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

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1
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3answers
47 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
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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
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0answers
8 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
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1answer
29 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
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7answers
672 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
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0answers
56 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 ...
5
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0answers
81 views

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

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
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2answers
1k 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
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1answer
56 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
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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
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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) ...
57
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1answer
673 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
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1answer
32 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
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0answers
10 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
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3answers
154 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....), ...
1
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1answer
294 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 ...
8
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2answers
187 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 ...
13
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1answer
180 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
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0answers
19 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
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6answers
117 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
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2answers
174 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 ...
29
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1answer
518 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
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2answers
246 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
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2answers
94 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
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1answer
355 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
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0answers
315 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
522 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
570 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
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1answer
68 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
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2answers
545 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
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1answer
37 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
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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 ...
10
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0answers
138 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
65 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
31 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
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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
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1answer
189 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
200 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
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0answers
15 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 ...
1
vote
2answers
58 views

Visualizing why rotations preserve orientation

It's clear geometrically that if you have two vectors in $\mathbb{R}^3$ a rotation will preserve their lengths and the angle between them. But how do you visualize that a rotation preserves ...
3
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0answers
117 views

Visualising surface integrals

For a current problem I am working on, I have run into angular surface integrals, i.e. the differential solid angle $\text{d}\Omega$. Specifically the surface integrals are defined by ...
0
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0answers
7 views

How common is it to plot linear transforms on continuous spaces?

I came across a question asking how the proof that the transform $f(g) = g(x-a) + g(x+a)$ was Hermitian worked. I was a bit surprised that the proof was symbolic, because if you plot out the ...
0
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0answers
33 views

Expressing Identities about Matrix of Linear Transformation

If $T\colon V\rightarrow W$ and $S\colon W\rightarrow U$ are linear transformations, then I will consider their composition in the order $S\circ T$ (i.e. $S\circ T(v)=S(T(v))$ ). Given basis $B_1$ ...
0
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1answer
37 views

How to represent a network of chemical reactions?

I am trying to figure out a suitable representation for given set of chemical reactions, which happen over an exhaustive set of chemicals. The chemicals are $A, B, C, D, E$ and the reactions are ...
80
votes
20answers
5k 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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3answers
56 views

Open Source Software for Creating Mathematical Diagrams

I work as a software engineer at a company developing navigation systems. As I have a mathematical background I normally get assigned the more mathematical problems and I find myself regularly having ...
2
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1answer
55 views

Visualising something geometrically

$W=B(x_{1},r)\cap B(x_{2},r)$. The boundary of the intersection is given by the union of $\delta_{1}W=\delta B(x_{1},r)\cap B(x_2,r)$ and $\delta_{2}W=B(x_1,r)\cap \delta B(x_{2},r)$. Let ...
1
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0answers
9 views

Visualization of the fact that the integers defining lens spaces must be coprime

This is related to this question I asked: Visualization of Lens Spaces and is also related to this question by @Earthliŋ: Why are the integers appearing in lens spaces coprime? I understand the ...
7
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1answer
135 views

Visualization of Lens Spaces

I am trying to visualize lens spaces geometrically. While I am aware of the fact that most manifolds which cannot be embedded in $\mathbb{R}^3$ are hard to visualize because of the obvious ...