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

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Book 2 of Visual Complex Functions

I am having a lot of fun in reading Visual Complex Functions by prof Wegert. (it is a very interesting read and well-recommended by me). Inside it, he regularly let things be and postpone until part ...
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1answer
93 views

How do you visualize $\mathcal{P}(1)$ in constructive mathematics?

If I understand correctly, constructive mathematics doesn't prove that the powerset $\mathcal{P}(X)$ of a set $X$ is a Boolean algebra; in general, all we can say is that its a Heyting algebra. This ...
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29 views

Visualizing Nash Equilibria of a 4 dimensional matrix

Are there any good ways to visualize Nash equilibria of a 4-d matrix? I have created an game theory model which consists of of four players (P1; P2; P3; P4) who can all choose between a set of 27 ...
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1answer
41 views

What is visual cryptography?

Question: 1. What is visual cryptography? 2. How does it work for secret image sharing? Attempt: I have tried to understand the concept of secret image sharing for black and white pixel from here ...
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1answer
24 views

Visualizing the set of points whose coordinates sum to zero

Consider the set $$S = \left\{(x_1, x_2, \dots, x_n) \in \mathbb{R}^n \;:\; \sum\limits_{i = 1}^n x_i = 0\right\}$$ I can see that in $1$D, we just have $x_1 = 0$. In $2$D, we have the line $x_2 = ...
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109 views

Adding grid on 3D surface in GeoGebra 5

I have made a file with GeoGebra where I visualize a conical intersection. The standard appearance of the surfaces is a dull, colored surface, like shown in the attached figure. To better get a sense ...
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3answers
141 views

How to visualize a $120^\circ$ (or $240^\circ$) rotation of a cube about its body diagonal?

I'm finding rotational symmetries of a cube and have some difficulties with visualizing $120^\circ$ or $240^\circ$ rotations. Any tips?
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3answers
243 views

What's the intuition behind the identities $\cos(z)= \cosh(iz)$ and $\sin(z)=-i\sinh(iz)$?

I'm trying to understand in an intuitive manner the relationship between the circular and hyperbolic functions in the complex plane, i.e.: $$\cos(z)= \cosh(iz)$$ $$\sin(z)=-i\sinh(iz)$$ where $z$ is ...
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2answers
51 views

How to test if vectors are equidistributed on the unit sphere

I can create a large collection of normalized real valued $n$-dimensional vectors from some random process which I hypothesis should be equidistributed on the unit sphere. I would like to test this ...
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1answer
55 views

Homeomorphism $\phi : T^2/A \to X/B$. What are $ T^2/A$ and $X/B$?

The question I am working on asks me to construct a homeomorphism $\phi : T^2/A \to X/B$ where $T^2$, $A$, $X$ and $B$ are given as follows: $T^2=S^1 \times S^1$ and $A \subset T^2$ is given by ...
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1answer
22 views

Graph a function of two variables online, with the output rendered as color

Are there any online utilities to graph a function of two variables online, with the output rendered as color in some way (for example, red to green, black to white, hue, etc...) Wolfram Alpha comes ...
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22 views

Parametric equations and specifications of a logarithmic triskelion (triple spiral)

There is a post in this forum that shows how to create an Archimedean triskelion: Parametric equations and specifications of a triskelion (triple spiral) ...
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1answer
44 views

Recursive formula for a visual pattern

I was looking at some of the examples at visualpatterns.org and coming up with explicit and recursive formulas for various aspects of the patterns. Consider the pattern below and the number of cubes ...
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2answers
31 views

Is this histogram considered bimodal?

Is this histogram bimodal? Because when I google what a bimodal histogram looks like, I keep getting images that say histograms like these are considered bimodal. Isn't it unimodal because the ...
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2answers
23 views

How to represent the following data in a pie chart.

