Questions tagged [visualization]

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

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5
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0answers
152 views

Distributions of prime numbers

When folding the number line not into a spiral (like Ulam did) but into a zig-zag pattern (like Cantor did) there are other patterns visible in the distribution of prime numbers: [To see these ...
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1answer
89 views

Why do we evaluate $-x^{z-1}e^{-x}$ as zero when explaining the gamma function through integration by parts?

The gamma function is the integral of $x^{z-1}e^{-x}$ If you integrate by parts you get two terms. The first one is $-x^{z-1}e^{-x}$ and this is bound by infinity and zero. If you plug in infinity, ...
3
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1answer
135 views

Distribution of triangular, square, and pentagonal numbers

I'm doing some research in visualizing arithmetic sets (resp. properties, resp. sequences of integers). I try to create patterns (in which I hope to observe some symmetries) by injectively mapping $\...
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1answer
76 views

How can one best visualize two dimensional manifolds in $\mathbb{R^4}$ (more specifically, $\mathbb{S}^2 \times \mathbb{R})$?

I'm trying to "get a picture", so to speak, of hypersurfaces in $\mathbb{S}^2 \times \mathbb{R}$. One example would be $\left(\dfrac{\cos(u)}{\sqrt{1+u^2}}, \dfrac{\sin(u)}{\sqrt{1+u^2}},\dfrac{u}{\...
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0answers
128 views

Geometrical visualization of Tensors

My question is about tensors. I have recently spent some time studying the various definitions of tensors and some tensor calculus. What I am missing now is an intuitive way to represent tensors and I ...
2
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0answers
60 views

Projecting 6D cartesian coordinates to lower dimension

I've got a noisy set of points that lie on/near a unit 6-sphere. I want to visualise their proximity to the sphere in some way. The only way I can come up with now is a histogram of the norm of the ...
0
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1answer
50 views

Would this suffice in a visual type theory to define an abstract List type?

See the image. I got that from: wikipedia article. In that, I don't understand the first function nil : () -> L. What is ()...
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0answers
72 views

Efficient visualisation of arrangements

First, I don't know the word for the set. I have n cases and the set I have is 2^n which lists all their "arrangements" present or absent, so a summation of combinations with k from 0 to n. Each ...
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0answers
94 views

Geometric intuition of a point in a flag manifold

From Wikipedia According to basic results of linear algebra, any two complete flags in an $n$-dimensional vector space $V$ over a field $F$ are no different from each other from a geometric point ...
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0answers
227 views

How can I use kernel density estimation for heat map visualization?

I want to know more about visualization and density estimations. Basically, I have a large sample of location data and every location object has a duration in milliseconds. I want to solve the ...
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1answer
222 views

Derivative of Square Root Visual

Derivative formulas through geometry | Essence of calculus, chapter 3 (3Blue1Brown): https://www.youtube.com/watch?v=S0_qX4VJhMQ There is a challenge at 12:23 asking the viewer to arrive at the ...
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1answer
942 views

Lagrange mean value theorem for two variables - visualization and intuition behind it

The two-variable version of the Lagrange mean-value theorem says that given a function $f(x,y)$, $$f(\vec{p_o} + \vec h)-f(\vec {p_o})=df(\vec {p_{\theta}})$$ Where $\vec p_{\theta}=\vec p_o + \theta ...
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0answers
53 views

Visualization of particular integration using trigonometric function

I know that when you're trying to integrate something, and by chance, appears somewhere something of the form : $$ \sqrt {1 - x^2} ; \sqrt {1 + x^2}$$ or something similar, it is known to replace x by ...
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1answer
127 views

How to interpret a Box-Percentile Plot?

How to interpret a box-percentile plot and find the outliers? I have been trying to find an example online but so far not been successful. Here is an example diagram:
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0answers
169 views

Understanding the definition of Sensitive dependence on initial conditions?

I was trying to understand the rigorous definition of sensitive dependence on initial conditions which is as follows - $f : X \mapsto X$ where $X$ is a metric space. If there exists $\epsilon > ...
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1answer
298 views

Row versus column picture

So, we have a system of linear equations.It's preferred that we visualize the column picture, since as dimensions go up we only have to think of vectors moving to more dimensions, instead of ...
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1answer
44 views

Visualizing $c-d<a-b \implies b<a+d-c$?

