For questions about visualizing mathematical concepts. This includes questions about visualization of mathematical theorems and proofs without words.

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11
votes
1answer
341 views

Kakeya Needle problem video

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid: What I'm searching for is a visualisation of the ...
1
vote
2answers
160 views

Software for visualizing partial derivatives?

I'm whipping up a set of notes, and I want to include a diagram or two showing some partial derivatives. Specifically, a diagram would include: a 3D surface of the form z=f(x,y), a plane of the form ...
4
votes
5answers
357 views

Geometric visualization of covector?

How could I geometrically visualize a linear functional?
4
votes
0answers
211 views

Visualization of immersed submanifold

I am trying to visualize the difference between immersed submanifold and embedded submanifold. At first, I thought that, for example, if I can embed manifold $M$ in $\mathbb{R}^4$ and if my friend can ...
2
votes
0answers
188 views

Looking for proof-without-words of Bezout's identity

I'm looking for a "proof-without-words" of Bezout's identity (for integers). Does anyone know of one?
0
votes
1answer
172 views

Visualise 3 simultaneous cubic equations

I have three equations of the form: $$\frac{i_1^3}{P_1}+i_1(Z_1+Z_2)+(i_2+i_3)Z_2-U_1=0$$ $$\frac{i_2^3}{P_2}+i_2(Z_1+Z_2)+(i_1+i_3)Z_2-U_2=0$$ $$\frac{i_3^3}{P_3}+i_3(Z_1+Z_2)+(i_1+i_2)Z_2-U_3=0$$ ...
3
votes
1answer
94 views

Visualizing the group operation in higher homotopy groups

I'm having trouble picturing the homotopy group operation of concatenation between two pointed spaces. For $n$-spheres, we have for $f,g: S^n \to X$ $$(f * g)(s_1,\ldots, s_n) = \begin{cases} ...
2
votes
2answers
1k views

Gallery of unlabelled trees with n vertices

Can anyone point me to a gallery (printed or online) of unlabelled trees, sorted according to their order (i.e., number of vertices)? That is, for each order n in oeis.org/A000055 (up to maybe n=11 ...
5
votes
0answers
63 views

An Auto-Generated Cartography of Mathematical Theories: Has it been done already?

While looking for a way to visualize the logical structure of mathematical theories a graph-like depiction came to my mind, where propositions are represented by vertices. An edge goes from ...
3
votes
0answers
283 views

Imagining four or higher dimensions and the difference to imagining three dimensions

I’m very interested in how people envision four or higher dimensions. And I’m especially interested in how geometers and topologists who actually work in four dimensions do. Now I know of the video ...
13
votes
6answers
452 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 ...
6
votes
4answers
355 views

Which Cross Product for the Desired Orientation of a Sphere ? [Stewart P1091 16.7.23]

P1086: For a closed surface, the positive orientation is the one for which the normal vectors point outward from the surface, and inward-pointing normals give the negative orientation. P1087: ...
0
votes
1answer
205 views

Visualizing Gauss-Legendre Quadrature

I'm creating a GUI interface for my Python Computing class that is supposed to showcase a few types of Numerical Integration. One of the ones I want to put in as an option is Gauss-Legendre ...
10
votes
0answers
401 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: ...
4
votes
1answer
158 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 ...
5
votes
2answers
265 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 ...
46
votes
4answers
2k 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 ...
0
votes
1answer
44 views

How to represent a bounded function

I am not completely sure whether this question belongs on mathematics.SE but I figured to give it a shot: I have a function which mathematically looks like this: $f(x)=\max(A,B\cos x)$ This will ...
2
votes
2answers
335 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 ...
1
vote
1answer
115 views

Visualizing diffeomorphisms

This is probably a really basic question (hence my asking it here as opposed to MO). In a comment to a question on mathoverflow ...
9
votes
4answers
192 views

Examples of familiar, easy-to-visualize manifolds that admit Lie group structures

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 $2$D blanket or a circle/curve or ...
1
vote
0answers
105 views

Which space this space drawn in this picture is homeomorphic?

Based in this question Why this space is homeomorphic to the plane? I would like to know which space this space is homeomorphic, any help or an intuitive idea are welcome. [Context of Image: ...
9
votes
7answers
513 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 ...
8
votes
2answers
1k 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 ...
1
vote
1answer
80 views

Vizualisation about line search in Linear Programming?

