The visualization tag has no wiki summary.
34
votes
4answers
840 views
Algebra: Best mental images
I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on ...
13
votes
3answers
884 views
Cutting a Möbius strip down the middle
Why does the result of cutting a Möbius strip down the middle lengthwise have two full twists in it? I can account for one full twist--the identification of the top left corner with the bottom right ...
12
votes
2answers
250 views
What can be gleaned from looking at a domain-colored graph of a complex function?
Functions from $\mathbb{C} \rightarrow \mathbb{C}$ are hard to visualize because of their 4-dimensional nature. One nice way of looking at them is by what's called domain coloring. An example from the ...
11
votes
3answers
716 views
Interesting implicit surfaces in $\mathbb{R}^3$
I have just written a small program in C++ and OpenGl to plot implicit surfaces in $\mathbb{R}^3$ for a Graphical Computing class and now I'm in need of more interesting surfaces to implement!
Some ...
11
votes
7answers
295 views
$\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$? [duplicate]
If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed ...
10
votes
2answers
262 views
What are all these “visualizations” of the 3-sphere?
a 2-sphere is a normal sphere. A 3-sphere is
$$
x^2 + y^2 + z^2 + w^2 = 1
$$
My first question is, why isn't the w coordinate just time? I can plot a 4-d sphere in a symbolic math program and ...
9
votes
6answers
478 views
Visualizations of some of the abstractions of algebraic geometry
Where, or do there exist, good visualizations of sheaves, stalks, stacks, and/or schemes? I'm a better visual thinker than I am a symbolic thinker, and it would be easier for me to follow some of the ...
8
votes
3answers
464 views
Is there a geometric interpretation of the exponential function of real numbers?
I can visualize the exponential function with the graph $y = e^x$, but I can do that for almost any function.
In addition to its graph, the function $f(x) = x^n$ can be visualized as the volume of a ...
8
votes
3answers
218 views
Three-dimensional art galleries
The well-known art gallery problem starts with an "art gallery" (a simple polygon in the plane, not necessarily convex) and asks for the minimum number of "guards" (points on the polygon) required to ...
8
votes
0answers
175 views
Visualizing the Partition numbers (suggestions for visualization techniques)
So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so ...
7
votes
1answer
111 views
Visualising a specific orbifold
Let $1 < k \in \mathbb N$ and $M = \{(z_1, z_2) \in \mathbb C^2 : k|z_1|^2 + |z_2|^2 = 1\}$. Let $S^1$ act on $M$ via $e^{i\theta}(z_1,z_2) = (e^{ik\theta} z_1, e^{i\theta} z_2)$. Then I am told ...
7
votes
3answers
349 views
Reconciling 'intersecting planes' and 'linear transformation' interpretations of matrices
I've learned in linear algebra class that an $n \times m$ augmented matrix can be thought of as a collection of n planes in $\mathbb {R}^m$ . If the matrix is invertible, the planes all intersect at a ...
7
votes
0answers
114 views
Visualizing a Calabi Yau
I would like to understand how I can visualize the quintic threefold
$$ z_1^5 + z_2^5 + z_3^5 + z_4^5 +z_5^5 - 5\psi z_1z_2z_3z_4z_5 = 0$$
For a similar problem, Hanson proposes the following:
...
6
votes
2answers
870 views
This quotient space is homeomorphic to the Möbius strip?
Let $G:\mathbb R \times [-1,1]\to \mathbb R \times [-1,1]$ be a map defined by $G(x,y)=(x+1,-y)$
This space $Q=\mathbb R\times [-1,1]/\sim$, where $(x_1,y_1)\sim (x_2,y_2)$ if and only if there is ...
6
votes
5answers
198 views
Cayley table group visualization
how can I make graphics like this? random colors. I got a script in GAP that prints rows of numbers but I want it colored
just random colors
...
6
votes
3answers
236 views
Favourite proofs with a visualization
As a fan of 'visual' proofs, I love the book Visual Complex Analysis by Tristan Needham.
For example, this picture http://en.wikipedia.org/wiki/File:Pythagoras_algebraic2.svg leads quickly to ...
6
votes
1answer
167 views
Visualize operator algebras?
