically, the first equation says that multiplying by $A$ is ...
3
votes
0answers
36 views
Where can I find 'Emily', the matrix visualization tool?
While searching for a tool to visualize sparse matrices I discovered this paper about 'emily', a piece of software which has everything I need. However, I cannot find a place to download or purchase ...
7
votes
0answers
65 views
Visualizing rational numbers as multiplication graphs
It's an interesting fact, that there's a straight forward way to visualize rational numbers. To each rational number – given as two integers $n<m$ – there corresponds a multiplication ...
4
votes
4answers
231 views
Proof without words of the Quadratic Formula?
As suggested by @Moti and @YvesDaoust in this post, a simple way to identify the roots (red dots) of a parabola (given focus and directrix, blue) by means of straightedge and compass is to draw the ...
3
votes
1answer
49 views
In this visual proof for the law of cosines, why are the products of subsegments of two intersecting chords equal?
The first line of the visual proof below states that
$$(2a\cos\theta-b)b=(a-c)(c+a)$$
I understand the line segments represented by each part of the equation, but what makes the equation true?
In ...
0
votes
0answers
39 views
Visualisation of an orientable surface bounded by the Möbius curve
I'm learning multivariable calculus on MIT OpenCourseWare. When the teacher explained Stokes theorem he mentioned the Möbius strip. He showed it was non-orientable. Then he showed a somehow twisted ...
2
votes
1answer
59 views
Graph property intuition
I was showing some basic graph property and though I could formally prove it, I can't wrap my head around the intuition of why this property holds. The property is the following:
Let $G$ be a graph ...
6
votes
0answers
118 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 ...
5
votes
0answers
56 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 ...
1
vote
2answers
45 views
Understanding Cones in general and the Ice cream Cone
Definitions
Let $\mathbb{R}^n$ be the n dimensional Eucledean space.
With $S \subseteq \mathbb{R}^n$, let $S^G$ be the set of all finite nonnegative linear combinations of elements of $S$.
A set $K$ ...
2
votes
2answers
127 views
Geometric intuition of the dimension of Grassmannians and flag manfolds
I wish to understand geometrically (not just algebraically) why the dimension of the Grassmanian $G(k,n)$ is $k(n-k)$ and the dimension of a flag manifold $F(k_{1},k_{2},...,k_{n},N)$ is $\sum_{i=1}^{...
1
vote
1answer
40 views
Geometric interpretation of transitive relations
This question provided a geometric interpretation for transitivity in equivalence relations, but what about just transitivity by itself, without reflexive and symmetric constraints?
For example, ...
0
votes
1answer
46 views
Do $4$ orthogonal lines exist in $\mathbb{R}^4$?
I just learned about the cross product in linear algebra.
I need some help with a mental image.
In math, obviously not in our $\mathbb{R}^3$ world, do there exist $4$ orthogonal lines in $\mathbb{R^...
0
votes
1answer
25 views
$z$ on the unit circle, how many parts does $z \to e^z$ have?
So say $z$ is on the unit circle $|z| = 1$. How many parts does the transformed region$$z \to e^z$$ have?
My work. Alright, so if something is on the unit circle, it's of the form$$z = a + bi,\text{ ...
1
vote
0answers
48 views
What are some visually appealing derivatives?
The derivative of a power tower made up of $e$ repeated $6$ times with an $x$ at the top, is, by the chain rule
$$\frac{d}{dx} \left(\displaystyle e^{\displaystyle e^{\displaystyle e^{\displaystyle ...
0
votes
1answer
71 views
Visual proof of Gauss Theorem?
I saw some proofs about Gauss Theorem here but I could not understand everything about it.
Is it possible to have a visual proof of Gauss Theorem, as it can be very interesting to understand and see....
1
vote
2answers
49 views
Comparing and classifying real and complex-valued graphs of cubic polynomials
There seem to be six essentially different types of cubic polynomials with real coefficients, giving rise to 1, 2 or 3 real roots in different ways.
Consider $f(z) = z^3 + a_2z^2 + a_1z + a_0$ and ...
3
votes
0answers
43 views
Symmetries in the equations and graphs of complex-valued polynomials
I tried to visualize the complex roots of a polynomial with real coefficients $a_i \in \mathbb{R}$:
$$f(z) = z^n + a_{n-1}z^{n-1} + \dots + a_1z + a_0$$
following some obvious thoughts:
For any ...
0
votes
1answer
22 views
Geometric proof that cross product is linear
I've been searching online quite a bit for a simple and elegant geometric proof that the cross product distributes over addition. That is,
$$\mathbf a \times (\mathbf b + \mathbf c) = \mathbf a \...
8
votes
3answers
83 views
Is there a visualization for inverse trig functions as indefinite integrals
Examining the indefinite integral formulations of inverse trig functions I notice some things
$$\arcsin(x)=\int_0^x \frac{1}{\sqrt{1-z^2}}dz$$
$$\arccos(x)=\int_x^1 \frac{1}{\sqrt{1-z^2}}dz$$
We ...
8
votes
1answer
166 views
Visualization of p-adic numbers
I try to understand and get a feeling which gaps p-adic numbers fill to complete $\mathbb{Q}$.
In the course of this I depicted (for $p = 2$) the "base" $\{p^k\}_{k\in\mathbb{Z}}$ with respect to ...
0
votes
3answers
55 views
Topological Spaces: What are they?
Let $X\subseteq \Bbb R$, $\tau$ is a topology on $X$.
What even is $(X,\tau)$? Is it an ordered pair? Is it $\{(x,y)|x\in X, y\in \tau\}$? Is it a subset of $\Bbb R^2$? I (pretty much) know what ...
0
votes
2answers
126 views
How could one geometrically visualize any given metric space $(X,d)$?
Example. Say $X=\mathbb{R}$ and $d(x,y)=\frac{d_0(x,y)}{1+d_0(x,y)}$ where $d_0(x,y)=|x-y|$ is the Euclidean metric.
The visualization of e.g. $\mathbb{R}$ with Eucledian distance $d_0(x,y)=|x-y|$ is ...
1
vote
0answers
44 views
How do you imagine linear transformations $\mathbb{F}_{p^n} \mapsto \mathbb{F}_{p^m}$
I am learning linear algebra (I know some introductory abstract algebra), and although I can imagine geometrically linear transformations from $\mathbb{R}, \mathbb{R}^2, \mathbb{R}^3$ to itself easily,...
