Questions tagged [visualization]

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

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25 views

Visualizing Eigenvectors in 2D

The stochastic matrix $$ A=\left( \begin{array} [c]{cc}% 0.9 & 0.2\\ 0.1 & 0.8 \end{array} \right) $$ has a dominant eigenvectors of $u_{1}=\left( \begin{array} [c]{c}% 2\,\\ 1\, \end{array} \...
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1answer
60 views

How to plot a complex function?

We cannot plot graph of a complex function $f:\mathbb {C\to C}$ as it requires $4$ dimensions.But we can show how the mapping transforms the domain plane into image plane.We can draw grid lines ...
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42 views

Generalized julia sets and simple connectedness

BACKGROUND: The Mandelbrot set is the set of complex numbers $c$ for which the sequence obtained by iterating $z \mapsto z^2 + c$ starting at $0$ remains bounded. For a complex number $c$, we can ...
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2answers
33 views

Geometrical explanation / visualisation as to why 2 vectors cannot span R3?

I have already understood as to why 2 vectors cannot span R3 in a simple algebraic manner. This is from an answer to this very question 3 years ago : Why two vectors cannot span ${\bf R}^3$? However, ...
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2answers
64 views

How to show graphically that $\sum_i(1/3)^i$ goes to $1/2$

As everybody knows, it is very easy to show: $$\sum_{i=1}^\infty \frac{1}{2^i} = 1$$ As follows: The coloured parts always show $\frac{1}{2^i}$ and it's easy to see they all come together to fill the ...
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1answer
112 views

Why does a spiral structure appear for this function?

Consider the following 2D vector field on the $xy$-plane $$\vec{V}=\begin{pmatrix} -(m^2-md+x^2)\cos{2d\,t}+xy\sin{2d\,t} \\ -xy\cos{2d\,t}+(m^2-md+y^2)\sin{2d\,t} \end{pmatrix}$$ where $d=\sqrt{x^2+y^...
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1answer
40 views

plotting 2 x 2 matrix visually. What it represents?

Hello I am engineering student. I need help understanding basic of matrix plotting. Currently I am learning about vectors and I wonder how they are represented by a matrix. Suppose we have a 2D vector ...
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1answer
42 views

Can you explain combination with tree data structure [closed]

I can use tree data data structure to understand permutation .Is it possible to understand combination using tree data structure.If so can you please explain it.
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1answer
57 views

Name for the surface formed by taking a filled in square and identifying diagonal vertices?

Consider the surface formed by taking a filled in square and identifying diagonal vertices. For example, $(0,0)\sim(1,1)$ and $(0,1)\sim(1,0).$ Image courtesy of @robjohn: Is there a name for this ...
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0answers
62 views

How do people visualise Algebraic varieties?

The usual way I used to imagine Differentiable Manifolds are, as locally Euclidean spaces, with some nice enough properties. Intuitively speaking, instead of just looking at the manifolds from '...
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1answer
93 views

Visual proof of $\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \frac{1}{4 \times 5} + … = 1$

I am trying to find a visual way of proving that $$\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \frac{1}{4 \times 5} + ... = 1$$ I am familiar with algebraic methods of proving ...
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1answer
29 views

Intuition or Geometric model for Dicyclic Groups?

When studying groups of order 12, I learned about dicyclic groups, of which the quaternion group is the first example, and there is one of every order $4n$ for $n > 1$. I can do the basic ...
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1answer
64 views

Visualizing $\mathbb{S}^{3}$ as consisting of two “polar” great circles together with a family of nested Clifford tori interpolating between them.

The following text describes a construction of the unit sphere $\mathbb{S}^{3}$. To consider the next example, it is helpful to practice visualizing $\mathbb{S}^{3}$ as consisting of two "polar&...
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1answer
51 views

Using PlotGraph command from JupyterViz in MyBinder [closed]

I am executing a program "Newprogram.gap" in GAP file in My Binder. There, I have an output which is an adjacency matrix, $A$ corresponding to a Cayley graph. 1). I need to add the command &...
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1answer
31 views

Calculating the rate of convergence from a plot of a limit.

The rate of convergence represents how quickly a sequence approaches its limit. $$ \lim _{n \rightarrow \infty} \frac{\left|x_{n+1}-x^{*}\right|}{\left|x_{n}-x^{*}\right|^{q}}=\mu $$ Here, the rate of ...
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2answers
39 views

Property of Determinant

If any two rows or columns of a determinant are interchanged, the sign of the value of the determinant is changed. What is the intuition of this? I can see from algebra perspective thats right, but I ...
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9 views

functions that are good “half way log scales” to visualize exponential distribution

I'm making graphs based on frequencies of data -- this is data from the internet and like most things on the internet frequency follows an exponential curve - the most frequent item is orders of ...
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2answers
77 views

3-variable 3-equations problem

For positive reals $x,y,z$, if $xy+yz+xz=\sqrt{\frac pq}$, given $$x^2+xy+y^2=2$$ $$y^2+yz+z^2=1$$ $$z^2+zx+x^2=3$$ find $p-q$, where $p$ and $q$ are relatively prime integers. I tried grouping the ...
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0answers
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Can you help me understand barycentric coordinates visually?