I am wondering if someone can provide some geometric intuition, or some simple way to visualize why $$ c-d<a-b \implies b<a+d-c $$ The way I have been trying to do this is to think of $a,b,c,d$ ...
0
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1answer
28 views

How to combine multiple indices?

I'm trying to combine various risk indices e.g. flooding, fire, burglary and structural damage. These values are scored from 1-3. For instance, I could have measures for two different areas as ...
0
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3answers
149 views

Visual proof of isosceles base-angle congruency?

A geometric proof (without algebra or trigonometry), and ideally presented visually (a proof without words). EDITED (I'm especially curious if it's possible without without using triangle congruency.)...
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1answer
44 views

Plotting lines, points, planes, triangles for technical documentation

I'm looking for a (free) tool to draw or plot graphics as the one linked below: Maybe a CAD solutions could be the easiest one. But which do you recommend for this certain case? I don't want to deal ...
1
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1answer
136 views

Software package for plotting 3-d splines

Given a finite point set $P \subset \mathbb{R}^2$ and a height function $h:P -> \mathbb{R}$ I want to produce a smooth surface that interpolates between the values $\left\{[p~h(p)]^\top \in \mathbb{...
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2answers
45 views

Visualising one dimension in real/physical world.

How can we visualize one-dimension in real/Physical world? Does any body have an example? Often people refer to one-dimension as motion being in a straight line. However motion in a straight line can ...
0
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1answer
145 views

A more direct way to see that the angle inscribed in a semicircle is $90^\circ$?

Like this inscribed angle proof, another proof enabled by this clever angle sum proof. Is there a simpler way to show this? Is this proof original?
3
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2answers
560 views

A way to directly see the Inscribed angle theorem? (i.e. central angle is twice the inscribed angle)

Dotted line is parallel to the diameter; radii are equal length. The upper angle is the inscribed angle; the double angle at the center is the central angle. The upper triangle is isosceles, ...
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13answers
6k views

A way to directly see that the interior angles of triangle sum to $180^\circ$?

I'm looking for a way to look at a triangle, and perhaps visualize a few extra lines, and be able to see that the interior angles sum to $180^\circ$. I can visualize that supplementary angles sum to $...
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0answers
39 views

Questions to test visualization in 4D

Some people claim to be able to visualize four dimensional objects. I have a friend who thinks that these folks are just blowing hot air. I thought it would be fun to test how well people actually ...
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0answers
116 views

(Soft) Why do our visual understandings of homotopy equivalence correspond to the definitions?

This may be too soft, and it may be a confusion only about formalism. But seeing why the homotopies equivalences I visualize in my head and that people describe are the same thing as the definition of ...
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0answers
90 views

Visualizing the minimization of $\|Ax-b\|_2^2$?

Consider the least-squares problem $$\text{minimize} \quad \|Ax-b\|_2^2$$ where $A$ is an $m \times n$ matrix and $b$ is an $m$-vector. How does it looks like geometrically? Could someone draw a ...
2
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1answer
107 views

Visualization vs memorization of mathematical knowledge [closed]

As far as math research is concerned, what kind of understanding level of mathematical knowledge is required in order to truly master a topic and leverage on it? Can top-level researchers fully ...
0
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1answer
24 views

A question regarding multi variate functions

When we have a simple one variable function $y = f(x)$, $f(x)$ gives the height of function/curve at an arbitrary point $x$, but if we have a multivariate function $u = f(x,y)$, what does it give us?
4
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1answer
229 views

Lagrange spectrum in diophantine approximation theory

Context. Hurwitz' theorem states that for every irrational $\xi$, there is infinitely many rationals $p/q$ such that $$\left\vert \xi-\frac pq\right\vert<\frac 1{q^2\sqrt 5}.$$ The number $\sqrt ...
22
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0answers
435 views

Geometric representation of Euler-Maclaurin Summation Formula

In Tom Apostol's expository article (here's a free link), upon seeing the figure below (or this from the Wolfram project) I was expecting more diagrams to come to continue the error decomposition of ...
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0answers
168 views

Online tool to visualize a curve of given curvature

For my differential geometry class it would be superuseful to have a tool to which I feed a curvature formula and that spits out a representation of a planar curve with such curvature. For example, ...
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4answers
317 views

