# Questions tagged [visualization]

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

641 questions
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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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
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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### 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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### 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 ...
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 ...
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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### 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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### 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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### 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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### 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 ...