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

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3
votes
7answers
888 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
1answer
44 views

Emil Artin on visualization of matrices

Someone called my attention to the fact that Emil Artin made very important remarks on the visual representation of matrices in some of his books. Could anyone tell me which precise book that is? ...
2
votes
0answers
30 views

Hyperplane in a complex vector space

This is my first question on MSE, I'm sorry if there already exists similar questions, I couldn't manage to find it. My friend, who studies Physics, asked me about the meaning of "functional" so I ...
4
votes
2answers
42 views

What does Dini continuity mean?

What does Dini continuity (the integral condition) mean visually? Description of Dini contuity: https://en.wikipedia.org/wiki/Dini_continuity
5
votes
3answers
344 views

Can one construct a “Cayley diagram” that lacks only an inverse?

My group theory text asks for an example of a Cayley-like diagram that exhibits all the properties of a group except (only) that at least some elements lack an inverse. Is it possible to construct ...
2
votes
1answer
135 views

Turning real roots into curves (for visualisation)

One can obviously map a set of real numbers $x_1, x_2, \ldots x_N$ to a curve in 2-D via $y=(x-x_1)(x-x_2)\ldots(x-x_N)$. Thinking about data visualisation, one can portray a set of $N$ observations ...
0
votes
1answer
53 views

Is there a way to visualize, like a picture in mind, the $n$-th derivative?

Is there a way to visualize (like a picture in mind) the $n$-th derivative ? For $n=1$ is the tangent line and we can visualize it quite well. More abstractly is it possible to see the geometric ...
0
votes
2answers
54 views

Visualisation of the reciprocal of an continued fraction?

If: $$a=\cfrac{l}{m+\cfrac{n}{o+\cfrac{p}{q+\cdots}}}$$ Then could you help me visualize $1/a$? I really don't understand it. Thank you so much!
6
votes
1answer
74 views

Animation of Weierstrass $\wp$-function as a map from a torus to the sphere?

I am wondering if there exists somewhere an "animation" of one such map (for some lattice / torus), in the style of the kind of $z \mapsto z^2$ maps one encounters in complex analysis classes (one can ...
0
votes
0answers
58 views

How do I visualize this quotient space?

If $V = [0,1] \times [0,1] \subset \mathbb{R}^2$. We define the equivalence relation $\sim$ on $V$ as follows: every element $(x,y) \in V$ is equivalent with itself and besides that the three ...
1
vote
1answer
295 views

Characterize normal subgroups - Find all subgroups of $S_3$ conjugate to $\{id, (1,3) \}$ - Fraleigh p. 143 14.29

(27.) A subgroup H is conjugate to a subgroup K of a group G (viz. p. 141 $K \le G$ is a conjugate subgroup of $H$), if $i_g[H] = gHg^{-1} =K$ for some $g \in G$. Show that conjugacy is an ...
1
vote
3answers
49 views

Need visualization advice for learning partial derivatives and calculus with more than one variable.

Okay so I just recently started learning calculus with more than one variable and whilst I'm coming to grips with many of the ideas and stuff I'm finding it difficult to visualize certain things for ...
0
votes
0answers
7 views

What is angular acuity?

I've been looking for the answer for a while now, but just can't seem to find it. Is it the same as FOV (Field Of View)?
0
votes
0answers
11 views

How to calculate viewing angles?

Forgive me if this is in the wrong forum, but I was not sure where to ask. Say I have a screen: 1920 x 1080 (27 inch diagonal) How do I work out the viewing angles (horizontal and vertical)? For ...
4
votes
1answer
42 views

Drawing an approximation to a circle in isometric projection

A circle viewed from from the side is an ellipse. A common approximation can be found on the web (eg do a google image search for isometric circle). This produces something like (with arc centers ...
13
votes
7answers
694 views

How should one picture a topology/ topological space?

