# Tagged Questions

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

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### Visualizing linear transformations on vector fields

I'm trying to figure out what it means to apply a linear transformation to a vector field geometrically. So I start with the easiest geometrically interesting transformation: a rotation. Using ...
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### An image suitable for a skyscraper sheaf?

This question relates to this thread: Skyscraper sheaf? Consider one of the diagramms for the representation of a sheaf (and stalks thereof) which are popular on the web: I just wanted to know ...
68 views

### Visualizing the geometric product?

The exterior product between blades has a relatively clear geometric interpretation: it gives the result of "extending" one factor along the other, with the direction pointing along the first factor ...
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### Skyscraper sheaf?

I was wondering whether someone could provide me with an easy definition of a skyscraper sheaf and, more importantly, with a visualization of it. Thanks in advance.
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### Is there a relation between isometric and orthographic measurements?

This image shows a couple of different isometric projections. In the black shows the figure's "true" dimensions in an orthographic projection while the red shows the dimensions in an isometric ...
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### Representing and Visualising Concrete Groups

What are some interesting and attractive ways to visualise specific Groups? Like: The Rubik's Cube, A set of Permutation Matrices, And.. What else?
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### How do I prove this point to line duality?

I need to prove this point to line duality. The thing is, I'm not sure what there is to prove. I guess that I have to prove that if two lines intersect to a point, then by duality their two points ...
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### Prove: In a finite geometry, if a line has $n$ points, then the total number of points is $n^2 - n + 1$.

Prove: In a finite geometry, if a line has $n$ points, then the total number of points is $n^2 - n + 1$. The following axioms define a finite geometry: ...
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### How do you draw/visualize a combination table?

This is somewhat related to a previous question I asked I have three variables in a programming function, and a 4th variable depends on these. I have to test the dependent variable against all ...
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### Can every math problem be visualised? [closed]

Basically, title. Can everything in mathematics be represented graphically one way or another?
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### Visual questions for 6th graders

I'm tutoring a 6th grader in math at the moment and because she never has a ton of homework I like to give her some interesting extra problems to do. It seems she really enjoyed a problem I showed ...
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### About the classification of linear autonomic differential equations of the plane.

For a linear autonomic differential equation of the plane $$\dot x = Ax,$$ with $A ∈ \operatorname{Mat}_{2×2} (ℝ)$, say we have a fundamental matrix $Φ \colon ℝ → \operatorname{Mat}_{2×2} (ℝ)$, that ...
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### Some explanation regarding a diagram of homotopy

I found this visualization regarding homotpy in wikipedia: https://commons.wikimedia.org/wiki/File:Homotopy_curves.png I would be very grateful if you could explai me all the abbreviations being ...
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### Interpretation of $a+b \ | \ a^n + b^n$ for odd $n$

It is not hard to show that $a+b \ | \ a^n + b^n$ for odd $n$. (because $f(x) = x^n - b^n = (x-b)h(x)$ we have $a - b \ | \ a^n - b^n$, so $a - (-b) \ | \ a^n - (-1)^n b^n$) Is there a nice ...
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### Orthogonality in Hilbert Spaces

For the sake of concreteness, let's say that our Hilbert space is the beloved $\mathscr L^2(\Bbb R)$. Suppose that we have $\psi,\phi\in\mathscr L^2(\Bbb R)$, what's the intuitive meaning to a semi-...
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### Slope in algebra I

What is a good project for teaching y=mx + b and having students discover slopes of lines in classroom or on the classroom buildings?
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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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### Name and explanation of triangular representation of 3 coordinates in 2 dimensions

There is a common two-dimensional graphical representation of three values that are required to sum to a constant. The three values are represented by a point inside an equilateral triangle. More ...
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### On the Hasse diagram for ideals

When consulting the wikipedia regarding prime ideals, the following Hasse diagram (is it also a lattice?) is provided as representation: https://en.wikipedia.org/wiki/Prime_ideal Any idea of who ...
132 views

### Topological definition of continuity (open set characterization)

I want to demonstrate the topologic definition of continuity, using the classical definition with epsilon's and delta's. So we have that: Classical definition - If the function $f:X \rightarrow Y$ ...
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### Is this a valid visualization of Euler's identity as a more generic pattern?