It seems to me that to study mathematics is to convert the abstract language into diagrams, graphs and images. It does depend on the subject how much this technique can ease the struggle yet most of ...
6
votes
2answers
103 views
An example of a Lie group
I have a trouble learning Lie groups --- I have no canonical example to imagine while thinking of a Lie group. When I imagine a manifold it is usually some kind of a 2D blanket or a circle/curve or a ...
5
votes
3answers
341 views
Visualizing quotient groups: $\mathbb{R/Q}$
I was wondering about this. I know it is possible to visualize the quotient group $\mathbb{R}/\mathbb{Z}$ as a circle, and if you consider these as "topological groups", then this group (not ...
5
votes
2answers
243 views
continuum between linear and logarithmic
A friend and I are working on a heatmap representing some population numbers. Initially we used a linear color scale by default. Then, because the numbers covered a wide range, I tried using a log ...
5
votes
2answers
382 views
How to Visualize points on a high dimensional (>3) Manifold?
Are there any ways to visualize(plot/draw) points on a high dimensional (ex: dimension = 5) manifold?
5
votes
2answers
168 views
linearly arranging the group tables of groups of order 128
I'm planning to make a video that shows color coded group tables for all 2328 groups of order 128 -- at 128$\times$128 pixels at 24 frames a second I think I get 97 seconds of video. Is there some ...
5
votes
2answers
433 views
How to draw a complex line bundle?
The most basic example of a topologically non-trivial real line bundle is the well-known Möbius strip. Everyone who learns about vector bundles will be confronted by it, if only because it has the ...
5
votes
0answers
120 views
Visualizing Exterior Derivative
How do you visualize the exterior derivative of differential forms?
I imagine differential forms to be some sort of (oriented) line segments, areas, volumes etc. That is if I imagine a two-form, I ...
4
votes
7answers
798 views
Techniques for visualising $n$-dimensional space
Can you guys point me to the types of things I should read about (I'm a maths lay person really) if I want to learn about the current thinking how $n$ dimensional spaces can be visualised and thus ...
4
votes
1answer
276 views
Intuition about the size of $\aleph_k$ with $k>1$
Assuming CH for simplicity, I know of some more or less intuitive way to think about difference in sizes of $\aleph_0$ and $\aleph_1$. The most straightforward is the distinction of natural/rational ...
4
votes
2answers
173 views
software tool for accurate visualization of algebraic curves
First of all, I apologize since this is not strictly speaking a "mathematical" question but I could not find a better place for it.
For a work presentation I need a tool for accurate visualization of ...
4
votes
3answers
98 views
Visualizing identity $m\le3n-6$ for simple connected finite planar graphs
How can I visualize the identity $m\leq3n-6$ (where $m$ is the number of edges, $n$ the number of vertices) for simple connected finite planar graphs?
4
votes
1answer
79 views
The complement of a torus is a torus.
Take $S^3$ to be the three-sphere, that is, $S^3=\lbrace (x_1,x_2,x_3,x_4):x_1^2+x_2^2+x_3^2+x_4^4=1\rbrace$. Using the stereographic projection, $S^3=\mathbb{R}^3\cup \lbrace \infty \rbrace.$ Can ...
4
votes
2answers
177 views
Computing the projection of an infinite 3D grid of points
Consider the subset $S$ of $\mathbb{R}^3$ consisting of points whose coordinates are integers (compare Gaussian integers, Euclid's orchard). The view from the origin has interesting structure; it has ...
4
votes
1answer
123 views
Intuitive understanding a theorem in analysis
Is there a way to intuitively understand/visualize the following theorem in analysis?
Let $(f_n)$ be a sequence of real functions differentiable in a finite/infinite open interval $(a,b)$. Suppose ...
4
votes
0answers
203 views
Visualizing the domain of the square root
I would like to show someone the domain of the complex square root function (the 2-sheeted riemann surface). Is there a good interactive visualization software for this?
I would like some sort of ...
4
votes
1answer
128 views
FLOSS tool to visualize 2- and 3-space matrix transformations
I'm looking for a FLOSS application (Windows or Ubuntu but preferably both) that can help me visualize matrix transformations in 2- and 3-space.
So I'd like to be able to enter a vector or matrix, ...