I can form a mental image of sets with structures like metrics or norms. But if I try to picture a topology/ topological space I fail every time. The information provided in Wikipedia confuses me ...
1
vote
0answers
59 views

How do I evaluate this sum :$\sum_{n=0}^\infty z^{n^3}$ and Is there a visual proof for it?

if $$\sum_{n=0}^\infty z^n = \frac1{1-z}, \quad z \in \mathbb{C},\; |z| < 1 .$$ then is there a way to deduce this sum:$$\sum_{n=0}^\infty z^{n^3}$$ from the above Identitie or any visual proof ...
2
votes
0answers
119 views

What are some of the iconic mathematical images ever done in 2D, 3D and 4D? I've mentioned just a few [closed]

Since I'm on my school break I'm trying to visualize mathematics, I encountered a visualization of the following(3D). Deformation of the Riemann surface of an Algebraic function by Anatoly ...
1
vote
2answers
2k 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.
3
votes
1answer
69 views

Visual approach to abstract algebra

I'm currently finding abstract algebra to be very fascinating. However, one of the things that pulls me back is that I sometimes find it hard to understand something visually. For example, one could ...
7
votes
1answer
2k views

How to visualize the Gaussian curvature of a 3D surface using color?

I have a 3D surface. I want to visualize color-coded Gaussian curvature. Is there any software (e.g. MATLAB, Mathematica) which can be used for calculating and visualizing the curvature in color code ...
4
votes
3answers
1k 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) ...
58
votes
1answer
679 views

Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles?

The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a ...
2
votes
1answer
34 views

In $\Bbb R^3$, is there a general principle governing these “visual” angles?

I believe most of you have drawn the xyz coordinate system hundreds of times and so have I. You may have drawn it like these, on various occasions: (the reverse directions of the axis are not shown.) ...
0
votes
0answers
11 views

A better way to visualize the change of output with respect to input of 3-dimentional data sets?

Background (1) I have a system $Ax=b$. Let us assume we do not know any information about $A$. (2) Both $x$ and $b$ are 3 dimensional data. In terms of physical meanings, both of them describes ...
5
votes
3answers
180 views

Visual representation of matrices

I am used to seeing most basic mathematical objects being visually represented (for instance, a curve in the plane divided by the xy axis; the same goes for complex numbers, vectors, and so on....), ...
8
votes
2answers
188 views

On visualizing the spaces $\Bbb{S}_{++}^n$ and $\Bbb{R}^n\times\Bbb{S}_{++}^n$ for $n=1,2,\ldots$

Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive-definite $n\times n$ real matrices. I am looking for hints concerning the visualization of such spaces for $n=1,2,\ldots$. I know that ...
14
votes
1answer
199 views

Textbooks for visual learners

Perhaps this question has already been asked (if so, please let me know) but I am looking for books that appeal to visual learners. I discovered that I am able to understand concepts much quicker ...
1
vote
0answers
24 views

How to visualize orientation of 3d objects

The way I visualize orientations of $1$- and $2$-dimensional objects is by an ant walking along a path. For a $1$d object (like a line/ line segment/ etc), just place the ant on the line and confine ...
4
votes
6answers
125 views

Visualize $z+\frac{1}{z} \ge 2$

As we know, always $$z+\frac{1}{z} \ge 2,~~~~~~~~~ z\in \mathbb{R}^+$$ However, is there any geometric way to visualize this equation for some one who is not that expert in math? I know this ...
9
votes
2answers
179 views

Visual Proofs of Series Summations

I'd like to put together a compilation of visually geometric proofs of series summations. I have three famous 2D examples to clarify what I mean below, but other "visually geometric" proofs of an ...
30
votes
1answer
537 views

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

In a 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 ...
17
votes
2answers
249 views

Enumerating Bianchi circles

Background: Katherine Stange describes Schmidt arrangements in "Visualising the arithmetic of imaginary quadratic fields", arXiv:1410.0417. Given an imaginary quadratic field $K$, we study the Bianchi ...
7
votes
2answers
98 views

How to visualize $\mathbb{C}^2$?

