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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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Visualization of skew-symmetric rank-2 tensor fields

Background I was reading Einstein's The Meaning of Relativity in which he points out that axial vectors are usually used in place of rank $2$ tensors for the sake of geometrical picturisation, as in ...
Awe Kumar Jha's user avatar
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1 answer
40 views

Geogebra specify length with brace

Is it possible to specify segment length with a brace or arrows, like below? Credit: https://mrchasemath.com/2018/01/17/geometric-proofs-of-trigonometric-identities/ Credit: https://senecalearning....
qwr's user avatar
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How to visualize the compound trigonometric function like sin(tan(x)) and find this limit?

This question comes out from the book Visual Differential geometry and Forms by Needham, the 4th exercise for Prologue and Act I, where we need to find the limit of $$ \lim_{x\rightarrow 0} \dfrac{\...
FireWOLF's user avatar
1 vote
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32 views

Degree of the neigbour verticies of a vertex degree 5 in a planar graph - visualization

I'm writing a lecture on 5-coloring planar graphs and I'm having trouble visualizing this inequality form the proof "$2n_5 \leq \sum_{d \geq 12} dn_d$" I want to make a simple drawing of ...
Intruder.guru's user avatar
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How to Draw the Function $f(x,y) = x^2 + y^2 - 1$ with Level Curves?

I'm currently working on a problem where I need to draw the function $f(x,y) = x^2 + y^2 - 1$. I've managed to figure out that the level curves of this function are circles with different radii. ...
cricket900's user avatar
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31 views

Visualizing Matrix Spaces

When thinking about vector spaces a common example is vectors in $\mathbb{R}^n$. It is also very common to visualize this space as a set of as arrows in some n-dimensional space, with the typical ...
GUT's user avatar
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How to morph enclosed mesh according to the enclosing mesh?

I am quite new to meshing and mesh manipulation. I am working on a problem consiting of meshes $A$, $B$, and $C$. The mesh $C$ completely encloses the meshes $A$ and $B$ as shown in the attached ...
Prakhar's user avatar
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How to analyze a vector with its entries following independent distributions?

Suppose let's say we have a vector $Y$ which is distributed as $Poisson(X)$, where $X$ is an $N \times 1$ vector. Does it mean that the individual elements in $Y$ are Poisson distributed with the mean ...
Aravind Muraleedharan's user avatar
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Why adding functional derivative of entropy doesn't get an uniform distribution?

From https://mbernste.github.io/posts/functionals/, the functional derivative of entropy is $−1−logP_x(x)$. From my understanding, adding this derivative to a distribution, it will get an uniform ...
Mark's user avatar
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Visualising the interior gluings of a 3D shape in 2D

I have a small triangulation of a 3-ball that I'm trying to form a nice 'visualisation' of for a paper/talk. The best I've got so far is a few rough sketches like the one below, where I've tried to ...
Finn T's user avatar
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11 votes
1 answer
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Beautiful errors in graph of $\sin(x^2+y^2)$

I was writing a simple program to help visualize inequalities based on 2 variables. The test inequality that I was using was this: $$\sin\left(0.1(x^2+y^2)\right)\geq0$$ Regions that satisfy the ...
Soham Saha's user avatar
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1 answer
70 views

Understanding and Visualizing Complex Roots According to the Fundamental Theorem of Algebra

I'm trying to deepen my understanding of the Fundamental Theorem of Algebra, which asserts that every non-constant single-variable polynomial with complex coefficients has as many roots as its degree, ...
user97662's user avatar
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2 votes
1 answer
106 views

What shape is an irregular cylinder flattened out?

A cylinder flattened out is a rectangle (ignoring the top and bottom faces). But what about a cylinder where the top face is smaller than the bottom face... ? What shape is that flattened out? (again ...
user3807846's user avatar
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1 answer
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Visualizing the Commutative Property Beyond Three Numbers

I've been pondering how to intuitively visualize the commutative property of multiplication when dealing with more than three numbers. For two numbers, we can easily conceptualize this with the area ...
Dimidri Sanchez's user avatar
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2 answers
123 views

Is there a visual proof for why this property of a matrix is true?

Let's say we have three equations written in their standard form: \begin{align} a_1x + b_1y + c_1 = 0 && (l_1) \\ a_2x + b_2y + c_2 = 0 && (l_2) \\ a_3x + b_3y + c_3 = 0 && (...
Fullk33's user avatar
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Visual proof of $\sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) = 1 $

Background Let $\zeta(\cdot)$ be the Riemann zeta function. I'm looking for a visual proof of the infinite series identity $$\sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) = 1. \tag{1}\label{1}$$ This ...
Max Muller's user avatar
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2 votes
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Reference Request: Visual Approach to Symmetric Groups

The symmetric group is a factor of the braid group (see e.g. Surjective Group Homomorphism From Braid Group Into Symmetric Group, Symmetric group, Braid Groups, and related groups). Consequently one ...
Alp Uzman's user avatar
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Can you have a graph/plot with more than 3 sides?