This is my first question here so I'm sorry if it is not a well written/formatted one. I love maths but I have always had a lot of trouble following long equations. It always helps me when visuals are ...
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51 views

Creating a complete number diagram once and for all

Background I have been recently searching for a good number diagram I could save for future reference. I was looking for a picture that would show different types or real numbers, and also ...
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4answers
79 views

Prove that the area of a right triangle is no more than the square of the hypotenuse divided by 4

I'm struggling with this proof. Let $a,b,c$ denote the base, height, and hypotenuse of a right triangle. Let $A$ denote area of this triangle We have the following relations: $$ a^2 + b^2 = c^2 \\ A =...
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2answers
66 views

How to imagine (or visualize) the convergence of $x,\; f(x),\; f(f(x)),\; f(f(f(x))), \dots\; $?

I found a problem (Calculus 3rd edition by Michael Spivak, Chapter 22 question 21) that goes as follows: 21. (a) Suppose that $f$ is continuous on $[0,1]$ and that $0\leq f(x) \leq 1$ for all $x$ in $...
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3answers
59 views

How to draw the graph of $x^2+(y-(x^2)^{1/3})^2=1$ without WolframAlpha? [closed]

How to draw the graph of $x^2+(y-(x^2)^{1/3})^2=1$ without WolframAlpha?
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1answer
46 views

How to visualize a Markov chain in the following way?

Nice to meet you, this is my first question in this site. I now can draw $A\to B\to C$ but stuck at $A\to B\to C\to D$. Let me show you my graph: $A\to B\to C$. It's easy to see that $A$, $B$ and $C$ ...
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1answer
53 views

Integral and differential calculus, after a good (and mathematically correct) explanation of the two.

My son is beginning his final year of secondary school this year and one of the major components is calculus. He's always been better at visual methods and I'm struggling to remember how I learnt this ...
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0answers
42 views

What Geometry Does This Mistaken Constrained Optimization Approach Suggest?

As this answer illustrates, a simple constrained optimization problem can be geometrically represented as finding the extrema of the intersection of two surfaces, where the surface representing the ...
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1answer
38 views

Calc 2 visualizing equations in 3D

Is there a way to create and visualize a 3D object given an equation? I am currently studying calc 2 and would really like to be able to interact or view the equations I am given. For example, I have ...
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3answers
129 views

Visual proof for $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

I've found visualizations of simple mathematical concepts to be a really useful tool in building intuition for more complex mathematical concepts. For example, this visualization of $(a + b)(c + d) = ...
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0answers
28 views

Visualizing convolution

I am taking a graduate level course in digital signal processing and am in the process of reviewing my notes from a past signals and systems course. One concept I struggled with conceptually is ...
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3answers
53 views

Visualizing infinitesimals

I am having problems visualizing infinitesimals. Like in a curve, when we integrate we can adjust in it a perfect infinitely thin rectangle. But how do I visualize it? Every time I try to visualize it,...
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0answers
12 views

Visualisation in Fourier Space

I was wondering...Just like points in Cartesian space represent solid bodies and other things, what do I make of a 3D object in Fourier space. I'm assuming Fourier space can also be visualised just ...
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0answers
69 views

How can I visualize this Compound Interest Chart with indefinite integrals?

How can I visualize the integrals below? Can someone draw on the chart and point to where $\int 1 \, dx = x$, $\int x \, dx = \frac 12 x^2 $, ... are ? I don't know which of these charts are more ...
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1answer
159 views

How can I visualize $\lim\limits_{x \rightarrow \pm \infty} f(x) = \lim\limits_{t \rightarrow 0^{\pm}} f(1/t)$?

I'm not asking about the proof, that I already understand. I'm itching to understand this graphically. I use James Stewart's graphs. As you can see, I wrote $1/t$. Then what? Also I'm trying to ...
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2answers
48 views

How can you behold that a 45 degree line's |x-intercept| = |y-intercept|?

I'm not asking for an algebraic explanation...I can do algebra. I know $f(x) = x - c = 0 \iff x = c$. Right now, I have to make my mind process four steps. First, a line with slope $|1|$ must angle 45°...
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1answer
28 views

How to determine shape of level curves of $V(y,z)= \ln{ \left( \frac{y^2 + (z+z_o)^2}{y^2 + (z-z_o)^2}\right)} $?

