Questions tagged [visualization]
For questions about visualizing mathematical concepts. This includes questions about visualization of mathematical theorems and proofs without words.
-1
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0answers
44 views
Does the “Group Explorer” software permit visualizing products of groups? [on hold]
Using the software called "Group Explorer" (SourceForge link), can one compute direct products or semidirect products of groups already in the software's group library, thus producing new groups, and ...
0
votes
0answers
30 views
In visualizing data, how to show multiple trajectories from simulations on a single plot? [closed]
I'm running some simulations, using a standard Matlab ODE solver, and the trajectories of ellipses evolve in an interesting way, when varying a control parameter. How could I plot these trajectories ...
0
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1answer
52 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
42 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
32 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
21 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,
$$\mathbf a \times (\mathbf b + \mathbf c) = \mathbf a \...
8
votes
3answers
78 views
Is there a visualization for inverse trig functions as indefinite integrals
Examining the indefinite integral formulations of inverse trig functions I notice some things
$$\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
116 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
48 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
108 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
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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
votes
1answer
22 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
132 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
515 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 $$
As $x_i \geq 0 $ is given, according ...
1
vote
3answers
39 views
Why do the (nonzero) vectors $x,y, x-y$ form a triangle? (can assume $\mathbb{R}^2$)
Let $x,y$ be any two nonzero vectors in $\mathbb{R}^2$ that are not scalar multiplies of eachother (i.e. are not linearly dependent), and $x-y$ be their difference.
I am wondering why these three ...
2
votes
4answers
68 views
Does the definition of the angle between two vectors require that they have the same “origin”?
I am thinking specifically about $\mathbb{R}^2$ so I can visualize things.
By "origin" I mean that they start at the same point.
When we graphically represnt vectors we don't care where the starting ...
1
vote
3answers
54 views
Is there a nice visualization of the length of a curve formula?
We know that if there is a curve $$\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
31 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
12 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
81 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
89 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
37 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
29 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
33 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
97 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
81 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
92 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
55 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
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0answers
59 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
40 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
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0answers
26 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
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0answers
51 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 ...
0
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0answers
47 views
Periodic functions wrapped into polygons: is this a transformation, a convolution or a projection? (or none of them)
I have been some days thinking about a visual manipulation of periodic functions wrapping them over polygons, and I am not sure if it could be considered a transformation, a convolution or a ...
0
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0answers
26 views
How can I use kernel density estimation for heat map visualization?
I want to know more about visualization and density estimations.
Basically, I have a large sample of location data and every location object has a duration in milliseconds.
I want to solve the ...
0
votes
1answer
115 views
Derivative of Square Root Visual
https://www.youtube.com/watch?v=S0_qX4VJhMQ
There is a challenge at 12:23 asking the viewer to arrive at the formula for $\frac{d}{dx}\sqrt{x}$ by considering small changes in the length and area of ...
0
votes
1answer
41 views
Lagrange mean value theorem for two variables - visualization and intuition behind it
The two-variable version of the Lagrange mean-value theorem says that given a function $f(x,y)$,
$$f(\vec{p_o} + \vec h)-f(\vec {p_o})=df(\vec {p_{\theta}})$$
Where $\vec p_{\theta}=\vec p_o + \theta ...
1
vote
0answers
35 views
Visualization of particular integration using trigonometric function
I know that when you're trying to integrate something, and by chance, appears somewhere something of the form :
$$ \sqrt {1 - x^2} ; \sqrt {1 + x^2}$$
or something similar, it is known to replace x ...
1
vote
1answer
46 views
How to interpret a Box-Percentile Plot?
How to interpret a box-percentile plot and find the outliers? I have been trying to find an example online but so far not been successful. Here is an example diagram:
1
vote
0answers
89 views
Understanding the definition of Sensitive dependence on initial conditions?
I was trying to understand the rigorous definition of sensitive dependence on initial conditions which is as follows -
$f : X \mapsto X$ where $X$ is a metric space.
If there exists $\epsilon > ...
0
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1answer
80 views
Row versus column picture
So, we have a system of linear equations.It's preferred that we visualize the column picture, since as dimensions go up we only have to think of vectors moving to more dimensions, instead of ...
0
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0answers
26 views
Tools for Visualizing Derivatives As Density
The Desmos application can be used to visualize and experiment with the standard, graphical presentation of derivatives (https://www.desmos.com/calculator/4pf1dxxzq2).
This alternative, "density of ...
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votes
1answer
40 views
Visualizing $c-d<a-b \implies b<a+d-c$?
I am wondering if someone can provide some geometric intuition, or some simple way to visualize why
$$
c-d<a-b \implies b<a+d-c
$$
The way I have been trying to do this is to think of $a,b,c,d$ ...
0
votes
1answer
21 views
How to combine multiple indices?
I'm trying to combine various risk indices e.g. flooding, fire, burglary and structural damage. These values are scored from 1-3. For instance, I could have measures for two different areas as ...
0
votes
3answers
102 views
Visual proof of isosceles base-angle congruency?
A geometric proof (without algebra or trigonometry), and ideally presented visually (a proof without words). EDITED
(I'm especially curious if it's possible without without using triangle congruency.)...
0
votes
1answer
32 views
Plotting lines, points, planes, triangles for technical documentation
I'm looking for a (free) tool to draw or plot graphics as the one linked below:
Maybe a CAD solutions could be the easiest one. But which do you recommend for this certain case? I don't want to deal ...
1
vote
1answer
61 views
Software package for plotting 3-d splines
Given a finite point set $P \subset \mathbb{R}^2$ and a height function $h:P -> \mathbb{R}$ I want to produce a smooth surface that interpolates between the values $\left\{[p~h(p)]^\top \in \mathbb{...
1
vote
2answers
34 views
Visualising one dimension in real/physical world.
How can we visualize one-dimension in real/Physical world? Does any body have an example? Often people refer to one-dimension as motion being in a straight line. However motion in a straight line can ...
0
votes
1answer
94 views
A more direct way to see that the angle inscribed in a semicircle is $90^\circ$?
Like this inscribed angle proof, another proof enabled by this clever angle sum proof.
Is there a simpler way to show this?
Is this proof original?
