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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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32 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 ...
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1answer
36 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 ...
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0answers
18 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
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1answer
60 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 ...
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0answers
31 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 ...
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45 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 ...
3
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4answers
212 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
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1answer
40 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 ...
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38 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 ...
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1answer
53 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 ...
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0answers
109 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 ...
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54 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 ...
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2answers
43 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 $K$ ...
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2answers
117 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}^{...
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1answer
35 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, ...
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1answer
45 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^...
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1answer
24 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{ ...
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0answers
46 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 ...
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1answer
62 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....
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2answers
47 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 ...
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0answers
38 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 ...
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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 \...
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3answers
81 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 ...
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1answer
142 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 ...
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3answers
51 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 ...
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2answers
121 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 ...
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0answers
43 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,...
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0answers
20 views

Deform surface along some flow

I am designing a software in which the user can cut a surface (such as a sphere or a torus) along some (closed) curve. I would then like the surface to 'unfold' in some way, for example cutting a ...
2
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1answer
28 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 ...
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4answers
135 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 ...
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4answers
525 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 ...
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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 ...
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4answers
109 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 ...
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3answers
55 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}...
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0answers
35 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 ...
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17 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\...
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1answer
83 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\...
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1answer
98 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 ...
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0answers
40 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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1answer
33 views

Visualization of open balls for different metric spaces

I've got a lot of problems imagining how open balls look like in metric spaces. This prevents me getting better insight in some proofs and exercises. An example is the $d_1$-metric defined as ...
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0answers
35 views

Venn diagram with ordered factors

I have three variables, each is categorical with three ordered levels. I'm trying to figure out if there is a way to represent the intersection of all possible combinations of the variables in 2D, ...
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0answers
103 views

Distributions of prime numbers

When folding the number line not into a spiral (like Ulam did) but into a zig-zag pattern (like Cantor did) there are other patterns visible in the distribution of prime numbers: [To see these ...
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1answer
84 views

Why do we evaluate $-x^{z-1}e^{-x}$ as zero when explaining the gamma function through integration by parts?

The gamma function is the integral of $x^{z-1}e^{-x}$ If you integrate by parts you get two terms. The first one is $-x^{z-1}e^{-x}$ and this is bound by infinity and zero. If you plug in infinity, ...
3
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1answer
98 views

Distribution of triangular, square, and pentagonal numbers

I'm doing some research in visualizing arithmetic sets (resp. properties, resp. sequences of integers). I try to create patterns (in which I hope to observe some symmetries) by injectively mapping $\...
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1answer
57 views

How can one best visualize two dimensional manifolds in $\mathbb{R^4}$ (more specifically, $\mathbb{S}^2 \times \mathbb{R})$?

I'm trying to "get a picture", so to speak, of hypersurfaces in $\mathbb{S}^2 \times \mathbb{R}$. One example would be $\left(\dfrac{\cos(u)}{\sqrt{1+u^2}}, \dfrac{\sin(u)}{\sqrt{1+u^2}},\dfrac{u}{\...
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0answers
66 views

Geometrical visualization of Tensors

My question is about tensors. I have recently spent some time studying the various definitions of tensors and some tensor calculus. What I am missing now is an intuitive way to represent tensors and I ...
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0answers
40 views

Projecting 6D cartesian coordinates to lower dimension

I've got a noisy set of points that lie on/near a unit 6-sphere. I want to visualise their proximity to the sphere in some way. The only way I can come up with now is a histogram of the norm of the ...
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1answer
43 views

Would this suffice in a visual type theory to define an abstract List type?

See the image. I got that from: wikipedia article. In that, I don't understand the first function nil : () -> L. What is ()...
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27 views

Efficient visualisation of arrangements

First, I don't know the word for the set. I have n cases and the set I have is 2^n which lists all their "arrangements" present or absent, so a summation of combinations with k from 0 to n. Each ...
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0answers
61 views

Geometric intuition of a point in a flag manifold

From Wikipedia According to basic results of linear algebra, any two complete flags in an $n$-dimensional vector space $V$ over a field $F$ are no different from each other from a geometric point ...