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

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2
votes
1answer
60 views

Visualising something geometrically

$W=B(x_{1},r)\cap B(x_{2},r)$. The boundary of the intersection is given by the union of $\delta_{1}W=\delta B(x_{1},r)\cap B(x_2,r)$ and $\delta_{2}W=B(x_1,r)\cap \delta B(x_{2},r)$. Let ...
2
votes
3answers
66 views

Open Source Software for Creating Mathematical Diagrams

I work as a software engineer at a company developing navigation systems. As I have a mathematical background I normally get assigned the more mathematical problems and I find myself regularly having ...
1
vote
0answers
10 views

Visualization of the fact that the integers defining lens spaces must be coprime

This is related to this question I asked: Visualization of Lens Spaces and is also related to this question by @Earthliŋ: Why are the integers appearing in lens spaces coprime? I understand the ...
7
votes
1answer
161 views

Visualization of Lens Spaces

I am trying to visualize lens spaces geometrically. While I am aware of the fact that most manifolds which cannot be embedded in $\mathbb{R}^3$ are hard to visualize because of the obvious ...
2
votes
0answers
85 views

Why are inter arrival times in the continuous version of discrete-time Markov chains always exponentially distributed?

I am curious whether there exist continuous time Markov processes for which the times between jumping times (which I call inter arrival times) are not exponentially distributed, but have some other ...
58
votes
1answer
686 views

Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles?

The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a ...
2
votes
3answers
61 views

Visualization of the quotient of $\mathbb{R}^2$ by an involution.

Consider $\mathbb{R}^2$ and let $\mathbb{Z}_2$ act by taking $(x,y) \rightarrow (-x,-y)$ and consider $\mathbb{R}^2/\mathbb{Z}_2$. I can, using algebraic machinery, show that the quotient is the same ...
2
votes
0answers
89 views

Visualising algebraic topology

I'm new to algebraic topology and although I can follow the arguments it would be nice to be able to visualise important concepts like homology and excision. Can anyone recommend a book or other ...
0
votes
1answer
20 views

Help with Diagram of the Standard Lift of Projective Plane

I am posting here because I need help finding (or making) a visual aid for a presentation. I am giving a short presentation about Projective Geometry next week, and I am building a beamer for it. One ...
2
votes
0answers
564 views

Intuition for gradient descent with Nesterov momentum

A clear article on Nesterov’s Accelerated Gradient Descent (S. Bubeck, April 2013) says The intuition behind the algorithm is quite difficult to grasp, and unfortunately the analysis will not be ...
0
votes
1answer
72 views

Visualizing a volume with MATLAB

I have a rational function $\phi: [0,1]^3 \to \mathbb R^3$ (A NURBS, to be precise) and I want to visualize the image of $\partial [0,1]^3$ using a surface plot (...
0
votes
0answers
18 views

Contours in $\mathbb{C}$ on the Riemann sphere

I'm looking for some sort of visualisation (either illustrations, or better yet some sort of applet) of how contours in the complex plane look on the Riemann sphere (the "actual" complex plane), for ...
1
vote
1answer
68 views

Interpreting a group homomorphism $f: \mathbb{Z}_{12} \to \mathbb{Z}_{3}$ visually

I am having a hard time studying and I am a visual learner. How could I visually imagine a (group) homomorphism $$\mathbb{Z}_{12} \to \mathbb{Z}_3?$$ Also, if the question states that the map $f$ is ...
21
votes
3answers
1k views

Can Number Theory be visualized?

So I was thinking about a hard euclidean geometry problem, when it hit me just how much more difficult it would become without the aid of a diagram. This got me thinking: Wouldn't it be great if we ...
1
vote
1answer
22 views

Visualizing cross product of points in 3-Space

If $p_0, p_1, p_2$ are three distinct points in space, then what does the cross product $$n = (p_0 - p_1) \times (p_0 - p_2)$$ mean geometrically? I'm having a little trouble visualizing this in ...
1
vote
1answer
115 views

Visualizing Ricci scalar curvature

I am trying to learn more about Ricci scalar curvature. I am trying to get an image in my head of what scalar curvature actually represents about the curvature of a manifold. The most familiar image I ...
2
votes
1answer
74 views

Are there 3D geometric proofs of Fibonacci identities?

