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

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0
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1answer
27 views

How to visualize change over time of multiple items at once?

I'm writing a recommendation system for a news website. At any point in time, there are between 50-100 articles that are candidates for recommendation based on the algorithm, etc. What I'm trying to ...
6
votes
1answer
93 views

what is the nature of a ball that goes over a “corner” of the real projective plane?

I'm make a little computer program to help me understand different 2d topological spaces, (such as torus and mobius band). I'm having issues with drawing balls that go over a corner of the real ...
3
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0answers
132 views

What methods are known to visualize patterns in the set of real roots of quadratic equations?

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations $ax^...
96
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20answers
20k views

Visually deceptive “proofs” which are mathematically wrong

Related: Visually stunning math concepts which are easy to explain Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually ...
0
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1answer
24 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?
2
votes
1answer
72 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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3answers
44 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 ...
0
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0answers
27 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 ...
2
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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
191 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 ...
12
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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 ...
97
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21answers
7k 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 ...
2
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0answers
28 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 ...
2
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0answers
39 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 ...
0
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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 ...
1
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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 ...
5
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0answers
138 views

Is there a way to visualize a group?

Is there a way to picture a group in ones head? I want to "see" the difference between abelian and non-abelian group. And if $f$ is a group homomorphism, is there a way to see that $\ker(f)=1\...
1
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2answers
157 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 ...
0
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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: $$ \...
1
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0answers
39 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 ...
3
votes
1answer
99 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 ...
0
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0answers
35 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 ...
2
votes
0answers
77 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 ...
1
vote
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 ...
0
votes
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 = ...
9
votes
3answers
263 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 ...
0
votes
0answers
188 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 ...
58
votes
2answers
1k views

Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?

In his gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$: Proof of equality ...
2
votes
3answers
215 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?
6
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1answer
84 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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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^...
2
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2answers
53 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 ...
3
votes
1answer
141 views

Infer distance from a point to a line, from the distance from a point to a plane [Stewart P793 12.4.44]

I'm able to prove $44$, but how would one deduce $43$ from it without further industry, forthwith? $43$ seems like a reduced, 2D version of $44$? I'm not enquiring about individual proofs. $44.$ ...
14
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2answers
3k views

What are all these “visualizations” of the 3-sphere?

a 2-sphere is a normal sphere. A 3-sphere is $$ x^2 + y^2 + z^2 + w^2 = 1 $$ My first question is, why isn't the w coordinate just time? I can plot a 4-d sphere in a symbolic math program and ...
1
vote
1answer
25 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 ...
0
votes
1answer
494 views

Determine Cross Product with Left Hand vs Right Hand

If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let: the 1st vector in the cross product = your index finger = in red ...
1
vote
0answers
25 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) ...
2
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2answers
765 views

Parametric equations and specifications of a triskelion (triple spiral)

I haven't been able to find the parametric equations and specifications to form a triskelion, a triple spiral (this is made of three interlocked couples of spirals). Using the parametric equation of ...
0
votes
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 ...
2
votes
2answers
41 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 ...
0
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2answers
35 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{...
4
votes
1answer
114 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
74 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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0answers
193 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 ...
1
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1answer
47 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 ...
0
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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 ...
0
votes
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|...
4
votes
1answer
2k 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 ...
2
votes
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 ...
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 ...