In the following diagram are reported the sports practiced by alumns of a school. $$ \begin{array}{c|lcr} \text{Sport} & \text{Alumns($n^\circ$)} \\ \hline \text{Soccer} &15 \\ ...
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73 views

Pictures of curves over finite fields with many points

At the manypoint page for $2^3$, genus=3, there is the note: "In his Harvard notes, Serre notes that a model of the Klein curve gives an example of a genus-3 curve with 24 points over $F_8$: $(x + y ...
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1answer
106 views

Visual references for the Riemann-Stieltjes integral.

I've seen a lot of excellent visual material (gifs, pictures) here, in topics like this, and I used many of them to understand/explain concepts (particularly gifs showing Riemann sums or fourier ...
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119 views

How are triangles oriented in a plane?

I was thinking about how the plane in geometry is defined and was wondering, obviously if I draw a triangle, for example, in $2$D I know what it is and looks like as I have defined an orientation ...
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1answer
42 views

Visualizing Normal Subgroups

I'm reading Nathan Carter's "Visual Group Theory" and I'm a little stuck on his choice of words and want to make sure I understand his statement correctly. It concerns normal subgroups and is stated ...
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1answer
31 views

How to visualize complex domains

I was hoping if someone can help me visualize complex domains. I know how simplex ones like $|z|<1$ or $\text{Re}z < 1$ look like but for the more complicated ones such as $$\text{Im } z < ...
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2answers
39 views

First isomorphism theorem visualisation on cyclic groups such as $C_4$?

I want to demonstrate the first isomorphism theorem on cyclic groups such as $C_4$. I find it hard to see how this map works. $C_4$ has the cycle $(13)(24)$. The cycle is a composition function so ...
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28 views

Are there useful visual representations of magmas?

In group theory we have Cayley graphs. Are there analogous or anyway useful visual representations of magma structures? I am unsure about how to construct a graph representing, for instance, a free ...
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22 views

Visual explanation of taylor polynomials

I've just been trying to understand taylor polynomials more intuitively from a visual perspective. But as soon as the terms start using second derivatives, it becomes unclear. 1, Some thought's I've ...
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13 views

How to visualize the product of two segments lengths? [duplicate]

So there seems to be easy ways to visualize addition. If three points a, b, and c are on the same straight line respectively, we can say that the sum of the lengths of ...
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1answer
36 views

How do you picture: $\Pr(B|A)$ shrunk down by $\Pr(A)$?

Though understanding these diagrams, I do not understand how to visualise the following explanation: $\color{green}{[P1.]}$ Suppose you were to grab the edges of $A$ and stretch it out so it ...
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1answer
72 views

The role of visualization and intuition in graduate and postgraduate math and developing it

In Visual Complex Analysis's preface, the author gives an analogy with pseudo-deaf musicians and follows the same to mathematics. Mathmatics today, he argues, is mostly build on abstract symbolic ...
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31 views

How can you picture Conditional Probability in 3D?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I modified the following's source for concision. $1.$ Now look at ...
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1answer
37 views

How can you picture Conditional Probability in a 2D Venn Diagram?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I pursue only intuition; do not answer with formal proofs. Which ...
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1answer
38 views

A good pictorial explanation of separation of variables?

I'm teaching ordinary differential equations for the first time, and I would like to give a compelling visual explanation of why it makes sense to "multiply by $dx$" and integrate when you want to ...
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2answers
72 views

How can I visualize a “span” of a set of vectors?

Can someone please help me visualize this concept, I know already what span of set of vectors is, but I am interested to know how it looks visually, thanks.
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3answers
55 views

Visualizing $180^\circ$ rotational symmetries of a tetrahedron

I am trying to learn about the symmetries of a regular tetrahedron. I understand the identity and all eight $120^\circ$ rotations that keep one vertex fixed, ...
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2answers
103 views

Intuitive or visual understanding of the real projective plane

If we take the definition of a real projective space $\mathbb{R}\mathrm{P}^n$ as the space $S^n$ modulo the antipodal map ($x\sim -x$), it is possible to see that $\mathbb{R}\mathrm{P}^1$ is ...
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4answers
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To prove $\frac{n!}{p!q!}$. [closed]

The number of permutaion of n objects, where p and are of one kind,q are of second kind,are of different kind is $\frac{n!}{p!q!}$ How can we proove above theorem $\frac{n!}{p!q!}$.I tried to prove ...
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43 views

Recommendations on visualizing basic linear algebra

I am teaching linear algebra this semester, and I would really like to recommend my students some cool youtube videos visualizing some simple stuff like the span of a set of vectors, linear ...
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Is there a program that receives as input your drawing of a curve and outputs a parametric curve tracing it (reasonably close)?

