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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21
votes
2answers
423 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
104 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
95 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/...
38
votes
1answer
1k 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
42 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
213 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
46 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
60 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
175 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
votes
2answers
268 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
69 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
119 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
546 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
128 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 ...
4
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4answers
156 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
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1answer
47 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 ...
6
votes
1answer
110 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
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0answers
46 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
109 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 graph $n/m$ ...
4
votes
4answers
306 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
143 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 ...
2
votes
1answer
90 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
214 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
58 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
182 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
183 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
104 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
50 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
43 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
54 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
112 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
65 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
105 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
76 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
98 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 ...
10
votes
1answer
676 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
74 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 ...
-1
votes
2answers
279 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
50 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,...
2
votes
1answer
89 views

How to describe curvilinear grid using coordinate functions?

A curvilinear grid around a cylinder has the following properties: The grid has $n_\varphi =20$ grid points in angular direction (along a circle in the xy-plane). The grid has $n_r =5$ grid points ...
3
votes
4answers
144 views

Visualizations of the (potential) irrationality of $\sqrt{2}$

The following statement is equivalent to Euclid's statement that $\sqrt{2}$ is irrational but has a rather different flavour. Consider the straight line through two points $0$ and $1$ with the ...
3
votes
4answers
810 views

how to draw the space of such linear combinations?

We have the linear combination $$ {2 \choose 1 } x_1 + {1 \choose 2} x_2 + {1 \choose -2} x_3 + {1 \choose 1} x_4 + {-1 \choose 0 } x_5 + {0 \choose -1 }x_6 $$ As $x_i \geq 0 $ is given, according ...
1
vote
3answers
40 views

Why do the (nonzero) vectors $x,y, x-y$ form a triangle? (can assume $\mathbb{R}^2$)

Let $x,y$ be any two nonzero vectors in $\mathbb{R}^2$ that are not scalar multiplies of eachother (i.e. are not linearly dependent), and $x-y$ be their difference. I am wondering why these three ...
2
votes
4answers
550 views

Does the definition of the angle between two vectors require that they have the same “origin”?

I am thinking specifically about $\mathbb{R}^2$ so I can visualize things. By "origin" I mean that they start at the same point. When we graphically represnt vectors we don't care where the starting ...
1
vote
3answers
58 views

Is there a nice visualization of the length of a curve formula?

We know that if there is a curve $$\Gamma=\{(x,y)\in\Bbb R^2\ :\ y=f(x), x\in[a,b]\}$$ then $$\text{length}(\Gamma)=\int\limits_a^b\sqrt{1+f'(t)^2}dt$$ and I get that this is because $$\text{length}...
1
vote
0answers
81 views

Intuition and visualization of area preserving maps?

I was trying to understand what is meant by "area preserving map"?. I was going through the Wolfram article about the area preserving map here but any motivation, intuition or visualization to ...
0
votes
0answers
32 views

Question about projective line and intersection

The book I'm reading through stated the following: Let $\mathbb P^1$ be the complex projective line, pick a random line that doesn't pass through the origin in $\mathbb C^2$. Then any point $p\...
7
votes
1answer
96 views

Visualizing and understanding the roots of $f(z) = z^2 - e^{i\varphi}$

[I've added another animated picture below, showing how the actions of the groups $Q_\alpha$, $R_\beta$ coincide every now and then.] The roots of $f(z) = z^2 - e^{i\varphi}$ are simply $\pm e^{i\...
2
votes
2answers
201 views

Visualizing complex functions $f: \mathbb{C} \rightarrow \mathbb{C}$

The graph of a complex function $f: \mathbb{C} \rightarrow \mathbb{C}$ is a 3-dimensional object in a 4-dimensional space and thus hard to visualize, even when it's a smooth 3-dimensional surface. A ...
0
votes
0answers
41 views

Dynamical systems with large number of attractors and their dependence on the parameters?

It is much important to study the attractors in a dynamical system as these indicate how the system behaves once the initial transients are discarded. Also, the study of systems with many numbers of ...

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