I was reading this nice question about a demonstration of Euler's identity, and tried to visualize how would look the left part of the identity in the complex plane by using the following function: ...
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### Parametrization of helicoid like surface for Faraday's law of induction of a solenoid?

I want to visualize with mayavi a possible surface for Faraday's law of induction in the electrodynamics of a solenoid. I.e. something like a helicoid with a smooth transition to a rectangular area, ...
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### Does the equality $\dfrac{10!}{6!} = 7!$ hold any special (geometric) meaning?

I've come across the following simple, but unexpected equality numerous times accidentally. $$\frac{10!}{6!} = 7!$$ which is the same as $$1*2*3*4*5*6*7 = 7*8*9*10$$ Does it hold any specific (...
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### Comparing a function and its estimate

What are some clever ways of comparing (visually) a function with its estimate? For regions where the function does not cross zero, plotting the ratio of the functions and plotting the relative error ...
1k views

### Visualizing the factorial

Often in basic mathematics, we can visualize things very easily, which I believe helps understanding (instead of just working out a number theoretical proof). For example: $$(n+1)^2 - n^2 = (n+1) +n$$ ...
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### Emil Artin on visualization of matrices

Someone called my attention to the fact that Emil Artin made very important remarks on the visual representation of matrices in some of his books. Could anyone tell me which precise book that is? ...
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### Hyperplane in a complex vector space

This is my first question on MSE, I'm sorry if there already exists similar questions, I couldn't manage to find it. My friend, who studies Physics, asked me about the meaning of "functional" so I ...
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### What does Dini continuity mean?

What does Dini continuity (the integral condition) mean visually? Description of Dini contuity: https://en.wikipedia.org/wiki/Dini_continuity
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### Is there a way to visualize, like a picture in mind, the $n$-th derivative?

Is there a way to visualize (like a picture in mind) the $n$-th derivative ? For $n=1$ is the tangent line and we can visualize it quite well. More abstractly is it possible to see the geometric ...
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### Visualisation of the reciprocal of an continued fraction?

If: $$a=\cfrac{l}{m+\cfrac{n}{o+\cfrac{p}{q+\cdots}}}$$ Then could you help me visualize $1/a$? I really don't understand it. Thank you so much!
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### Animation of Weierstrass $\wp$-function as a map from a torus to the sphere?

I am wondering if there exists somewhere an "animation" of one such map (for some lattice / torus), in the style of the kind of $z \mapsto z^2$ maps one encounters in complex analysis classes (one can ...
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### How do I visualize this quotient space?

If $V = [0,1] \times [0,1] \subset \mathbb{R}^2$. We define the equivalence relation $\sim$ on $V$ as follows: every element $(x,y) \in V$ is equivalent with itself and besides that the three elements ...
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### How do I evaluate this sum :$\sum_{n=0}^\infty z^{n^3}$ and Is there a visual proof for it?

if $$\sum_{n=0}^\infty z^n = \frac1{1-z}, \quad z \in \mathbb{C},\; |z| < 1 .$$ then is there a way to deduce this sum:$$\sum_{n=0}^\infty z^{n^3}$$ from the above Identitie or any visual proof ...
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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 ...
188 views

### Visual approach to abstract algebra

I'm currently finding abstract algebra to be very fascinating. However, one of the things that pulls me back is that I sometimes find it hard to understand something visually. For example, one could ...
270 views

### Drawing an approximation to a circle in isometric projection

A circle viewed from from the side is an ellipse. A common approximation can be found on the web (eg do a google image search for isometric circle). This produces something like (with arc centers T,U,...
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### In $\Bbb R^3$, is there a general principle governing these “visual” angles?

I believe most of you have drawn the xyz coordinate system hundreds of times and so have I. You may have drawn it like these, on various occasions: (the reverse directions of the axis are not shown.) ...