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

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0answers
12 views

Visual explanation of taylor polynomials

I've just been trying to understand taylor polynomials more intuitively from a visual perspective. But as soon as the terms start using second derivatives, it becomes unclear. 1, Some thought's I've ...
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0answers
12 views

How to visualize the product of two segments lengths? [duplicate]

So there seems to be easy ways to visualize addition. If three points a, b, and c are on the same straight line respectively, we can say that the sum of the lengths of ...
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1answer
24 views

How do you picture: $\Pr(B|A)$ shrunk down by $\Pr(A)$?

I do not understand how to picture and visualise the following explanation: $\color{green}{[P1.]}$ Suppose you were to grab the edges of $A$ and stretch it out so it covers all of $\Omega$. $B$ ...
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1answer
52 views

The role of visualization and intuition in graduate and postgraduate math and developing it

In Visual Complex Analysis's preface, the author gives an analogy with pseudo-deaf musicians and follows the same to mathematics. Mathmatics today, he argues, is mostly build on abstract symbolic ...
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0answers
23 views

How can you picture Conditional Probability in 3D?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I modified the following's source for concision. $1.$ Now look at ...
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1answer
23 views

How can you picture Conditional Probability in a 2D Venn Diagram?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I pursue only intuition; do not answer with formal proofs. Which ...
3
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1answer
36 views

A good pictorial explanation of separation of variables?

I'm teaching ordinary differential equations for the first time, and I would like to give a compelling visual explanation of why it makes sense to "multiply by $dx$" and integrate when you want to ...
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2answers
63 views

How can I visualize a “span” of a set of vectors?

Can someone please help me visualize this concept, I know already what span of set of vectors is, but I am interested to know how it looks visually, thanks.
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3answers
45 views

Visualizing $180^\circ$ rotational symmetries of a tetrahedron

I am trying to learn about the symmetries of a regular tetrahedron. I understand the identity and all eight $120^\circ$ rotations that keep one vertex fixed, ...
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2answers
72 views

Intuitive or visual understanding of the real projective plane

If we take the definition of a real projective space $\mathbb{R}\mathrm{P}^n$ as the space $S^n$ modulo the antipodal map ($x\sim -x$), it is possible to see that $\mathbb{R}\mathrm{P}^1$ is ...
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4answers
93 views

To prove $\frac{n!}{p!q!}$. [closed]

The number of permutaion of n objects, where p and are of one kind,q are of second kind,are of different kind is $\frac{n!}{p!q!}$ How can we proove above theorem $\frac{n!}{p!q!}$.I tried to prove ...
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0answers
40 views

Recommendations on visualizing basic linear algebra

I am teaching linear algebra this semester, and I would really like to recommend my students some cool youtube videos visualizing some simple stuff like the span of a set of vectors, linear ...
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0answers
21 views

Is there a program that receives as input your drawing of a curve and outputs a parametric curve tracing it (reasonably close)?

From what I know, B-Splines is the closest thing that we have to drawing curves and having them defined by the computer. I have some B-Spline code that does this interactively. However, those are a ...
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0answers
27 views

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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3answers
44 views

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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1answer
35 views

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 ...
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0answers
51 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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1answer
66 views

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.
3
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1answer
23 views

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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0answers
23 views

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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0answers
19 views

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 ...
0
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1answer
49 views

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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2answers
47 views

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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2answers
69 views

Can every math problem be visualised? [closed]

Basically, title. Can everything in mathematics be represented graphically one way or another?
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0answers
68 views

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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0answers
20 views

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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1answer
25 views

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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2answers
48 views

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 ...
0
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1answer
38 views

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 ...
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1answer
19 views

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?
3
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1answer
81 views

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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0answers
7 views

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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1answer
40 views

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 ...
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3answers
81 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 $f:X \rightarrow Y$ ...
2
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1answer
70 views

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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0answers
35 views

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, ...
3
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1answer
68 views

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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0answers
15 views

Can't get figure out how to get coordinates back in MDS (multidimensional scaling)

I have a 3,5 matrix $X$. I have 5 items in 3 dimensional space. Let $K= X^TX$. $D$ is the squared distance between any 2 points. (So it is a $5x5$ matrix). $H$ = $I - \frac{1}{5} *ones(5,5)$ $B = ...
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0answers
53 views

Grothendiek and singular points

My question relates to the following thread I opened some weeks ago: A question regarding Grothendieck , topos and (adelic??) points Specifically, consider this paragraph: At 1:14:30 and after, ...
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0answers
48 views

Stellating the Octahedron

I have a few related questions and I'd be happy to get some help with any one of them. Is the stellation of a polyhedron generally a 'messy' affair that involves trimming away portions of the ...
0
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1answer
69 views

Visualising a 1-(50,15,15) design.

The problem I have is the visualisation of a 1-(50,15,15) design. That is a set of 50 points and 50 blocks (lines), so that each point is on 15 lines, and each line contains 15 points. My attempts ...
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1answer
50 views

How to calculate optimal sizes of rectangles for this type of array visualization?

Given array of positive numbers, I would like to draw this diagram and be able to put descriptions inside: There should be no empty space left, consider that these numbers represent % of total. Do ...
3
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0answers
109 views

Is there a way to visualize a group?

Is there a way to picture a group in ones head? I want to "see" the difference between abelian and non-abelian group. And if f is a group homomorphism, is there a way to see that Ker(f)=1<=>f ...
2
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0answers
100 views

What methods are known to visualize patterns in the set of real roots of quadratic equations?

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations ...
2
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0answers
57 views

Ideals in a ring as geometric objects?

I am interested in learing about the possibility of (one-sided) ideals in a ring being repreented geometrically. In other words, about their status as geometric objects (after all, they can be dealt ...
2
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0answers
56 views

What type of diagram could this be?

Some years ago, I saw a diagram somewhat like this somewhere on Wikipedia. I remember that it was supposedly used in some branch of mathematics. Unfortunately, neither Google Image nor trawling ...
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1answer
54 views

What is the correct equation for “Normal distribution function of continuous random variable”?

I was reading a book and came across with a equation which gives the normal distribution function of continuous random variable. It was used in a software called ...
2
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1answer
59 views

Alternative geometric interpretation for big-o and little-o

I understand that, in big-o notation, when we say that a function $f$ is $O(x^2)$ we're basically saying that $$|f(x)|\le M |x^2|$$ for some constant $M>0$ and for all $x>x_0$ for some ...
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1answer
12 views

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 ...
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5answers
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: $$(n+1)^2 - n^2 = (n+1) +n$$ ...