Using Visualization for Learning: $a^0=1$ [closed]

Honestly, this was a question from a student. I show $\forall a\in \mathbb{R}-\{0\}:a^0=1$ for the class. $$1=\frac{a^m}{a^m}=a^{m-m}=a^0$$or $$a^{m}=a^{m+0}\to a^m=a^m\times a^0\\(a\neq 0) \to cancel ...
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3answers
1k views

How to visualise positive and negative tangents

A quick internet search of simple trigonometry methods returns a whole bunch of acronyms for remembering whether sin, cos and tan functions yield positive or negative results in the four quadrant of a ...
0
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1answer
86 views

Visualizing Riemann surface

On reading about the construction of a Riemann surface for algebraic functions, I am having difficulties visualizing why the construction produces a manifold over points that are multiple roots of the ...
0
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0answers
49 views

Error made if we consider the whole globe as the coordinate chart?.

I was looking at this article at mathworld where it has mentioned - "This coordinate chart is not valid on the whole globe, since it doesn't give unique coordinates at the north or south pole (which ...
141
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15answers
19k views

What's new in higher dimensions?

This is a very speculative/soft question; please keep this in mind when reading it. Here "higher" means "greater than 3". What I am wondering about is what new geometrical phenomena are there in ...
4
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0answers
65 views

Graph layout that reflects graph symmetries

I am looking for practical computational methods to lay out graphs in such a way that the geometry of the drawing reflects some of the symmetries of the graph. Here are two example drawings of two ...
2
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2answers
1k views

What is the difference between a trace and a contour in calculus?

As far as I can tell they're exactly the same thing, but the notes here discuss them as if they are separate: The final topic in this section is that of traces.  In some ways these are similar to ...
3
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1answer
153 views

Does this dynamical system show an “absorbing area” or a “chaotic area”?

I am following the technical report by C.Mira: "Noninvertible maps: notion of chaotic area vs that of strange attractor" in order to characterize the behavior some dynamical systems of my own. In the ...
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0answers
159 views

Program that shows action of Möbius transformations

This is not directly a mathematical question, but I think this is the best place to ask it. I discovered that I'd find it very helpful to have some sort of a program where I can plot the coefficients ...
0
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2answers
126 views

Making something a control parameter or a variable when analysing a dynamical system

I am writing down a draft trying to accurately characterize some nonlinear/noninvertible discrete dynamical systems (of a former question here) and due to my lack of knowledge I am having doubts here ...
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0answers
56 views

open-source illustrations of Riemann surfaces

I'd like to generate publication-quality images of Riemann surfaces using open source software. Is Inkscape the best tool? If so, are there libraries of vector images of Riemann surfaces one can ...
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1answer
138 views

Visualizing what part of the surface is integrated by surface integral

I'm trying to draw the region of the surface area of the cylinder, $x^2+y^2 \le 2x$, limited by the cone $z=\sqrt{(x^2+y^2)}$ and the plane $z=0$ . I know that the cylinder's center is at $(1,0)$ ...
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2answers
227 views

How does derivative of definite integral make sense

Derivative is taken at a point and hence is value at a point. But definite integral is the value over a domain. Then how come derivative of definite integral make sense.
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3answers
119 views

Calculate different $RGB$ colors based on slope

I'm trying to plot a graph and color the lines between points based on their slope. The points on my graph only consist of positive values between $0$ and $9$ for both the $X$ and $Y$ axis. The only ...
6
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2answers
431 views

Is there a good way of visualizing Grassmann manifolds?

I am trying to explain $Gr_{\mathbb{R}}(m,n)$, the Grassmann manifold of the collection of $n$-dimensional linear subspaces of the $\mathbb{R}^m$ space. After I used the projective space ...
4
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1answer
66 views

Can the integration of $x^2$ be diagramed to show it's $\frac{x^3}{3}$?

It's visually easy to see how the integral of a constant is $x=y$ as the area is growing linearly as you sum the dx boxes. Similarly the integral for $y=x$ is $\frac{x^2}{2}$ which is the triangle ...
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0answers
36 views

Visualizing n-dimensional landscape versus collection

Much has been written on the topic of visualizing n-dimensional spaces (where n is in general much larger than 3). T-SNE and parallel coordinates are two ways to do it rather effectively. (I'm not ...

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