# Tagged Questions

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

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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 ...
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### Algebra: Best mental images

I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on ...
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### Visually deceptive “proofs” which are mathematically wrong

Related: Visually stunning math concepts which are easy to explain Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually ...
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### How to Visualize points on a high dimensional (>3) Manifold?

Are there any ways to visualize(plot/draw) points on a high dimensional (ex: dimension = 5) manifold?
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### 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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### 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 ...
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### $\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$? [duplicate]

If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed ...
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### What's the name of these two surfaces?

I've plot two implicit surfaces which are shown in the above, I only know their expression, but I don't know how to call them.
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### Why is it that I cannot imagine a tesseract?

I try hard to "visualise" (say "imagine") a tesseract but I can't. Why is it that I can't? This may be a question for a scholar of some other discipline and not for a mathematician, e.g. ...
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### Visualizing a homotopy pull back

I am currently taking a course in algebraic topology, which also covers a lot of category theory. My question is pretty straightforward: How do you visualize the (homotopy) pull back of a diagram B-->...
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### Min-Max Principle $\lambda_n = \inf_{X \in \Phi_n(V)} \{ \sup_{u \in X} \rho(u) \}$ - Explanations

In general, I am generally someone who like to solve questions with visual support. With this idea in mind, is it someone could explain to me, with a visual support if possible, how is it possible to ...
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### Problem using Stokes's Theorem - Boundary Curve, Unit Normal Vector [Stewart P1097 16.8.5]

$\Large{1.}$ How does one determine the boundary curve, denoted as C, to be the plane $z = -1$? I’m flummoxed because $S$ here is given as bottomless. I'm not enquiring about formal or rigorous ...
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### Fourier Series and epicycles - How to extract the radii and angular velocities from the Fourier Series expansion of a function.

NOTE: I am attaching Mathematica code for those who may want to check it out and understand what I'm asking for. The rest of the question is pretty mathematical in nature, I'll also try the ...
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### 0 to the power of 0, what does the essential discontinuity actually look like?

So having watch this clip by Numberphile which explains why $0^0$ is undefined https://www.youtube.com/watch?v=BRRolKTlF6Q And also this http://mathforum.org/dr.math/faq/faq.0.to.0.power.html And ...
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### Intuition about the size of $\aleph_k$ with $k>1$

Assuming CH for simplicity, I know of some more or less intuitive way to think about difference in sizes of $\aleph_0$ and $\aleph_1$. The most straightforward is the distinction of natural/rational ...
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### Intuition - Fundamental Homomorphism Theorem - Fraleigh p. 139, 136

Let $\phi: G \to H$ be a group homomorphism with $K = \ker\phi$. Then $G/K \simeq \phi[G].$ The hinge to the proof is to define $\Phi: G/K \to \phi[G]$ given by $\Phi(gK) = \phi(g)$. Then we must ...
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### Cauchy's Generalized Mean Value Theorem. Required function. (S.A. pp 140 t5.3.5)

Cohen, Henle. Calculus pp 827, (http://www.vias.org/calculus/09_infinite_series_10_06.html) I revised the footnote in pp 14 http://math.uga.edu/~pete/2400calc2.pdf. This theorem can be illustrated ...
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### Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles?

The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a ...
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### Can I represent groups geometrically?

I have just taken up abstract algebra for my college and my professor was giving me an introduction to groups, but since I like geometric definitions or ways of looking at stuff, I kept thinking, "How ...
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### Are there any visual proofs for $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$?

I was flipping through Proofs Without Words (PWW) and saw many visual proofs for sequences and series. However, I saw none for $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$ Are there any ...
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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 ...
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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 ...
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### Can one construct a “Cayley diagram” that lacks only an inverse?

My group theory text asks for an example of a Cayley-like diagram that exhibits all the properties of a group except (only) that at least some elements lack an inverse. Is it possible to construct ...
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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 ...
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### What's the intuition behind the identities $\cos(z)= \cosh(iz)$ and $\sin(z)=-i\sinh(iz)$?

I'm trying to understand in an intuitive manner the relationship between the circular and hyperbolic functions in the complex plane, i.e.: $$\cos(z)= \cosh(iz)$$ $$\sin(z)=-i\sinh(iz)$$ where $z$ is ...
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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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### 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 ...
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### Computing the projection of an infinite 3D grid of points

Consider the subset $S$ of $\mathbb{R}^3$ consisting of points whose coordinates are integers (compare Gaussian integers, Euclid's orchard). The view of $S$ from a perspective camera within the space ...
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### FLOSS tool to visualize 2- and 3-space matrix transformations

I'm looking for a FLOSS application (Windows or Ubuntu but preferably both) that can help me visualize matrix transformations in 2- and 3-space. So I'd like to be able to enter a vector or matrix, ...
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### Geometric visualization of covector?

How could I geometrically visualize a linear functional?
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### Intuition for gradient descent with Nesterov momentum

A clear article on Nesterov’s Accelerated Gradient Descent (S. Bubeck, April 2013) says The intuition behind the algorithm is quite difficult to grasp, and unfortunately the analysis will not be ...
I find this picture of the ordinal numbers up to $\omega^\omega$ rather hard to grasp: I wonder if the following might be a more compelling way to visualize ordinal numbers up to $\omega^\omega$: ...