It may be a stupid question but I was working on a ternary plot (example image attached) and the idea about the maximum number of sides for a graph (in 2D, 3D and ND) came up. I'm curious as to the ...
Ahmed Tayee's user avatar
2 votes
2 answers
108 views

How to train myself to not think visually for simple math problems?

I have a problem where it is often very difficult for me to solve simple math problems without thinking about them visually, but at the same time, I have poor spatial reasoning and so I'm unable to ...
asdfasdf's user avatar
-1 votes
1 answer
67 views

Visualisation Tool for Coordinate Systems [closed]

I'd like to use a tool or a program for the visualisation of different coordinate systems such as: Spherical coordinates Cartesian coordinates Local tangent plane coordinates I am not familiar with ...
Sylvia's user avatar
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3 votes
0 answers
62 views

Problems with an interesting visual solution? [closed]

I am looking for questions which can be simplified and solved with a visual/geometric proof. For example, solving Buffon's needle problem using a circular needle. Preferably understandable for 16-18 ...
cat's user avatar
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Why the angle between Two Vectors in a plane or in a space is not measured from head to tail?

Let us assume two vector a and b such that they lie on the same plane. Then if we want to measure the angle between a and b, For now, let us assume it to be θ Then why is the angle measured from the ...
3b1b aimer's user avatar
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1 answer
48 views

Developing visual intuition for proofs involving cartesian product and sets

I am beginning to learn set theory proofs. It has been extremely useful to draw Venn diagrams for proofs just involving union, intersection, complement, e.t.c. However with cross product involved, how ...
sharkleberryfin's user avatar
0 votes
3 answers
156 views

Stumped on $\int_{-1}^1\int_{-\sqrt{1-x^2}}^\sqrt{1-x^2} \int_{x^2+y^2}^{2-x^2-y^2} dz\;dy\;dx$

I cannot express the following integral in spherical coordinates. It as though I am finding the volume between the solids. Any help will be appreciated. $$\int_{-1}^1\int_{-\sqrt{1-x^2}}^\sqrt{1-x^2}...
Jason Broadway's user avatar
1 vote
1 answer
116 views

How to plot spherical harmonics? [closed]

Let me start by saying that I am only interested in the mathematical aspect of the thing. I would like to plot just for the fun of it the spherical harmonics that are used to plot the electronic ...
Charlie's user avatar
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0 answers
27 views

How to construct a harmonic mapping

How to construct a harmonic mapping on the complex plane using the method of "Shear Construction"?
Aleh Kryshchuk's user avatar
1 vote
1 answer
47 views

If the direct product of two semilattices exists, what does its Hasse diagram look like in terms of its constituent semilattice Hasse diagrams?

This is likely to be a quick question. Definition: A semilattice $(L,\lor)$ is a commutative, idempotent semigroup. The Hasse diagram $H$ of $L$ is with respect to the order $x\le y$ iff $x\lor y=y$....
Shaun's user avatar
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2 votes
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62 views

How to assign a color to each point on the real projective plane?

I have a function $f:\mathbb{R}^3\to \mathbb{R}P^2$ and I'd like to visualize it as an assignment of a color to each point in space. But I'm having trouble coming up with a good mapping from $\mathbb{...
user7530's user avatar
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0 votes
1 answer
97 views

Visual explanation for the shape of $e^{2\pi i z}$

To be clear from the beginning, I am looking to develop my visual intuition for complex functions. I can explain algebraically why the plot below for $q(z) = e^{2\pi i z}$ looks the way it does, but I ...
stillconfused's user avatar
4 votes
4 answers
502 views

Dodecahedral number visualization

The dodecahedral numbers, 0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, ... numbers of the form ${3 n \choose 3}$ (A006566). Does anyone have a good visualization of these? In particular, I'd ...
Ed Pegg's user avatar
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1 vote
0 answers
76 views

Examples of a classifying space using the bar resolution

While studying group (co)-homology, I've learned about how to construct a chain complex such that its cell chain complex coïnide with the bar resolution of a certain group $G$. For that we first ...
bml64's user avatar
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0 votes
1 answer
123 views

The parabola paradox - how can this be thought of or visualized

One thing that has been hard to wrap my mind around- take a power function such as $x^2$ or $x^8$. I know the domain of x is infinite, unbounded. On the one hand such power functions increase in slope ...
gcr's user avatar
  • 127
3 votes
0 answers
52 views

Real elliptic curve embedded into complex torus

Main goal: Visualize the "real slice" of elliptic curve from the complex torus perspective Basic setup: for $\tau \in \mathbb H$ (the upper half plane $\{z\in \mathbb C: \Im(z)>0\}$), one ...
D.R.'s user avatar
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1 vote
1 answer
81 views

From eigenvectors, eigenvalues and the determinant of a matrix to the visual transformation

I found the image below illustrating the transformations by several matrices with given eigenvectors, eigenvalues and determinants. For matrix in the second row ($\lambda_1=\lambda_2=1$ and $det(A)=1$)...
Tran Khanh's user avatar
1 vote
0 answers
199 views

