Questions tagged [visualization]
For questions about visualizing mathematical concepts. This includes questions about visualization of mathematical theorems and proofs without words.
934
questions
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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.
...
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53
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Assistance in recognizing & visualizing a specific topology
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 ...
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2
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69
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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 ...
1
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1
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63
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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 ...
0
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1
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115
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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 ...
0
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1
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55
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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 ...
0
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0
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34
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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 ...
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31
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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 "...
2
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45
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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-...
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1
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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 ...
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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 ...
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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 ...
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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}=\...
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1
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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 ) ...
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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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What is the most diagrammatic proof that $gf = 0 \implies \text{im } f \subset \ker g$?
The proof of the following fact is trivial to most Arrow theorists / Linear algebraists, but I'm developing software that needs to "understand" in a sense this basic fact, because it is ...
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87
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How to picture $x^n−y^n \equiv (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$?
Many students fail to intuit $x^n−y^n \equiv (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$ , as substantiated by the glut of duplicates. How can students pictorialize it?
After substituting $z = \...
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Big picture: Geodesics on a spherical surface embedded in $\Bbb R^3$
Consider the closed surface $M$ embedded in $\Bbb R^3$ with $g:=ds^2=dx^2+dy^2+dz^2$ and with $M:=\log^2 x+ \log^2 y +\log^2 z=1.$
Then restrict the metric to $M$. Here is a 3D plot of $M$ embedded in ...
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Suggestion for algebraic topology books that will increase my visualization
I am student of mathematics and I have started to learn algebraic topology . I want to learn visual things . Can you please suggest books that can help me to increase visualization ?
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Visualising trigonometry Identity
Can someone help me visualize this trigonometry Identity
Prove that :
$$1+\tan A\times\tan \frac{A}{2} = \sec A$$
I got the answer by manipulating the lhs substitution lots of mundane stuff but ...
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53
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Visual representation of Zeta Series in Real numbers
There are lots of visual representations of geometric series. Take for instance a $1$ by $1$ square and apply some action iteratively. Say that the final area is a piece of this square so you can ...
2
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1
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Representing a group with a 3D polyhedron?
I'm reading "From Spinors to Quantum Mechanics" by Gerrit Coddens. In Chapter 2.9 he shows diagrams representing two different groups as nodes and edges making up a polyhedron in 3D space.
...
3
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2
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240
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Why exactly does this 2 get larger and smaller? (superimposed rotating patterns forming a random moiré)
While searching for something very different, I stumbled upon these links by Emin Gabrielyan whom I assume is also the author of the arXiv preprint The basics of line moire patterns and optical ...
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45
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How to visualize an affine hyperplane in complex spaces?
I have seen this question and detailed answers about complex hypersurfaces here. However, I want to ask how to visualize an affine hypersurface.
For example, in $\mathbb{C}^2$, if $H$ is a complex ...
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82
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Visual representation of multivectors and forms
I read the beautiful (in my opinion) “Geometrical Vectors” by Gabriel Weinreich. In it, he offers a nice geometrical visualization of multivectors (elements of $\bigwedge_pV$, where $V$ is a finite ...
2
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78
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Tool for drawing and colouring shapes easily
I want a tool to easily make geometry diagrams. For example, say I wish to overlay two circles, and then fill their overlap with the color red, whilst coloring other areas with other colors? Let's say ...
3
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91
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Intuitive interpretation of entropy
I'm trying to understand entropy and KL divergence. While it makes sense in a simplistic case, such as the case of a coin flip, I struggle wrapping my head around it when it is a more complicated case ...
3
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how to improve my 3d visualisation skill
when I solve a volume problem in calc3 the hardest part is to visualise the solid in 3d but I recently noticed that after solving many problems I can visualise solids more clearly and I figured that ...
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Python Simulation: Values of Uniform Distribution overshoots the range
I am trying to plot the uniform distribution. It is supposed to be looking like the following.
Source:https://www.youtube.com/watch?v=UC-CBUSQXAo
However, when I try to plot it myself using the ...
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30
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latex question: comutative diagrams side by side
I would like to put the following two diagrams side be side:
{\Large$$
\begin{tikzcd}
X\arrow{r}{i}\arrow[swap]{dr}{j} & M \arrow{d}{j'}\\
& N
\end{tikzcd}
$$}
and
{\Large$$
\begin{tikzcd}
X\...
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Correct visualization of a probability density function of 3 continuous random variables
I am reading Rice's book of mathematical statistics, and I have understood that if we visualize a joint probability density function (pdf) of 2 continuous random variables, then in 3D space they ...
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Understanding projective change of coordinates.
An affine change of coordinates on $\mathbb A^n$ ,the affine $n$-space is defined to be a polynomial map $T:\mathbb A^n\to \mathbb A^n$ which is given by linear homogeneous polynomials and is also ...
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How to plot/visualize the inverse image of a set using computer software?
