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

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1answer
13 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
52 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 ...
0
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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 ...
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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
38 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 ...
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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 ...
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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. ...
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 ...
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0answers
18 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
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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 ...
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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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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 ...
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 ...
14
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0answers
222 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
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? $$(A\setminus B)\times (A\setminus B) = (A\times A)\setminus (B\times B)$$
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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 ...
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 ...
1
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2answers
255 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
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1answer
47 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. ...
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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 ...
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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 ...
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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 ...
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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} ...
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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 ...
2
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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 ...
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 ...
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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 ...
4
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5answers
287 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?
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0answers
27 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
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2answers
70 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
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1answer
74 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 \\ ...
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2answers
62 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 ...
3
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3answers
77 views

Viewing an abelian group using cayley diagram

I cannot understand this way of viewing whether a group is abelian using cayley's diagram: (from Visual group theory book) What I can't understand is that while checking being abelian we check ...
1
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1answer
37 views

Order of an element in direct product using cayley's diagram

How can I find the order of element (1,1) of the group $C_4\times C_3$ visually in the diagram below :
0
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1answer
36 views

Analysing/Visualising shape of multi-variate function.

I have an unknown function $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ for which I'm determining a first order Taylor approximation through a non-linear optimization process in six variables (the ...
2
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1answer
50 views

Criteria of regularity that a Cayley's Diagram should meet .

As referred in the Visual group theory Book by Nathan Carter- The unofficial definition of a group says that : A group is a collection of actions satisfying the rules: 1. there is a predefined list ...
2
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2answers
279 views

Visualization of Eratosthenes’ sieve

In otherwise great paper on prime numbers, I found following visualization of Eratosthenes’ sieve: I found it somewhat scary and confusing. Is there any better visualization of Eratosthenes’ sieve ...
0
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1answer
98 views

Line integrals of vector fields-positive, negative, or zero

I have a question about line integrals of vector fields being positive, negative, or zero. If you are measuring the work it takes to "push" a point on the curve through the vector field, does this ...
1
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0answers
78 views

Geometrical application of generation function for permutation

It is quite well known that the generation function for permutations is represented as $$(1+x)(1+x+x^2)\dots(1+x+x^2+x^3...+x^{n−1})$$ (See, e.g., question The generating function for permutations ...
1
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1answer
96 views

Plotting the intersection of multiple surfaces with WolframAlpha

I want to plot the intersection of two surfaces like in this post. But if I enter the much simplified expression ContourPlot3D[{x^2 + y^2 + z^2 - 4=0, xy=1}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}] ...
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: ...
2
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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 ...
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2answers
47 views

$2\times$ distance to a line${} = {}$distance to a point, in geogebra

While preparing for my math exams, I got this question: give the locus (if thats the right word) where $2$ times the distance to the line l $x=8$ equals $1$ time the distance to point F $2,0$. I was ...
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0answers
28 views

Nyquist limit explanation

Kindly explain Nyquist in easy words. The actual question is as follows. We can attempt to display sampled data by simply plotting the points and letting the human visual system merge the points into ...
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0answers
23 views

Different ways of visualizing a certain classes of single variable complex functions

Single variable complex functions of a real variable are ubiquitous in engineering contexts such as control engineering and signal processing, and visualizing them is of utmost importance in designing ...
1
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1answer
78 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 ...
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10answers
3k views

Visualizing the square root of 2

A junior high school student I am tutoring asked me a question that stumped me - I was wondering if anyone could shed some light on it here. We were talking about how the square root of 2 is an ...