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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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I attempted to visualize dot product of complex vectors. What do you advice?

I am an electronics undergraduate student currently learning wavelets. In the book A First Course in Wavelets with Fourier Analysis authors first introduce complex vectors and their dot products. Then ...
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0answers
23 views

Visual Intuition: Gaussians closed under addition

I'm trying to develop some intuition for the fact that the family of Gaussian distributions is closed under addition. I.e. if $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$, then $Y = \sum_iX_i$ is also ...
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1answer
65 views

Closed expression and physical interpretation of the median

Opposed to the arithmetic mean there is no immediate closed expression for the median of a distribution $n(x)$ of a variable $x\in\mathbb{N}$ over a population of $N$ items, at least not when ...
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1answer
19 views

How can I visualize points saved in csv file in Graphing Calculator 3D [closed]

I am new to Graphing Calculator 3D, I have some data files with lines of data points look like: pointA (1.0, 0.0, 1.0), pointB (2.0, 0.0, 2.0), time:19:32:22 pointA (1.1, 0.0, 1.0), pointB (2.1, 0.0,...
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0answers
45 views

What's the geometric interpretation of the square root of a matrix?

Question: If I have a matrix $A$, I know that its square root is a matrix that has the same eigenvectors as $A$ but its eigenvalues are the square roots of the eigenvalues of $A$. What does this ...
151
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23answers
12k 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 interpretation ...
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2answers
20 views

Area of revolution of a square

A square of side length 1 is rotated 360 degrees about one of its vertices. What is the area of the region the square covers while rotating? I don't know how to visualize this as a geometric shape ...
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3answers
538 views

Easy visualizations of small countable ordinals

The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as ...
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1answer
3k 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 ...
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2answers
83 views

Asymptotic expansion of $Li^{-1}$ and zeros of $F(s)$ and $G(s)$

If you downvote please leave some constructive feedback. I would like to compare and visualize/gain insight about the zeros of two functions, $F(s)$ and $G(s).$ $\pi(m)$ is the prime counting ...
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0answers
37 views

Educational gif-animations on mathematical analysis [closed]

I drew some simplified gif-animations by math. Do you think, they will help in the study of mathematics to people, who have difficulties with this?
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0answers
72 views

A summation of the multiplication of reflected Mobius functions and their behavior for different values of $k$

$$ \Psi_k(N)=\sum_{n=1}^{N} \mu(n)\mu(k-n)$$ where $\mu(n)$ is the Mobius function. This function is interesting to me because for the case of $k=N$ it has the symmetric property of being odd with ...
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2answers
100 views

How to see if a subgroup is normal from Cayley graph

Let be a Cayley diagram of group $G$. Let $H$ be the orbit of element p. Is $H$ a normal subgroup of $G$? Is there a simple way to check that because going by definition seems complicated. I tried ...
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0answers
54 views

Visualising connectedness in topology

Recently started a topology course, I am struggling to visualise connected topological spaces in topology, I am understanding the definition when it comes to clopen subsets and disjointness, ie $[0,1) ...
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1answer
787 views

Proof without words of $\oint zdz = 0$ and $\oint dz/z = 2\pi i$

I found this visual "proof" of $\oint zdz = 0$ and $\oint dz/z = 2\pi i$ quite compelling and first want to share it with you. But I have a real question, too, which I will ask at the end of this post,...
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4answers
2k 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....), ...
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0answers
42 views

Visually suggestive way to present a finite group

When studying about groups, we can often grasp the structure of a small group "internally". For instance, we can "see" that $\mathbb Z/3\mathbb Z$ as a three-fold symmetric shape;more complicated ...
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187 views

Figures and Numbers: Relating properties of geometric shapes and their Fourier series

Consider two types of parametrized curves $\gamma:[0,2\pi]\rightarrow \mathbb{R}^2$ open curves $\gamma_\sim(t) = (t,a(t) + b(t))$ closed curves $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$ ...
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1answer
218 views

