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

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1answer
22 views

What is a geometric interpretation of multiplication/division in the complex plane? [duplicate]

How can one visualize the multiplication/division of a complex number, z, by a real number, an imaginary number, or another complex number?
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1answer
64 views

Real life illustration of the fact that rationals have measure zero

I wonder if there's any real world phenomenon that reflects the mathematical fact that $\Bbb Q^k$ has Lebesgue measure zero in $\Bbb R^k$, or put another way, the likelihood that we get a rational ...
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0answers
25 views

How can I visualize Quaternion Linear Interpolation?

It’s hard enough to visualize a quaternion, geometrically speaking. A complex number is simple: it’s a point in a plane. Suppose we had a number like this: a + bi + cj I supose you can visualize ...
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1answer
19 views

Non-trivial 3D curve that projects as a line or a segment onto the faces of the quadrant

I want to illustrate how high dimensional objects may have misleading projections. Examples are for instance given with HiSee software, with nD bouquets of circles. Are there non-trivial (not a 3D ...
2
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1answer
187 views

Min-Max Principle $\lambda_n = \inf_{X \in \Phi_n(V)} \{ \sup_{u \in X} \rho(u) \}$ - Explanations

In general, I am generally someone who like to solve questions with visual support. With this idea in mind, is it someone could explain to me, with a visual support if possible, how is it possible to ...
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6answers
2k views

Visual representation of the fact that there are more irrational than rational numbers.

Would anybody know of a visual or even (preferably) geometric representation of this? To make it more specific: Text, symbols and written numbers are predominantly used as labels, and and less to ...
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3answers
40 views

Graphical explanation of the difference between $C^1$ and $C^2$ function?

We are all aware of the intuitive (graphical) explanation of the concepts of continuous and differentiable function. Whenever these two concepts are formally defined, the following elementary ...
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0answers
27 views

How do I visualise the contour integral of a complex function? [duplicate]

I've just learnt about the contour integral of a complex function, but I'm having trouble figuring out what it is calculating visually. I understand it is somewhat analogous to the line integral for ...
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0answers
34 views

Hopf map visualization (animation request)

Let $\phi:D^3\to S^2$ be the composition $D^3\to S^3\to S^2$, the first map being the quotient by the boundary and the second map being the Hopf map. Then: $$f_t:x\mapsto(1-t)x+t\phi(x)$$ is a ...
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0answers
8 views

Why the light contrast of a set $\mathcal{E}$ of VCRG produced by an encryption scheme for a binary image B is defined as follows

In the paper "Image encryption by multiple random grids, Shyong Jian Shyu, 42(7):1582-1596 · July 2009" here, the light contrast of a set $\mathcal{E}$ of VCRG produced by an encryption scheme for a ...
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0answers
19 views

How to sort a list of tuples?

I am neither a mathematician, nor a computer scientist, but I have the following problem, which I cannot solve myself. I have $k$ $i$-tuples $(x_1, x_2, …, x_i)$, where $x ∈ [0,1]$. I need to order ...
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3answers
74 views

Meaning of mathematical symbol $\pm$

What is the meaning of the $\pm$ symbol in relation to this expression? For example, the perceived area of a circle probably grows somewhat more slowly than actual (physical, measured) area: $$ \...
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0answers
38 views

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
97 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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0answers
32 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
43 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
26 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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0answers
163 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
192 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
255 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 $A=S^...
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1answer
24 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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0answers
24 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
46 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
36 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
33 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 \\ \text{...
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0answers
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 +...
4
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1answer
110 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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0answers
170 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
44 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 < 2|...
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2answers
41 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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0answers
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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0answers
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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0answers
14 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 ...
0
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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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0answers
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
41 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 ...
3
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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
73 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
57 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, $(123),(132),(243),(234),(134),(143),(124)...
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2answers
152 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
96 views

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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0answers
45 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 dependence,...
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0answers
23 views

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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0answers
48 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 ...
4
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3answers
80 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 ...