Context: I expect that a circle will be written as $$R^2 = (x-a)^2 + (y-b)^2,$$ where $R$ is the radius, and $(a,b)$ is the center of the circle. Question: How to determine analytically that the level ...
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1answer
39 views

optimizing 3 variable function

I have the following function: m = f(x,y,z) = (a * (1 + x/100) * b - d * (1 + y/100)) / (h * (1 + z/100)) where a, d, h, b are positive constants and the variables ...
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1answer
81 views

Web based app for differential geometry visualization

Is there by any chance a web based app for viewing parametric curves and surfaces that arise in differential geometry to help visualization? Thanks.
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2answers
204 views

Simple physical intuition for differential 1-form of a scalar function acting on a tangent vector?

I'm trying to gain some very basic physical intuition as to how the differential 1-form $df$ of a smooth function $f$ acts on a tangent vector $\mathbf{v}$. Say the temperature of a hot plate is given ...
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0answers
9 views

What's the best measure of central tendency for speed graphs.

In my program, I'm getting the network speed measurement every second, to save the memory, we average (mean) like 10 continuous measurement and only store mean values in the data model. For example ...
2
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3answers
115 views

Genus of a graph and planarity

In 'Introduction to graph theory' by Trudeau it is proved the theorem Every graph has a genus by first embedding the graph onto a sphere , and then adding handles, say we use $k$ of them, where the ...
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1answer
116 views

How to intuitively understand the $n$-dimensional cube as the dimension grows large [duplicate]

So I read* that for the convex body, i.e. the cube $[-1,1]^n$ in $\mathbb{R}^n$, the smallest ball containing it has radius $\sqrt{n}$, while the largest ball inside the cube has radius $1$. Also, &...
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1answer
28 views

How to sketch $f(z) = \frac{1}{64}i \times z$ on the complex plane?

How do I sketch a plane of points on the complex plane that is multiplied by $\frac{1}{64}i$? I mean $f(z) = \frac{1}{64}i\times z$. Do I just rotate my existing plane by $\frac{1}{64}\times\frac{\pi}{...
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2answers
116 views

What does it mean for the sun to subtend to a solid angle? [closed]

I am reading Feynman's Six Easy Pieces and he says the following in the "Theory of Gravitation" chapter (emphasis mine): When the sun is nearby, the particles coming toward the earth ...
2
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1answer
60 views

Visual understanding of $\frac{\mathbb R} {\mathbb Q} $

Consider the quotient set $\frac{\mathbb R} {\mathbb Q} $ obtained by the equivalence relation $$ x \equiv y \mod \mathbb Q \quad \text{iff} \quad x-y \in \mathbb Q $$ I was wondering if there exists ...
2
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1answer
58 views

Seeking a specific book with graphs of interesting functions, but can't remember the name

I am looking for a specific book that was recommended on youtube in either a Numberphile or Matt Parker video. it was a visual dictionary of pictures of interesting plots of math functions. the cover ...
2
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2answers
56 views

Visualizing complex roots of a quadratic equation

Is there some way to visualize or plot Complex roots of a quadratic equation ( of real coefficients): $$ y= x^2 + v x + c =0 $$ whose product is constant? .. in a complex plane? Shown are cases ...
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2answers
90 views

Is there a graphical proof that $9n+1$ is triangular if $n$ is?

It is straightforward to prove that if $n$ is a triangular number then $9n+1$ must be. Is there a systematic decomposition of a triangle made of $9n+1$ dots into nine triangles of $n$ dots plus one ...
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1answer
110 views

How to visualize $\dfrac{a}{b} \div \dfrac{c}{d} \equiv \dfrac{a}{b} \times \dfrac{d}{c}$?

I read https://matheducators.stackexchange.com/q/7837, but it doesn't answer my question because the Multiplicand is a whole integer there. The trick is to rationalize the denominator, to enable me ...
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0answers
15 views

How to visualize division as splitting Dividend into B equal “partial groups”, then rounding up A partial groups to get a full group?

Because $10 \div \dfrac{4}{3}$ isn't an integer, I changed the numbers in this Reddit post. My $X = 6, A = 2, B = 3$. Undoubtedly, I know $6 \div \dfrac{2}{3} = 6 \times \dfrac{3}{2} = 9$, but don'...
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0answers
35 views

How can I visualize dividing by a fraction as partial and full groups?

I can't visualize all these "full groups" and "partial groups." Can someone please picture them? Please see my questions red in-line. Please don't use the identity that $\dfrac{a}{...

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