In a homework question I had to do, the rotational matrix $A = \begin{pmatrix} 0&-1\\1&0 \end{pmatrix}$ was given. Its eigenvalues in $\mathbb{C}$ are $i$ and $-i$. The set of all eigenvectors ...
11
votes
1answer
374 views

Kakeya Needle problem video

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid: What I'm searching for is a visualisation of the ...
2
votes
2answers
33 views

Inverse mapping on a set $U_1\times U_2$, wrong intuition?

Let $f(x) = (f_1(x),f_2(x))$ where $f: X\to Y_1\times Y_2$. And $f_1:X\to Y_1, f_2: X\to Y_2$ where $X,Y_1,Y_2$ are topological spaces. I want to prove some continuity properties, but my ...
5
votes
0answers
320 views

Visualizing the domain of the square root

I would like to show someone the domain of the complex square root function (the 2-sheeted riemann surface). Is there a good interactive visualization software for this? I would like some sort of ...
9
votes
7answers
529 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 ...
22
votes
2answers
572 views

Can I represent groups geometrically?

I have just taken up abstract algebra for my college and my professor was giving me an introduction to groups, but since I like geometric definitions or ways of looking at stuff, I kept thinking, "How ...
2
votes
1answer
74 views

Are there 3D geometric proofs of Fibonacci identities?

There is a significant number of identities involving Fibonacci numbers that can be proven in a sort of geometric way, as it is shown in the following picture: However, I couldn't find any such ...
24
votes
2answers
549 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 ...
3
votes
1answer
44 views

No extremals satisfying the Euler equation - what does it mean?

Consider the functional $J[y] = \int_{0}^{1}xyy^{'}dx$. If I want to find extremals (a function $y=y(x)$ that makes the functional stationary) with boundary condition $y(0)=0$ , $y(1)=1$ for this ...
0
votes
0answers
16 views

How to visualise the step where $T(\mathbf{e_j})=\sum_i^n{a_{ij}\mathbf{f_i}}$?

Consider a linear map $T:\mathbb{F}^n \rightarrow \mathbb{F}^m$ and let $\{\mathbf{e_j}:j\in[1,n]\}$ be the standard basis for $\mathbb{F}^n$,$\{\mathbf{f_i}:i\in[1,m]\}$ be the standard basis for ...
11
votes
0answers
147 views

Is it a good approach to heavily depend on visualization to learn math?

I am a third year undergraduate and I am a beginner on these "real mathematics" (no pun intended). Before contacting the "real math", my math level should be considered to be "good", although I was ...
2
votes
1answer
66 views

Soft Question: Difficult to Visualize Areas of Mathematics

Yesterday I came across this webpage, which describes a recent (successful) attempt to visualize isometric embeddings of flat tori in 3D Euclidean space. The webpage and associated paper discuss the ...
1
vote
1answer
33 views

Is it possible to estimate the number of primes between 0 and a 128 bit number?

I'm attempting to visualize an RSA public/private key pair, or a SHA2 hash. In order to reduce that massive number that is meaningful to humans I'm looking at this SHA2 visualization function to ...
0
votes
0answers
18 views

Spaces of visual patterns, but not recurse/chaos.

I'm looking for information on existing/notable, spaces of visual patterns, that do not rely on, or appear to make much use of, recursion/chaos to function, as a cellular automata or fractal would. ...
0
votes
1answer
194 views

Cauchy's Generalized Mean Value Theorem. Required function. (S.A. pp 140 t5.3.5)

Cohen, Henle. Calculus pp 827, (http://www.vias.org/calculus/09_infinite_series_10_06.html) I revised the footnote in pp 14 http://math.uga.edu/~pete/2400calc2.pdf. This theorem can be illustrated ...
2
votes
2answers
259 views

How to read a cycle graph?

As an important tool for visualizing some small finite groups it is useful to know how read such graph, and with time trying to make sketch of them by my own. I would like to know, for a start, how ...
0
votes
0answers
18 views

Why the gradient of the r vector is the identity map, geometrically speaking?

When doing some simple quantum mechanics problem involving commutators, I forgot the result of this expression $$\left[\vec{r} ,\hat{p}\right]$$ Thus I then brute force it using the definition of ...