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

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1
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1answer
12 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
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1answer
38 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 ...
89
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5answers
10k 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
23 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. ...
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0answers
23 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
21 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 ...
2
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0answers
61 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
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2answers
70 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
37 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
11 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
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2answers
92 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 ...
3
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2answers
703 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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0answers
17 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
452 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
52 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 ...
75
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18answers
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 ...
0
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1answer
56 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
27 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
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1answer
52 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 ...
14
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0answers
218 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 ...
0
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2answers
59 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
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2answers
343 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 ...
3
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1answer
66 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 ...
1
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2answers
145 views

Does a Möbius strip have only one shape? Or may it have different shapes?

I'm reading a book about geometry, and after thinking and viewing the Möbius strip, I want to know whether the book is right or not. The book says with a little description (that I can't write here ...
12
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0answers
294 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 ...
34
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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 ...
2
votes
1answer
48 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
140 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 ...
1
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2answers
244 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.
4
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1answer
103 views

How can I visualize what open sets “look” like in unfamiliar topological spaces?

The question is extremely general, but I do have a specific case I'd like to look at, and I'm hoping that some combination of specific pointers and general advice will help me out. Consider the ...
0
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1answer
46 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. ...
2
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1answer
230 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 ...
2
votes
1answer
44 views

Usefulness of alternative constructions of the complex numbers

Complex numbers $\mathbb{C}$ are usually constructed as $\mathbb{R}^2$ together with a suitable multiplication. But this is not the only possible way, one can get to the complex numbers. One ...
1
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0answers
48 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
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2answers
68 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 ...
3
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1answer
116 views

Intuition - Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection ...
0
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0answers
10 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
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1answer
25 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
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0answers
67 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} ...
1
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1answer
77 views

Visualizing second-order Markov chain

You can visualize a first-order Markov chain as a graph with nodes corresponding to states and edges corresponding to transitions. Are there any known strategies to visualize a second-order Markov ...
0
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0answers
19 views

Draw an ellipse corresponding to a bivariate normal distribution

Let $$\mu=\left(\begin{array}{c}\mu_1 \\ \mu_2\end{array}\right)$$ and $$\Sigma=\left(\begin{array}{cc}\Sigma_{1,1} & \Sigma_{1,2} \\ \Sigma_{2,1} & \Sigma_{2,2}\end{array}\right)$$ be ...
3
votes
7answers
653 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 ...
3
votes
3answers
1k 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
98 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 ...
9
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2answers
922 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?
0
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1answer
253 views

Visualization of 2-dimensional function spaces

As a follow-up question to what is the norm measuring in function spaces I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...
4
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5answers
309 views
1
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1answer
30 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 ...
6
votes
3answers
331 views

Proofs without words of some well-known historical values of $\pi$?

Two of the earliest known documented approximations of the value of $\pi$ are $\pi_B=\frac{25}{8}=3.125$ and $\pi_E=\left(\frac{16}{9}\right)^2$, from Babylonian and Egyptian sources respectively. ...
0
votes
1answer
19 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 ...