Strangely but closely related parametrized curves

Compare the following two parametrized curves for $k \in \mathbb{N}^+$: $$x_r(t) = \cos(t)(1 + r\sin(kt))$$ $$y_r(t) = \sin(t)(1 + r\sin(kt))$$ with $0 \leq t < 2\pi$ and $0 \leq r \leq 1$ (being ...
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0answers
51 views

Circle group $S^1$

Can someone describe the circle group $S^1$ in a easily understandable way? Are elements in $S^1$ in the form of $e^{2i\theta\pi}$? What does this look like pictorially? Doesn't necessarily need a ...
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6 views

Unit vector Identicon

Is there a way to visualize multiple n-dimensional ($n\approx300$) unit vectors with the property that large changes to the vector result in large visual changes, but small changes result in little to ...
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1answer
181 views

Derivative of Square Root Visual

Derivative formulas through geometry | Essence of calculus, chapter 3 (3Blue1Brown): https://www.youtube.com/watch?v=S0_qX4VJhMQ There is a challenge at 12:23 asking the viewer to arrive at the ...
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1answer
101 views

The Chaos Game and Benford's law: will I notice the bias?

If I construct a Sierpinsky Gasket using The Chaos Game algorithm, but the random data I am using to decide the vertex of the next step is ruled by Benford's law (so it is partially biased), how would ...
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1answer
26 views

If I have n posts that don't occur at the end of the fence, why are there n + 1 sections? [closed]

https://en.wikipedia.org/wiki/Off-by-one_error#Fencepost_error: More generally, the problem can be stated as follows: If you have n posts, how many sections are there between them? ...
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66answers
493k views

Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are ...
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2answers
126 views

Visualizing complex functions $f: \mathbb{C} \rightarrow \mathbb{C}$

The graph of a complex function $f: \mathbb{C} \rightarrow \mathbb{C}$ is a 3-dimensional object in a 4-dimensional space and thus hard to visualize, even when it's a smooth 3-dimensional surface. A ...
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0answers
16 views

How to display a 2d manifold of points?

I have created a manifold for a set of discrete (integer) points which lie in 2 dimensions. The manfiold has dimensions of $N\times N$ and I have a metric which dictates the seperation between the ...
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1answer
92 views

An Auto-Generated Cartography of Mathematical Theories: Has it been done already?

While looking for a way to visualize the logical structure of mathematical theories a graph-like depiction came to my mind, where propositions are represented by vertices. An edge goes from ...
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0answers
44 views

Is this a new visual for finding the genralized Sum of Positive Integer Powers formulas?

In 2012-2013, I (with experimentation and hard work) founded this image to iteratively compute the formulas for the sum of powers. In 2015, I uploaded this three part video series on it. (In the ...
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1answer
133 views

Intuition behind Well Ordering Principle and Axiom of Choice

I am learning about Axiom of Choice and Well Ordering Principle from Munkres's Topology book, but I can't quite wrap my head around it properly. I have these questions: [Munkres 0.4.3] If $A = A_1 \...
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1answer
124 views

Proof that $n$ planes cut a solid torus into a maximum of $\frac16(n^3+3n^2+8n)$ pieces

Question: How many pieces can a solid torus be cut into with three (affine) planar cuts? A google search will quickly reveal that the answer is thirteen, as can be read about here. The picture below ...
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3answers
121 views

Enigmatic patterns in Archimedean spirals

Distributing the natural numbers as circles evenly along the Archimedean spiral yields surprising patterns when changing the radius of the circles: they cover more and more of the plane, finally ...
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0answers
88 views

The quadrature of the circle: comparing Archimedean and Ulam spirals

There are two closely related arrangements of the natural numbers that allow to show patterns in the distribution of some sets of numbers (multiples of 2, 4, 8, square numbers, prime numbers): the ...
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2answers
82 views

Parametrizing the square spiral

Related to this question concerning number spirals I have another one, more specific. While it is rather easy to arrange the natural numbers along an Archimedean spiral by $$x(n) = \sqrt{n}\cos(2\pi\...
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1answer
155 views

