Questions tagged [visualization]
For questions about visualizing mathematical concepts. This includes questions about visualization of mathematical theorems and proofs without words.
2
votes
3answers
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 ...
0
votes
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 ...
0
votes
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
votes
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 ...
3
votes
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 ...
6
votes
0answers
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
votes
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
votes
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 ...
0
votes
0answers
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 ...
2
votes
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 ...
6
votes
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 ...
5
votes
0answers
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 ...
1
vote
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 $K$ ...
2
votes
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}^{...
1
vote
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, ...
0
votes
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^...
0
votes
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$$z = a + bi,\text{ ...
1
vote
0answers
46 views
What are some visually appealing derivatives?
The derivative of a power tower made up of $e$ repeated $6$ times with an $x$ at the top, is, by the chain rule
$$\frac{d}{dx} \left(\displaystyle e^{\displaystyle e^{\displaystyle e^{\displaystyle ...
0
votes
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....
1
vote
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 ...
3
votes
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 ...
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,
$$\mathbf a \times (\mathbf b + \mathbf c) = \mathbf a \...
8
votes
3answers
81 views
Is there a visualization for inverse trig functions as indefinite integrals
Examining the indefinite integral formulations of inverse trig functions I notice some things
$$\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 ...
8
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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,...
0
votes
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
votes
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 ...
3
votes
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 ...
3
votes
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 $$
As $x_i \geq 0 $ is given, according ...
1
vote
3answers
40 views
Why do the (nonzero) vectors $x,y, x-y$ form a triangle? (can assume $\mathbb{R}^2$)
Let $x,y$ be any two nonzero vectors in $\mathbb{R}^2$ that are not scalar multiplies of eachother (i.e. are not linearly dependent), and $x-y$ be their difference.
I am wondering why these three ...
2
votes
4answers
109 views
Does the definition of the angle between two vectors require that they have the same “origin”?
I am thinking specifically about $\mathbb{R}^2$ so I can visualize things.
By "origin" I mean that they start at the same point.
When we graphically represnt vectors we don't care where the starting ...
1
vote
3answers
55 views
Is there a nice visualization of the length of a curve formula?
We know that if there is a curve $$\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}...
1
vote
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 ...
0
votes
0answers
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\...
7
votes
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\...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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, ...
5
votes
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 ...
2
votes
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
votes
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 $\...
1
vote
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}{\...
1
vote
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 ...
2
votes
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 ...
0
votes
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 ()...
0
votes
0answers
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 ...
0
votes
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 ...
