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

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2
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1answer
58 views

Soft Question: Difficult to Visualize Areas of Mathematics

Yesterday I came across this webpage, which describes a recent (successful) attempt to visualize isometric embeddings of flat tori in 3D Euclidean space. The webpage and associated paper discuss the ...
1
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1answer
23 views

Is it possible to estimate the number of primes between 0 and a 128 bit number?

I'm attempting to visualize an RSA public/private key pair, or a SHA2 hash. In order to reduce that massive number that is meaningful to humans I'm looking at this SHA2 visualization function to ...
0
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0answers
17 views

Spaces of visual patterns, but not recurse/chaos.

I'm looking for information on existing/notable, spaces of visual patterns, that do not rely on, or appear to make much use of, recursion/chaos to function, as a cellular automata or fractal would. ...
1
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2answers
866 views

Direct proof. Square root function uniformly continuous on $[0, \infty)$ (S.A. pp 119 4.4.8)

(http://math.stanford.edu/~ksound/Math171S10/Hw8Sol_171.pdf) Prove for all $e > 0,$ there exists $d > 0$ : for all $x, y \ge 0$, $|x - y| < d \implies |\sqrt{x} - \sqrt{y}| < e$. (a) ...
0
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1answer
175 views

Cauchy's Generalized Mean Value Theorem. Required function. (S.A. pp 140 t5.3.5)

Cohen, Henle. Calculus pp 827, (http://www.vias.org/calculus/09_infinite_series_10_06.html) I revised the footnote in pp 14 http://math.uga.edu/~pete/2400calc2.pdf. This theorem can be illustrated ...
1
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2answers
54 views

How to read a cycle graph?

As an important tool for visualizing some small finite groups it is useful to know how read such graph, and with time trying to make sketch of them by my own. I would like to know, for a start, how ...
0
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0answers
9 views

Why the gradient of the r vector is the identity map, geometrically speaking?

When doing some simple quantum mechanics problem involving commutators, I forgot the result of this expression $$\left[\vec{r} ,\hat{p}\right]$$ Thus I then brute force it using the definition of ...
1
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2answers
45 views

Visualizing why rotations preserve orientation

It's clear geometrically that if you have two vectors in $\mathbb{R}^3$ a rotation will preserve their lengths and the angle between them. But how do you visualize that a rotation preserves ...
3
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0answers
104 views

Visualising surface integrals

For a current problem I am working on, I have run into angular surface integrals, i.e. the differential solid angle $\text{d}\Omega$. Specifically the surface integrals are defined by ...
0
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0answers
6 views

How common is it to plot linear transforms on continuous spaces?

I came across a question asking how the proof that the transform $f(g) = g(x-a) + g(x+a)$ was Hermitian worked. I was a bit surprised that the proof was symbolic, because if you plot out the ...
0
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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
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1answer
29 views

How to represent a network of chemical reactions?

I am trying to figure out a suitable representation for given set of chemical reactions, which happen over an exhaustive set of chemicals. The chemicals are $A, B, C, D, E$ and the reactions are ...
80
votes
20answers
5k views

Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction

I recently proved that $$ \sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2 $$ Using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial ...
1
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1answer
290 views

Characterize normal subgroups - Find all subgroups of $S_3$ conjugate to $\{id, (1,3) \}$ - Fraleigh p. 143 14.29

(27.) A subgroup H is conjugate to a subgroup K of a group G (viz. p. 141 $K \le G$ is a conjugate subgroup of $H$), if $i_g[H] = gHg^{-1} =K$ for some $g \in G$. Show that conjugacy is an ...
2
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3answers
51 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 ...
22
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1answer
436 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
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1answer
47 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 ...
1
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0answers
8 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 ...
6
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1answer
102 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 ...
9
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4answers
190 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
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0answers
43 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 ...
2
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0answers
49 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 ...
23
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0answers
273 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
57 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 ...
20
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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 ...
2
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0answers
81 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
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1answer
16 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 ...
1
vote
0answers
118 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
47 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 (...
45
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13answers
12k views

Interesting math-facts that are visually attractive

To give a talk to 17-18 years old (who have a knack for mathematics) about how interesting mathematics (and more specifically pure mathematics) can be, I wanted to use nice facts accompanied by nice ...
3
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1answer
90 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 ...
0
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0answers
16 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
63 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 ...
1
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1answer
79 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 ...
1
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1answer
20 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 ...
92
votes
5answers
11k views

How were 'old-school' mathematics graphics created?

I really enjoy the style of technical diagrams in many mathematics books published in the mid-to-late 20th century. For example, and as a starting point, here is a picture that I just saw today: ...
0
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0answers
25 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. ...
2
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0answers
82 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
98 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 ...
3
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0answers
61 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
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0answers
16 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 ...
4
votes
2answers
135 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
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0answers
29 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 ...
8
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6answers
480 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 ...
2
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0answers
58 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 ...
0
votes
1answer
62 views

What kind of graph was github's impact graph?

Github used to have a graph called an impact graph. It feels almost like a Sankey diagram and almost like a stacked area chart. What is the name of this kind of graph? I couldn't find a better ...
1
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0answers
31 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
1answer
56 views

logic - how to model or diagram conditional probabilities of multiple related scenarios.

I am interested in modeling questions and specific outcomes so that i can evaluate conditional probabilities and mathematical expectation. I am looking for a way to diagram or otherwise describe the ...
0
votes
2answers
63 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 ...
6
votes
2answers
396 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 of $S$ from a perspective camera within the space ...