Question
Let $f: \mathbb C \rightarrow S^2$ be a holomorphic map and $E\subset S^2$ be a geodesic line segment in $S^2$. I would like to plot/visualize $f^{-1}(E)\subset \mathbb C$ on the plane. What ...
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How can I improve my comprehension of mathematical notation and avoid getting lost in the details, when there's no obvious visualization to rely on?
For context, I'm an engineering student nearing the end of my degree and I've been tutoring algebra, calculus, and statistics for years, so I'm not a newbie to math. It's only really been an issue in ...
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Plotting one standard deviation from a curve fit when one optimised value is negative?
I wasn't sure whether to ask this on a math site or SO since it's related to SciPy and Matplotlib. But I think my question is more mathematical than anything.
I fit a second-order polynomial ($a + b x ...
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104
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Drawings of high-dimensional hypercubes?
Do you guess what you see here?
It's the graph of the 8-dimensional hypercube which is drawn as 16 copies of the 4-dimensional hypercube (or tesseract).
All $4n$-dimensional hypercubes can be drawn ...
2
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1
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234
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Visualization of a Standard Two Dimensional Polyhedron
I am reading this book on linear programming, and the authors give an excellent exposition of the topic by the interplay between the underlying algebra and geometry. Their main approach is to motivate ...
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57
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visualization of triple integrals with iterated bounds in different coordinate systems
Lately i've been trying rather hard to write a code in cpp or python that not only calculates the result of a given triple integral, but also depicts the shape of the volume defined by the integral. ...
0
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Minmax principle/theorem and hyperbolic paraboloid
This question is completely out of curiosity:
While reading this page Min-max theorem and Hyperbolic paraboloid and thinking about the picture of hyperbolic paraboloid presented therein. I have a ...
2
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Resources that help build intuition about mathematics
I am looking for any books or resources that help build intuition at a fundamental level about mathematics.
I have a newfound interest in maths, after being away from it for years. Recently I started ...
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Geometric proof that muliplication in the complex numbers $\mathbb C$ is commutative
It is beyond well-known that multiplication in the complex plane has a geometric interpretation, namely we can interpret multiplication by $z\in \mathbb C$ as acting on the plane by fixing $0$, and ...
6
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2
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Geometric interpretation of integral $-\int\frac{1}{\sqrt{a+2bx-hx^{2}}}dx=\frac{1}{\sqrt{h}}\arccos\frac{b-hx}{\sqrt{b^{2}+ah}}$
The following formula is given as "the familiar arc-cosine form" by Joos, in his Theoretical Physics. The German language original has $e$ in place of $h$.
$$-\int\frac{1}{\sqrt{a+2bx-hx^{2}...
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Visualizing the Poincare homology sphere
I know that past a certain point, one should graduate from the view that homology/homotopy groups "count holes" in any realistic, grounded, real-life meaning of the word "hole". ...
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How to picture the number of $p$-permutations of $t$ things with $k$ kinds, where $n_1, n_2, n_3, \cdots , n_k$ = the number of each kind of thing?
Some students can't even grok this problem statement, as they are (informationally) overloaded by the number of variables : $p, t, k, n_1, \cdots, n_k \;$. Kindly improve my picture, or draw a better ...
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Affine change of coordinates.
I am reading Fulton's book on algebraic curves.In the second chapter they have defined what they call affine coordinate change map between two affine spaces.It is defined in the following manner:
Let ...
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0
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72
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Visualising an inconsistent $3\times 3$ linear system
I am working with the linear system
$$
\left\{ \begin{aligned}
x_1 + x_2 + x_3 &= 1 \\
2x_1 + x_2 + x_3 &= 3 \\
3x_1 + x_2 + x_3 &= 2
\end{aligned} \right.
$$
(Taken from David Towers:...
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0
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35
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Can $S^3$ be distinguished from spaces like $S^2\times S^1$ or $S^1\times S^1\times S^1$ using only the number of holes and connectedness
I am trying to have a mental picture of topological spaces like $S^3, T^3$ etc. I know about fact that the topological invariants are the essential tools to do this. But the topological invariants ...
0
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0
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11
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Least Squares 2D plot of Multidimentsional Set of Points from Table of Pairwise Distances
I have a set of thirteen points in a many-dimensional space and have a table of distances between each pair of points. I want to plot the points on the plane in a way that best preserves the pairwise ...
4
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1
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123
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Is this a thing? Visualizing complex functions with 3D animation
I have read about various ways to visual complex functions, e.g. colored graphs, vector fields, conformal maps, etc.
Here is another way: 3D animation. Picture a 3D plot with three axes for $\text{Re}(...
0
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1
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47
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Is there a way to create a 3-circle Ven diagram where the area of each intersection is a preset value?
I'm trying to create a program that takes NBA lineup data, and creates a ven diagram to show the net rating for each combination of a certain number of players (either 2 or 3). For example, with two ...