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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0answers
53 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 ...
0
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0answers
12 views

Stellating the Octahedron

I am trying to create a very primitive animation/demonstration that shows the stellation of an octahedron to yield the stella octangula. Unfortunately, it seems that the mental image I have for ...
-1
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0answers
30 views

Visualising a 1-(50,15,15) design.

The problem I have is the visualisation of a 1-(50,15,15) design. That is a set of 50 points and 50 blocks (lines), so that each point is on 15 lines, and each line contains 15 points. My attempts ...
0
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0answers
24 views

How to calculate optimal sizes of rectangles for this type of array visualization?

Given array of positive numbers, I would like to draw this diagram and be able to put descriptions inside: There should be no empty space left, consider that these numbers represent % of total. Do ...
3
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0answers
88 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<=>f ...
2
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0answers
48 views

Ideals in a ring as geometric objects?

I am interested in learing about the possibility of (one-sided) ideals in a ring being repreented geometrically. In other words, about their status as geometric objects (after all, they can be dealt ...
0
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1answer
10 views

Comparing a function and its estimate

What are some clever ways of comparing (visually) a function with its estimate? For regions where the function does not cross zero, plotting the ratio of the functions and plotting the relative error ...
2
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0answers
49 views

What type of diagram could this be?

Some years ago, I saw a diagram somewhat like this somewhere on Wikipedia. I remember that it was supposedly used in some branch of mathematics. Unfortunately, neither Google Image nor trawling ...
0
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1answer
30 views

What is the correct equation for “Normal distribution function of continuous random variable”?

I was reading a book and came across with a equation which gives the normal distribution function of continuous random variable. It was used in a software called ...
2
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1answer
44 views

Alternative geometric interpretation for big-o and little-o

I understand that, in big-o notation, when we say that a function $f$ is $O(x^2)$ we're basically saying that $$|f(x)|\le M |x^2|$$ for some constant $M>0$ and for all $x>x_0$ for some ...
17
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0answers
388 views

Visualizing the Partition numbers (suggestions for visualization techniques)

So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so ...
2
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1answer
99 views

Visualize meaning of quotient in quotient map, group - etc?

What are the reasons for the name "Quotient" in Quotient map, group - etc? Overhead picture shows each of the three cosets in $A_4$ is mapped to a single - gray - node. But this isn't division? ...
21
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4answers
1k views

Visualizing the factorial

Often in basic mathematics, we can visualize things very easily, which I believe helps understanding (instead of just working out a number theoretical proof). For example: $$(n+1)^2 - n^2 = (n+1) +n$$ ...
1
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1answer
298 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 ...
3
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7answers
917 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
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1answer
52 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
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0answers
35 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
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2answers
46 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
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3answers
349 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
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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
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1answer
54 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
58 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
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1answer
79 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
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0answers
59 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
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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
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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
12 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
54 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
712 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
60 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
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0answers
120 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
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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
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1answer
79 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) ...
58
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1answer
686 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
37 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
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
205 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
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2answers
189 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
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1answer
214 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
31 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
131 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
186 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 ...
32
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1answer
559 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
251 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
100 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 ...
10
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1answer
380 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
328 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 ...