From what I know, B-Splines is the closest thing that we have to drawing curves and having them defined by the computer. I have some B-Spline code that does this interactively. However, those are a ...
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45 views

Fourier Series and epicycles - How to extract the radii and angular velocities from the Fourier Series expansion of a function.

NOTE: I am attaching Mathematica code for those who may want to check it out and understand what I'm asking for. The rest of the question is pretty mathematical in nature, I'll also try the ...
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3answers
70 views

Visualizing linear transformations on vector fields

I'm trying to figure out what it means to apply a linear transformation to a vector field geometrically. So I start with the easiest geometrically interesting transformation: a rotation. Using ...
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1answer
43 views

An image suitable for a skyscraper sheaf?

This question relates to this thread: Skyscraper sheaf? Consider one of the diagramms for the representation of a sheaf (and stalks thereof) which are popular on the web: I just wanted to know ...
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65 views

Visualizing the geometric product?

The exterior product between blades has a relatively clear geometric interpretation: it gives the result of "extending" one factor along the other, with the direction pointing along the first factor ...
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1answer
79 views

Skyscraper sheaf?

I was wondering whether someone could provide me with an easy definition of a skyscraper sheaf and, more importantly, with a visualization of it. Thanks in advance.
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1answer
31 views

Is there a relation between isometric and orthographic measurements?

This image shows a couple of different isometric projections. In the black shows the figure's "true" dimensions in an orthographic projection while the red shows the dimensions in an isometric ...
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28 views

Representing and Visualising Concrete Groups

What are some interesting and attractive ways to visualise specific Groups? Like: The Rubik's Cube, A set of Permutation Matrices, And.. What else?
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22 views

How do I prove this point to line duality?

I need to prove this point to line duality. The thing is, I'm not sure what there is to prove. I guess that I have to prove that if two lines intersect to a point, then by duality their two points ...
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1answer
64 views

Prove: In a finite geometry, if a line has $n$ points, then the total number of points is $n^2 - n + 1$.

Prove: In a finite geometry, if a line has $n$ points, then the total number of points is $n^2 - n + 1$. The following axioms define a finite geometry: ...
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2answers
81 views

How do you draw/visualize a combination table?

This is somewhat related to a previous question I asked I have three variables in a programming function, and a 4th variable depends on these. I have to test the dependent variable against all ...
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2answers
75 views

Can every math problem be visualised? [closed]

Basically, title. Can everything in mathematics be represented graphically one way or another?
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1answer
83 views

Visual questions for 6th graders

I'm tutoring a 6th grader in math at the moment and because she never has a ton of homework I like to give her some interesting extra problems to do. It seems she really enjoyed a problem I showed ...
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About the classification of linear autonomic differential equations of the plane.

For a linear autonomic differential equation of the plane $$\dot x = Ax,$$ with $A ∈ \operatorname{Mat}_{2×2} (ℝ)$, say we have a fundamental matrix $Φ \colon ℝ → \operatorname{Mat}_{2×2} (ℝ)$, that ...
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1answer
27 views

Some explanation regarding a diagram of homotopy

I found this visualization regarding homotpy in wikipedia: https://commons.wikimedia.org/wiki/File:Homotopy_curves.png I would be very grateful if you could explai me all the abbreviations being ...
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Interpretation of $a+b \ | \ a^n + b^n$ for odd $n$

It is not hard to show that $a+b \ | \ a^n + b^n$ for odd $n$. (because $f(x) = x^n - b^n = (x-b)h(x)$ we have $a - b \ | \ a^n - b^n$, so $a - (-b) \ | \ a^n - (-1)^n b^n$) Is there a nice ...