3
votes
4answers
152 views
Proof-without-words for $\bar a\times (\bar b\times\bar c)=\bar b (\bar a\cdot\bar c)-\bar c (\bar a\cdot \bar b)$ or some visual-biased explanation?
Griffiths' Introduction to Electromagnetism -book has equations called 20.10 below.
I have proved this equation d) pretty much on the first mathematics -course I had but I have not yet understood a ...
3
votes
1answer
734 views
How to visualize the Gaussian curvature of a 3D surface?
I have a 3D surface. I want to visualize color-coded Gaussian Curvature. Is there any software (e.g. MATLAB, Mathematica) which can be used for calculating and visualizing the curvature in color code ...
3
votes
1answer
225 views
Ways to visualize the real numbers?
I was just wondering if there are any diagrams for visualizing subsets of the real numbers, or totally 'radically' different ways of looking at them as a real line? The model of the line relies on ...
3
votes
3answers
387 views
Trying to draw the tautological line bundle ($\subseteq \mathbb{CP}^1\times \mathbb{C}^2$)
In order to learn about vector bundles, I would like to draw the tautological vector bundle over the complex projective line
$$ E = \{(x,v) \in \mathbb{CP}^1 \times \mathbb{C}^2 : v \in x \} .$$
...
3
votes
2answers
133 views
Programs for visualizing math designed for laymen?
I'm looking for interactive applications that are for university topics (designed for windows) in mathematics only , preferably something that doesn't include programming languages to use because ...
3
votes
0answers
105 views
Illustrations of a line and a curve intersecting for complex field
Are there nice illustrations on the Net of say $y=a·x+b$ and $y=x^2$ intersecting where x and y are complex? I'm thinking of the amplitude of y being depicted as height above the complex plane with ...
3
votes
1answer
80 views
Visualizing SPD cone for $3\times3$ matrices
Can anyone see a good way to visualize the SPD cone for 3x3 symmetric matrices?
I'm interested in something that would highlight it's special structure, like non-smoothness.
Here's one attempt, ...
2
votes
5answers
237 views
Software to display 3D surfaces
What are some examples of software or online services that can display surfaces that are defined implicitly (for example, the sphere $x^2 + y^2 + z^2 = 1$)? Please add an example of usage (if not ...
2
votes
2answers
229 views
Why is it that I cannot imagine a tesseract?
I try hard to "visualise" (say "imagine") a tesseract but I can't.
Why is it that I can't?
This may be a question for a scholar of some other discipline and not for a mathematician, e.g. ...
2
votes
1answer
108 views
2
votes
2answers
555 views
calculate location on an isometric world map
I want to display a point with a latitude and a longitude value in an isometric world map using JavaScript and jQuery.
For a rectangular map, I use the following code to calculate the x and y ...
2
votes
2answers
162 views
Fuzzy Venn diagram regions labeled in ternary
I have a couple of questions about the Venn diagrams object :
Words from the binary alphabet with n letters label each region of an order-n Venn diagram. Is there any more profound connection ...
2
votes
2answers
145 views
Lie group and SO3 visualisation
Maybe I'm asking a very vague question but I'd like to know if there are some visualisation tools available already that explain lie algebra exponential map or logarithm? I'd like to be able to ...
2
votes
1answer
158 views
general (asymmetric) real rank-2 tensor visualization in 3d
I have general rank-2 real tensor in 3d space represented as a 3x3 real matrix $M$ (it is gradient of a vector field). I am writing some code to visualize it in several isolated points, this is what I ...
2
votes
1answer
386 views
Can we put the mathematics of paradoxes in visual art into perspective?
(Pun definitely intended.)
Dear MSE-Community,
If I were to choose one artist that has made interesting works of art not only because of their beauty but also because of their connections to ...
2
votes
1answer
42 views
visualizing functions invariant (or almost) under modular transformation
In the spirit of Möbius Transformations Revealed, I'd like to make a pair of movies depicting how Klein's absolute invariant $j(\tau)$ and the Dedekind eta function $\eta(\tau)$ transform when ...
2
votes
1answer
161 views
visual proof of the chinese remainder theorem?
I have seen visual proofs for fermats little theorem, gauss sum and many other things. I find them very useful. Is there a visual proof for the chinese remainder theorem? Thanks in advance.