Visualization of groups with a normal subgroup_rev#1

Let $G$ be a group and $H \unlhd G$. In general, $H=H_Z \sqcup H_{G \setminus Z}$, where $H_Z:=H \cap Z(G)$ and $H_{G \setminus Z}:=H \cap (G \setminus Z(G))$. I'm investigating on a plausible visual ...
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1answer
35 views

Adding vector fields

Consider two vector fields: $$ \vec F_1=(\sin(x),\sin(y)) $$ $$ \vec F_2=(\sin(1-x),\sin(y)), $$ where $x,y \in(0,\pi).$ Does adding the two superimposed vector fields produce a net vertical flow, ...
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0answers
26 views

How to plot ideals of rings

Im trying to better understand ideals of rings and I think being able to visualize what I'm working with would help. I want to graph them (I'm talking mostly about quadratic rings), but I don't know ...
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1answer
33 views

Calculate points of a tesseract (hypercube)

I would like to know how to calculate the points of a hypercube. I am trying to use the mac app Grapher to simulate what one would look like. Does anyone know the equation I could use to generate the ...
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0answers
43 views

Animate this Moire pattern. What mathematical tools could be used to analyze this moving pattern?

For a mathematical art project, I want to animate the following pattern I made on desmos. It seems to be a Moire pattern. However I cannot make the pattern move smoothly and continuously because ...
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1answer
8k views

Plot Individual Vectors Online

I'm looking to plot individual vectors (not a field) for an equilibrium lab using some type of free online site or tool. I've googled for a while and found nothing. Any ideas?
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1answer
135 views

Visualization of groups with a normal subgroup

Suppose $G$ a group and $H \triangleleft G$ (proper normal subgroup). The simplest way to visualize this basic setup is that (Venn-wise) of a bubble ($H$) into a bigger one ($G$), sharing the unit and ...
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1answer
25 views

Visualization / sketch for this basic proof about subspace topology

Let $(X,d)$ be a metric space and $A\subset X$ a subset equipped with the induced metric $d_{A}$. Then the open subsets of $(A,d_{A})$ are exactly the intersections of open subsets of of $(X,d)$ ...
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0answers
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Wild visualization of higher dimensions [closed]

I have a very sophisticated mental picture of higher dimensions and I really need some guidance in correcting my wild imagination. Is it ok to visualize $ \mathbb{R}^4 $ like a regular 3D space ...
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0answers
88 views

Visual representations of groups (in their symmetric groups)_part#2

Background In this post, I have shown that a plausible visual representation of a group $K$ in $\operatorname{Sym}(K)$ can be established, where $\operatorname{Aut}(K) \setminus \lbrace \iota_{\...
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92 views

What “tools” are available within pure mathematics to visualize more advanced topics? [closed]

What "tools" exist within mathematics to visualize concepts for more advanced areas of mathematics, particularly within Analysis, Topology and Algebra ? Furthermore, can one rigorously integrate such "...
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1answer
41 views

Explicit construction and proving or disproving expander graph for this family

In combinatorics, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. In an expander graph most vertices are far apart ...
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4answers
5k views

Interesting implicit surfaces in $\mathbb{R}^3$

I have just written a small program in C++ and OpenGl to plot implicit surfaces in $\mathbb{R}^3$ for a Graphical Computing class and now I'm in need of more interesting surfaces to implement! Some ...
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2answers
188 views

Compute fundamental group _visually_ by the polygonal representation of the space

Some time ago, before I learnt about covering spaces and Seifert-Van Kampen theorem, I tried to compute visually the fundamental group of some spaces. For example I figure out by myself that the ...
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2answers
925 views

Mathematics and the art of linearizing the circle

[I edited the question and put stronger emphasis on "constant curvature" than on "naturalness".] One of the most prominent problems of ancient mathematics was the squaring of the circle: to construct ...
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0answers
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Graph intersection of two 3d loci

I'm trying to graph a 1 dimensional object that curves in 3 dimensions. The only way I've ever been able to do this is with parametrics in Mac Grapher, or by graphing two 2 dimensional surfaces in 3 ...