I am trying to visualize this recursive algorithm in LP, Wikipedia here. I am looking for references about in which kind of problems is this used and what does it really look like? I am also ...
1
vote
0answers
64 views

Visualising the derivative/slope $f'(x_0)$ of $f:\mathbb{R} \rightarrow \mathbb{R}$ as a line segment

A function $f:I\subseteq \mathbb{R} \rightarrow \mathbb{R}$ that is differentiable at $x_0 \in I$ obeys the following equality for all $h\in (I-x_0)$ (i.e. for all $h\in \mathbb{R}$ such that $x_0+h ...
2
votes
2answers
336 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 ...
3
votes
0answers
151 views

Visualizing and manipulating 4-dimensional data with 3D technology

It is possible to visualize 3 dimensional data (like a scatter plot) by projecting it on a 2 dimensional screen in a way that allows to interact with it in an intuitive way. Is it possible to ...
2
votes
1answer
266 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.
3
votes
3answers
2k 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
417 views

Kerning on the fly -algorithm [closed]

Do anyone know any algorithm which would calculate automatically kerning of characters based on glyph shapes when user types text? I don't mean trivial calculation of advance widths or similar, I ...
10
votes
1answer
415 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 ...
2
votes
0answers
95 views

Help me to vizualise this falling ball on spinning Earth

The earth rotates. The ball falls in an latitude, not equator, let say in Germany. I am trying to understand how to express the ball in terms of the angular velocity on the planet. The constant ...
3
votes
1answer
349 views

Visualizing Infinity discerning countable and uncountable

This is rather a philosophical question. Although it uses topological notions, it isn't any precise mathematics, so maybe one cannot take it very seriously. Sometimes I try to picture an infinite set ...
3
votes
4answers
225 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
0answers
139 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 ...
0
votes
2answers
220 views

Visualizing $L_2$ or an infinite dimensional Hilbert space

I need to explain abstract objects in an infinite dimensional Hilbert space. What is the best way to visualize it for an engineering audience? Does anybody have a good example?
2
votes
1answer
233 views

Is there a geometric projection for every complex function?

I was wondering about the best way to visualize complex functions. As they're $$ {\mathbb R}^2 \rightarrow {\mathbb R}^2\ ,$$ I think best way are complex plane image/grid transforms like they used in ...
4
votes
2answers
256 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 ...
7
votes
3answers
2k 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 ...
2
votes
2answers
467 views

Visualization of Cantor set by Mathematica or Maple!

We all know what it is the Cantor set. The Cantor set is created by repeatedly deleting the open middle thirds of a set of line segments. One starts by deleting the open middle third $(\frac{1}{3}, ...
2
votes
1answer
82 views

What do you call a graph of value vs rank of the value?

What do you call a graph of value vs rank of the value? For example, say I make a census of the 2010 net income for a bunch of people. The graph of net income vs age gives me a: scatterplot The ...
5
votes
3answers
299 views

Can one construct a “Cayley diagram” that lacks only an inverse?

My group theory text asks for an example of a Cayley-like diagram that exhibits all the properties of a group except (only) that at least some elements lack an inverse. Is it possible to construct ...
1
vote
1answer
1k views

Visualize a difference equation with Matlab [closed]

I have a difference equation for a Single Pole Infinite Impulse Response Filter, defined on a discrete time-series: $y[n]-(1-\alpha)*y[n-1]=\alpha*x_n$ While the []s brackets refer to a position n ...
0
votes
0answers
186 views

The geometry of functions $\mathbb{R}^2\rightarrow \mathbb{R}$ that satisfy the norm axioms

What are constraints on the "looks" of a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$, if $f$ satisfies $$f(x_1+x_2)\leq f(x_1) + f(x_2), \ \quad x_1,x_2\in \mathbb{R}^2 \quad \quad (1)$$i.e. the ...
11
votes
2answers
1k 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 ...
0
votes
1answer
438 views

Visualizing solids of revolution

Can someone shed a little bit of light on the problem of volumes of revolution about the $x$-axis and the $y$-axis of the same shapes? Take for example $f(x)=x^2$. If we want to find the volume ...
8
votes
3answers
389 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 ...
4
votes
3answers
145 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
162 views

Is there any way to give sense to a geometric/visual proof?

Suppose one is given the following visual proof that $$\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{1}{2^k} = 1$$ which is the following construction over $[0,1]\times[0,1]$ What this is ...