# 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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### Polar plots of $\sin(kx)$

The plots of $\sin(kx)$ over the real line are somehow boring and look essentially all the same: For larger $k$ you cannot easily tell which $k$ it is (not only due to Moiré effects): But when ...
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### Is there a visualization for inverse trig functions as indefinite integrals

Examining the indefinite integral formulations of inverse trig functions I notice some things $$\arcsin(x)=\int_0^x \frac{1}{\sqrt{1-z^2}}dz$$ $$\arccos(x)=\int_x^1 \frac{1}{\sqrt{1-z^2}}dz$$ We ...
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I try to understand and get a feeling which gaps p-adic numbers fill to complete $\mathbb{Q}$. In the course of this I depicted (for $p = 2$) the "base" $\{p^k\}_{k\in\mathbb{Z}}$ with respect to ...
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### Topological Spaces: What are they?

Let $X\subseteq \Bbb R$, $\tau$ is a topology on $X$. What even is $(X,\tau)$? Is it an ordered pair? Is it $\{(x,y)|x\in X, y\in \tau\}$? Is it a subset of $\Bbb R^2$? I (pretty much) know what ...
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### How could one geometrically visualize any given metric space $(X,d)$?

Example. Say $X=\mathbb{R}$ and $d(x,y)=\frac{d_0(x,y)}{1+d_0(x,y)}$ where $d_0(x,y)=|x-y|$ is the Euclidean metric. The visualization of e.g. $\mathbb{R}$ with Eucledian distance $d_0(x,y)=|x-y|$ is ...
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### How do you imagine linear transformations $\mathbb{F}_{p^n} \mapsto \mathbb{F}_{p^m}$

I am learning linear algebra (I know some introductory abstract algebra), and although I can imagine geometrically linear transformations from $\mathbb{R}, \mathbb{R}^2, \mathbb{R}^3$ to itself easily,...
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### How to describe curvilinear grid using coordinate functions?

A curvilinear grid around a cylinder has the following properties: The grid has $n_\varphi =20$ grid points in angular direction (along a circle in the xy-plane). The grid has $n_r =5$ grid points ...
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### Visualizations of the (potential) irrationality of $\sqrt{2}$

The following statement is equivalent to Euclid's statement that $\sqrt{2}$ is irrational but has a rather different flavour. Consider the straight line through two points $0$ and $1$ with the ...
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### how to draw the space of such linear combinations?

We have the linear combination $${2 \choose 1 } x_1 + {1 \choose 2} x_2 + {1 \choose -2} x_3 + {1 \choose 1} x_4 + {-1 \choose 0 } x_5 + {0 \choose -1 }x_6$$ As $x_i \geq 0$ is given, according ...
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### Why do the (nonzero) vectors $x,y, x-y$ form a triangle? (can assume $\mathbb{R}^2$)

Let $x,y$ be any two nonzero vectors in $\mathbb{R}^2$ that are not scalar multiplies of eachother (i.e. are not linearly dependent), and $x-y$ be their difference. I am wondering why these three ...
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### Does the definition of the angle between two vectors require that they have the same “origin”?

I am thinking specifically about $\mathbb{R}^2$ so I can visualize things. By "origin" I mean that they start at the same point. When we graphically represnt vectors we don't care where the starting ...
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### Is there a nice visualization of the length of a curve formula?

We know that if there is a curve $$\Gamma=\{(x,y)\in\Bbb R^2\ :\ y=f(x), x\in[a,b]\}$$ then $$\text{length}(\Gamma)=\int\limits_a^b\sqrt{1+f'(t)^2}dt$$ and I get that this is because \text{length}...
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### Intuition and visualization of area preserving maps?

I was trying to understand what is meant by "area preserving map"?. I was going through the Wolfram article about the area preserving map here but any motivation, intuition or visualization to ...
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### 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 ...