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

# Questions tagged [visualization]

864 questions

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### Why is this family of dynamical systems able to produce spirals and clusters of points?

I have found by trial and error an interesting family of dynamical systems giving some nice strange attractors. They are chaotic complex systems based on the digamma function. It is defined by a complex discrete map as follows:
(Disclaimer, tl:dr:…

iadvd

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### The famous prime race and generalizations

So I was messing around with the famous prime race that comes down to this:
We make a list of primes. The list has two rows; the top row is for primes $1\mod 4$ and the bottom row for primes $3\mod 4$. Our list, up to the 10th prime:
5 13 17 …

vrugtehagel

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### Geometric interpretation of Young's inequality

Is there a geometric interpretation of Young's inequality, $$ab \leq \frac{a^{p}}{p} + \frac{b^{q}}{q}$$ with $\dfrac{1}{p}+\dfrac{1}{q} = 1$?
My attempt is to say that $ab$ could be the surface of a rectangle, and that we could also say…

Frank

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### Intuitive Way To Understand Principal Component Analysis

I know that this is meant to explain variance but the description on Wikipedia stinks and it is not clear how you can explain variance using this technique
Can anyone explain it in a simple way?

Jack Kada

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### Visualising finite fields

I'm interested in finding visual and/or physical approaches to understanding finite fields. I know of a few: V. I. Arnold has a few pictures of 'finite circles' and 'finite tori' in his book Dynamics, Statistics and Projective Geometry of Galois…

Marius Kempe

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### Visualizing quotient groups: $\mathbb{R/Q}$

I was wondering about this. I know it is possible to visualize the quotient group $\mathbb{R}/\mathbb{Z}$ as a circle, and if you consider these as "topological groups", then this group (not topological) quotient is topologically equivalent to a…

The_Sympathizer

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### Infinite area under curve without using derivatives and integrals

I am looking for a function $f$ with the following properties:
$f$ is continuous on $[0,\infty[$
$f(0)=1$
$f(x)\to0$ as $x\to\infty$
$\int_0^{\infty} f(x) \,\mathrm{d}x = \infty$
It is not difficult to find a such a function, for example $f(x) =…

Julia

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### Cutting a Möbius strip down the middle

Why does the result of cutting a Möbius strip down the middle lengthwise have two full twists in it? I can account for one full twist--the identification of the top left corner with the bottom right is a half twist; similarly, the top right corner…

AndJM

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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 plotting $\sin(kx)$ over the unit circle by
$$x(t) =…

Hans-Peter Stricker

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### Handbook of mathematical drawing?

My drawing skills are pretty awful, and although I haven't yet had to teach multivariable calculus, someday I will. (And next semester in calculus II we're already doing some volumes by integrating cross sections, volumes and surface areas of…

Dave Gaebler

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### What are all these "visualizations" of the 3-sphere?

a 2-sphere is a normal sphere. A 3-sphere is
$$
x^2 + y^2 + z^2 + w^2 = 1
$$
My first question is, why isn't the w coordinate just time? I can plot a 4-d sphere in a symbolic math program and animate the w parameter, as w goes from .1 to…

bobobobo

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### 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 the plot of the sine function with amplitude $r$ over…

Hans-Peter Stricker

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### 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 quite a bit since the concept of topology is new to…

Arthur

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### Visual explanation of the following statement:

Can somebody fill me in on a visual explanation for the following:
If there exist integers $x, y$ such that $x^2 + y^2 = c$, then there also exist integers $w, z$ such that $w^2 + z^2 = 2c$
I know why it is true (ex. take $w = x-y, z = x+y$), but I…

MCT

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### Visualizing Exterior Derivative

How do you visualize the exterior derivative of differential forms?
I imagine differential forms to be some sort of (oriented) line segments, areas, volumes etc. That is if I imagine a two-form, I imagine two vectors, constituting a…

Yrogirg

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