Finding points on any function at an equal distance

I'm developing an app to visualize mathematical functions in a coordinate system. I'm calculating $y$ for every pixel along the $x$ axis. This method works well with basic functions, like $f(x) = x$ ...
TLeo's user avatar
  • 19
1 vote
1 answer
40 views

5-cell "4D tetrahedron"

I do not know if this is the right place to ask this. But if I have a 3D object (tetrahedron) I can make a projection of it into 2D space where I use colour as the third dimension to determine height. ...
Loksorr's user avatar
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1 vote
1 answer
58 views

Assistance in recognizing & visualizing a specific topology [closed]

I’m a bit out of my element asking math questions here, but hopeful someone can point me in the right direction. Thanks in advance! Questions: Do the two tables illustrated below each form a shape in ...
Nate S's user avatar
  • 13
0 votes
2 answers
105 views

Is there a visualization tool that shows matrices transforming (multiplying) ellipses?

Matrices send ellipses to ellipses. Is there any online visualization tool where I can enter a matrix $M$ and see a plot of ellipse $e$ being sent to $Me$? This would be very helpful to visualize ...
SRobertJames's user avatar
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1 vote
1 answer
75 views

Drawing isosceles right triangle on unit circle for intuitive meaning of √2/2

Have you wondered about the meaning of $\sqrt{2}/2$ on the unit circle? Drawing a unit length line at $45^\circ$ from the origin, reflecting it on the $x$ axis and connecting the points where the ...
Attila Vajda's user avatar
0 votes
1 answer
127 views

What is going on with this Taylor Series?

I created a program to find high degree Taylor series approximations for any function, but I notice an interesting behavior as higher order polynomials are included. My understanding is that a ...
Colter's user avatar
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0 votes
1 answer
65 views

Visualizing the last two paragraphs of 3.2.3(b) solution.

Here is the question I am trying to understand its solution: Finish the proof of Borsuk-Ulam theorem (Hatcher) I did not understand this part of the solution: Let $\gamma:[0,1]\to \Bbb RP^n$ be a ...
user avatar
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0 answers
42 views

Is this visualization of arc connectedness of SO(3) wrong?

The article "Anti-twister mechanism" at Wikipedia has an animation showing Dirac's theorem. The rotation axis of the cube is vertical, so it's the distinguished direction. The four belts ...
ByteEater's user avatar
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1 vote
0 answers
43 views

Understanding a figure showing the wedge product

On the wikipedia article explaining exterior algebra, this figure is shown, visualizing the relationship between the wedge product of 2 vectors (or differential forms for that matter), and the "...
Noiv's user avatar
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2 votes
0 answers
64 views

Immersion of $\mathbb RP^2$: explanation of Kirby's article on the Boy's surface

I am trying to understand Rob Kirby's AMS notice https://www.ams.org/notices/200710/tx071001306p.pdf on Boy's surface. From this blogpost https://divisbyzero.com/2020/04/08/make-a-real-projective-...
D.R.'s user avatar
  • 8,885
0 votes
1 answer
40 views

Intuition of Homotopy in Non-Euclidean Spaces

Please note that I am a beginner in Algebraic Topology. I am struggling with the intuition of homotopy equivalence in non-Euclidean spaces. To be more specific, we know that any two loops with the ...
Carson Newman's user avatar
0 votes
1 answer
28 views

Visualisation of the cubic curve $C: y^2 z − x^3 − x^2z − xz^2 − z^3$

To get some insight on the zero locus of the cubic curve, I've tried a couple of online visualisers on Google and mostly failed to generate the plot (run time errors...) except Wolfram Alpha which ...
Rowing0914's user avatar
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0 answers
54 views

Determine shape of marginal gaussian given 2D gaussian contour lines

Given a 2D multivariate Gaussian contour lines and a point $x_0$, is it possible to draw the marginal distribution $f(y)=\mathbb{P}((X,Y)=(x,y)|x=x_0)$. From the picture, it seems the mean of the ...
edamondo's user avatar
  • 1,397
7 votes
0 answers
143 views

Visual or intuitive proof of $\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}=\frac{\pi}{4}$

It is well known that the alternate sum of the reciprocals of the odd numbers adds up to $\frac{\pi}{4}$. That is $$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots = \sum_{k=0}^\infty\frac{(-1)^k}{2k+1}=\...
A. Bellmunt's user avatar
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2 votes
1 answer
90 views

How can I improve my picture proof of Reverse Triangle Inequality?

Diagram beneath reappears on standardized tests IN BLACK AND WHITE with different lengths, letters, and orientation that require students to label in terms of $\vec{b}, \vec{r}$ ( = circle's radius ) ...
user avatar
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0 answers
172 views

While pictorializing $|x - y| < |x + y|$, how can solely 1 picture simultaneously prove (Reverse) △ Inequalities, $|x-y| ≤ |x|+|y|, |x|-|y| ≤ |x-y|$?

On p. 12, Michael Spivak's Calculus (2008 4 edn) proved $|x + y| ≤ \color{darkgoldenrod}{|x| + |y|}$ (Triangle Inequality). Ibid, exercise 12, p. 16. (iv) ${\color{red}{|x-y|}} ≤ \color{goldenrod}{|x|...
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