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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### Is it a good approach to heavily depend on visualization to learn math?

I am a third year undergraduate and I am a beginner on these "real mathematics" (no pun intended). Before contacting the "real math", my math level should be considered to be "good", although I was not the best. (I am always too lazy, and not that…

MonkeyKing

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### Visual research problems in geometry

I am considering doing research in mathematics to be my career (and my life) someday.
I'm a visually oriented person in general; for example, I prefer chess over cards because when I play chess, I do all my thinking by looking at the board and…

Akram Hassan

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### Visualizing the square root of 2

A junior high school student I am tutoring asked me a question that stumped me - I was wondering if anyone could shed some light on it here.
We were talking about how the square root of 2 is an irrational number, and that means you can't write that…

dvanaria

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### Is there a relationship between $\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n} = 1$ and $\int_{1}^{\infty} \frac{1}{x^2} \,dx = 1$?

A classic example of an infinite series that converges is:
${\displaystyle {\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n}=1.}$
A classic example of an infinite integral that…

Toph

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### Visualized group tables for $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

Let me share a kind of visualization of group tables which is especially well suited for cyclic groups like $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$. In these groups you can easily give colors to each group member $k$ with
hue = red if $k > 0$…

Hans-Peter Stricker

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### Enumerating Bianchi circles

Background: Katherine Stange describes Schmidt arrangements in "Visualising the arithmetic of imaginary quadratic fields", arXiv:1410.0417. Given an imaginary quadratic field $K$, we study the Bianchi group $\mathrm{PSL}_2(\mathcal{O}_K)$, which is…

Chris Culter

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### Visualizing Hopf fibration $S^3\to S^2$ as a based map $S^1\to \mathrm{Map}(S^2,S^2)$

A fiber bundle $F\to E\to B$ may be interpreted as $E$ being a bunch of $F$s arranged in the shape of a $B$. For instance, a Mobius band $M$ is a bunch of line segments $[0,1]$ arranged in the shape of a circle $S^1$, so we may write $[0,1]\to M\to…

arctic tern

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### How do I visualize differential equations?

OK, I got an exam in about a week, and there is a point that I don't really got my head around yet. Our professor likes to give for example three pictures and one differential equation.
The question now is which of these pictures approximates the…

joachim

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### continuum between linear and logarithmic

A friend and I are working on a heatmap representing some population numbers. Initially we used a linear color scale by default. Then, because the numbers covered a wide range, I tried using a log color scale (as shown here). My friend said it…

LarsH

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### What can be gleaned from looking at a domain-colored graph of a complex function?

Functions from $\mathbb{C} \rightarrow \mathbb{C}$ are hard to visualize because of their 4-dimensional nature. One nice way of looking at them is by what's called domain coloring. An example from the wiki article is shown below.
When we look at the…

I. J. Kennedy

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### 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 $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$
with $a(t)$, $b(t)$ being $2\pi$-periodic…

Hans-Peter Stricker

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### Visualizing quadratic residues and their structure

[I corrected the pictures and deleted one question due to user i707107's valuable hint concerning cycles.]
Visualizing the functions $\mu_{n\ \mathsf{ mod }\ m}(k) = kn\ \ \mathsf{ mod }\ \ m$ as graphs reveals lots of facts of modular…

Hans-Peter Stricker

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### This quotient space is homeomorphic to the Möbius strip?

Let $G:\mathbb R \times [-1,1]\to \mathbb R \times [-1,1]$ be a map defined by $G(x,y)=(x+1,-y)$
This space $Q=\mathbb R\times [-1,1]/\sim$, where $(x_1,y_1)\sim (x_2,y_2)$ if and only if there is $n\in \mathbb Z$ such that $G^n(x_1,y_1)=(x_2,y_2)$…

user42912

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### Visual intuition for direct sum vs. tensor product of vector spaces

I completely understand the formal mathematical distinction between the direct sum and the tensor product of two vector spaces. I also understand that the direct sum has a nice visual interpretation (especially the direction sum of two 1D vector…

tparker

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### Gradient is NOT the direction that points to the minimum or maximum

I understand that the gradient is the direction of steepest descent (ref: Why is gradient the direction of steepest ascent? and Gradient of a function as the direction of steepest ascent/descent).
However, I am not able to visualize it.
The Blue…

The Wanderer

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