There is a significant number of identities involving Fibonacci numbers that can be proven in a sort of geometric way, as it is shown in the following picture: However, I couldn't find any such ...
1
vote
0answers
81 views

I seek Visual illustrations of Concepts of Mathematics as animated videos for students of Higher Mathematics

There are some very good animated videos explaining concepts of mathematics on youtube, like videos of website " why U" but most of these deal with elementary mathematics. I am searching for videos ...
3
votes
1answer
114 views

How to visualize bilinear transform of the form $S(z) = \frac {T}{2} \frac {z+1}{z-1}$

Note that this is the bilinear transform from a z-domain as appears in Z-transform to s-domain in Laplace transform Recall that bilinear transform has form $M(z) = \frac{az+b}{cz+d}$ with and has to ...
3
votes
0answers
151 views

Is this a legit way to visualize complex functions?

I am doing laplace transform in a class and I hate how there seems to be no graphical support when things are transformed to laplace domain i.e. nobody cares what they look like in laplace domain But ...
3
votes
2answers
128 views

How to visualize the gradient as a one-form?

I just finished reading the proof that the gradient is a covariant vector or a one-form, but I am having a difficult time visualizing this. I still visualize gradients as vector fields instead of the ...
4
votes
0answers
96 views

How to visualize cotangent spaces.

I was wondering how to intuitively and visually understand dual vector spaces and one-forms. So my question is (1), how to visualize cotangent spaces and (2), how to intuitively understand them? My ...
0
votes
0answers
20 views

Visualize Gaussian curvature

I have calculated Gaussian curvature and I have values in interval [-a, b]. I need to recompute this to interval [0, 1], with 0,5 = zero curvature. If I use standard scale, I have a problem because ...
0
votes
0answers
28 views

About teaching an advanced principle mainly with pictures.

There is a concept I have seen on Pinterest called Infographics. The use of colourful pictures and graphics and diagrams that can show in a pictorial way some explanation of advanced math principles. ...
4
votes
2answers
201 views

How to actually use the Weierstrass-Enneper parameterization to draw a minimal surface?

I'm interested in drawing (with Mathematica for example) the generalized Scherk saddle tower with threefold symmetry, a shape that I find very attractive. In an article (see here) I found the ...
0
votes
0answers
57 views

Diagram to depict dependencies/prerequisites of activities

A long time ago in a business course I was taught about diagrams that help plan activities, and determine estimates of time required to complete a project: activities in the project were drawn as ...
2
votes
0answers
66 views

Is there a way to graphically show that a solution is the minimum or stationary solution to a functional?

I'm looking for the functional analogue to the visual representations of function optimization you most commonly see. To illustrate, if we have some function: $$ f(x) = (x-1)^2+1 $$ We can look at ...
1
vote
0answers
32 views

Visualizing Riemannian surface

Given a multi-valued complex function $f: z = x+\mathrm{i}\,y\rightarrow w=u+\mathrm{i}\,v$ with $x,y,u,v\in\mathbb{R}$, we know the image $\{f(z)\,|\,z\in\mathbb{C}\}$ is a Riemannian surface. How to ...
0
votes
0answers
33 views

Expressing Identities about Matrix of Linear Transformation

If $T\colon V\rightarrow W$ and $S\colon W\rightarrow U$ are linear transformations, then I will consider their composition in the order $S\circ T$ (i.e. $S\circ T(v)=S(T(v))$ ). Given basis $B_1$ ...
0
votes
2answers
66 views

How to visualize(inside ones brain) the Four-dimensional_space

Can the fourth dimension https://en.wikipedia.org/wiki/Four-dimensional_space be visualized intuitively by the humans. Does the professional mathematicians can do this ? If so what are the things to ...
4
votes
1answer
79 views

Visualising this CW structure for the $S^3$

I'm asked to prove that the following is a CW structure for the 3-sphere, (as a part of an exercise involving defining the Cw structure of the Lens Spaces) I'm asked to prove that the following is a ...
24
votes
2answers
555 views

Are there any visual proofs for $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$?

I was flipping through Proofs Without Words (PWW) and saw many visual proofs for sequences and series. However, I saw none for $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$ Are there any ...
2
votes
1answer
51 views

How would you draw $(A\setminus B)\times (A\setminus B) = (A\times A)\setminus (B\times B)$?

I know it's useful to prove set equalities to make a quick sketch of the sets described. How can I draw this one? $$(A\setminus B)\times (A\setminus B) = (A\times A)\setminus (B\times B)$$
8
votes
2answers
167 views

0 to the power of 0, what does the essential discontinuity actually look like?

So having watch this clip by Numberphile which explains why $0^0$ is undefined https://www.youtube.com/watch?v=BRRolKTlF6Q And also this http://mathforum.org/dr.math/faq/faq.0.to.0.power.html And ...
35
votes
3answers
6k views

The Stupid Computer Problem : can every polynomial be written with only one $x$?

