Questions tagged [visualization]
For questions about visualizing mathematical concepts. This includes questions about visualization of mathematical theorems and proofs without words.
934
questions
1
vote
1
answer
23
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.
...
- 11
1
vote
1
answer
53
views
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 ...
- 13
0
votes
2
answers
69
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 ...
- 2,790
1
vote
1
answer
63
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 ...
- 11
0
votes
1
answer
115
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 ...
- 1
0
votes
1
answer
55
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 ...
- 893
0
votes
0
answers
34
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 ...
- 297
1
vote
0
answers
31
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 "...
- 31
2
votes
0
answers
45
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-...
- 8,363
0
votes
1
answer
38
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 ...
0
votes
1
answer
26
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 ...
- 576
0
votes
0
answers
14
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 ...
- 1,187
6
votes
0
answers
121
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}=\...
- 1,572
2
votes
1
answer
81
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 ) ...
user1211588
0
votes
0
answers
169
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|...
user1211588
6
votes
2
answers
170
views
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 ...
- 21k
1
vote
1
answer
87
views
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 = \...
user1211588
2
votes
0
answers
45
views
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 ...
- 892
3
votes
1
answer
100
views
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 ?
- 99
5
votes
1
answer
97
views
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 ...
- 85
1
vote
0
answers
53
views
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 ...
- 1,009
2
votes
1
answer
56
views
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.
...
- 1,200
3
votes
2
answers
240
views
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 ...
- 1,760
1
vote
1
answer
45
views
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 ...
- 1,353
0
votes
1
answer
82
views
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 ...
- 175
2
votes
0
answers
78
views
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 ...
- 1,057
3
votes
1
answer
91
views
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 ...
- 143
3
votes
1
answer
372
views
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 ...
- 2,174
0
votes
0
answers
30
views
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 ...
- 141
0
votes
0
answers
30
views
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\...
- 742
1
vote
0
answers
103
views
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 ...
- 197
0
votes
1
answer
95
views
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 ...
- 3,324
0
votes
0
answers
31
views
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 ...
0
votes
2
answers
184
views
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 ...
0
votes
0
answers
55
views
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 ...
- 179
0
votes
0
answers
104
views
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 ...
- 17.9k
2
votes
1
answer
234
views
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 ...
- 14.7k
0
votes
0
answers
57
views
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
votes
0
answers
41
views
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 ...
- 29
2
votes
0
answers
74
views
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 ...
- 161
1
vote
0
answers
85
views
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 ...
- 8,363
6
votes
2
answers
236
views
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 $h$.
$$-\int\frac{1}{\sqrt{a+2bx-hx^{2}...
- 3,270
8
votes
0
answers
207
views
Visualizing the Poincare homology sphere
I know that past a certain point, one should graduate from the view that homology/homotopy groups "count holes" in any realistic, grounded, real-life meaning of the word "hole". ...
- 8,363
-3
votes
1
answer
76
views
How to picture the number of $p$-permutations of $t$ things with $k$ kinds, where $n_1, n_2, n_3, \cdots , n_k$ = the number of each kind of thing?
Some students can't even grok this problem statement, as they are (informationally) overloaded by the number of variables : $p, t, k, n_1, \cdots, n_k \;$. Kindly improve my picture, or draw a better ...
- 85
-1
votes
1
answer
264
views
Affine change of coordinates.
I am reading Fulton's book on algebraic curves.In the second chapter they have defined what they call affine coordinate change map between two affine spaces.It is defined in the following manner:
Let ...
- 3,324
1
vote
0
answers
72
views
Visualising an inconsistent $3\times 3$ linear system
I am working with the linear system
$$
\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:...
- 593
0
votes
0
answers
35
views
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 ...
- 305
0
votes
0
answers
11
views
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 ...
- 101
4
votes
1
answer
123
views
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}(...
- 16.2k
0
votes
1
answer
47
views
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 ...