When I was a child, I wanted to be a mathematician so I asked my parents to buy me a computer to make super complex calculations. Of course, they were not crazy enough to buy an expensive super ...
1
vote
2answers
2k views

What is the fourth dimension of a Tesseract?

Is the fourth dimension of the Tesseract time? That is why it is represented as a moving 3D structure on Wikipedia? I am asking because I have trouble understanding what it is.
0
votes
1answer
58 views

Books on the visual/graphical aspects of geometry

Are there any books providing a general overview of the visual/graphical aspects of geometry? For example, Tilings (e.g. hyperbolic) and tessellations Plane/space filling shapes/objects (e.g. ...
1
vote
0answers
60 views

Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
0
votes
2answers
88 views

Looking for intuïtive explanation why contour integral of $\frac{dz}{z} $equals $2\pi i$ in complex analysis

$$\oint \frac{dz}z = 2\pi i$$ I've seen the derivation of it using the parametrisation. Since this result is used all the time in my complex analysis course, i'd like to understand this ...
0
votes
0answers
21 views

Are there tools for presentation and vizualization of deduction?

I read that Kalish and Montague introduced a natural deduction method (http://en.wikipedia.org/wiki/Donald_Kalish), which can be easily implemented in software. Any other tools who can help a logician ...
0
votes
1answer
38 views

How to visualize implicit functions

I have a task of visualizing few implicit functions. Firstly lets say I have the following function of $N$: $$\epsilon = \sqrt{\frac{8}{N}\ln \left( \frac{4(2N)^{50}}{0.05} \right)}$$ Now this is ...
4
votes
0answers
89 views

How to visualise Bollobas' 1965 theorem?

Theorem $[n]=\{1,\ldots,n\}$. Let $\lbrace (R_i, S_i), i \in I \rbrace, R_i, S_i \subset [n]$ be such that $R_i \cap S_i = \emptyset, R_i \cap S_j \ne \emptyset (i \ne j)$. Then $$\sum_{i \in I} ...
2
votes
1answer
231 views

Why is it not possible to visualise a 4th dimension object? [duplicate]

By drawing a cube on a paper or by seeing it on a screen (a 2D surface - see Figure below), we can sort of visualise how a 3D cube would look like. I was wondering whether we will be able to ...
1
vote
1answer
63 views

How to visualize probability distributions in terms of sets - joint and marginal?

Let there be two sets, $\mathcal{X},\mathcal{Y}$, both finite, and they represent the set of values that the discrete random variables, $X,Y$ can take. $\mathcal{P}_{Y|X}$ be all possible ...
0
votes
1answer
22 views

visualizing a function from the plane into the reals

If $$(x_1,y_1), (x_2,y_2),(x_3,y_3)$$ are points in the plane and if $a,b$ are fixed real numbers, how can I visualize $$f=(ax_1+b-y_1)^2+(ax_2+b-y_2)^2+(ax_3+b-y_3)^2$$ as a function from the plane ...
4
votes
5answers
296 views

How to graph/visualize complicated inequalities

I'm having trouble visualizing areas defined by for example, $$ x^2 + y^2 \leq 2y $$ Or $$ (x^2 +y^2)^2 \leq 2a^2(x^2 - y^2) $$ What is the thought process in picturing these regions?
0
votes
0answers
51 views

Affine Transformation and Continuous Deformation

How do these two concepts relate? Thus far I have a (what I think is a) good intuitive idea of a continuous deformation- the visual basically looks like the boundary being stretched so that it never ...
2
votes
2answers
81 views

How can I visually imagine the area of a circle divided by $\pi$?

If I have a circle with an area of 100 units^2, and I divide it by $\pi$, how can I imagine that visually in my mind? Since 100 / $\pi$ =~ 31.83, and the square of that is =~ 5.64, I currently ...
4
votes
1answer
88 views

Visualizing Lie algebra of SO(3)

Let $SO(3)$ be the Lie group of 3D rotations. Rotation about z-axis by an angle $\phi$ is represented in standard basis by this matrix: $$ \begin{pmatrix} \cos \phi & -\sin\phi & 0 \\ ...
1
vote
2answers
98 views

Parallel Lines Intersecting in the Projective Plane

My question is about visualizing projective space, in particular the real projective plane $\mathbb{P}^2(\mathbb{R})$. I know there are different ways to define this space, but in each we can say that ...