Articles of visualization

Usefulness of alternative constructions of the complex numbers

Complex numbers $\mathbb{C}$ are usually constructed as $\mathbb{R}^2$ together with a suitable multiplication. But this is not the only possible way, one can get to the complex numbers. One alternative would be via $\mathbb{R}[x]/{\left< x^2+1\right>}$, but I have never seen this one be used in practice. When visualizing complex number one usually uses the fact […]

Intuition – $fr = r^{-1}f$ for Dihedral Groups – Carter p. 75

Name $r$ = clockwise 90 deg. rotation and $f$ = flip across the square’s vertical axis = the brown $\color{brown}{f}$ in my picture underneath. Zev Chonoles’s $f$ is different. Carter fleshes out why $frf = r^{-1} $ intuitively: (1.) Can someone please unfold, like Carter, why $fr = r^{-1}f $? I see why for this […]

Sketch Saddle Point of a function of two variables $ f(x, y) = 4 + x^3 + y^3 – 3xy$

As regards $ f(x, y) = 4 + x^3 + y^3 – 3xy$, I computed that (0,0) is a saddle point, and (1,1) is a local minimum. So I’m not asking about this, and am asking only about sketching contours. $1.$ Here, I’m referring to the middle sketch entitled “Why not this?” in red. How […]

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 topologically equivalent to the circle. It is equivalent to the upper half of the circle where the two end points are glued together – i.e. another […]

Direct proof. Square root function uniformly continuous on $[0, \infty)$ (S.A. pp 119 4.4.8)

(http://math.stanford.edu/~ksound/Math171S10/Hw8Sol_171.pdf) Prove for all $e > 0,$ there exists $d > 0$ : for all $x, y \ge 0$, $|x – y| < d \implies |\sqrt{x} – \sqrt{y}| < e$. (a) Given $\epsilon>0$, pick $\delta=\epsilon^{2}$. First note that $|\sqrt{x}-\sqrt{y}|\leq|\sqrt{x}+\sqrt{y}|$. Hence if $|x-y|<\delta=\epsilon^{2}$, then $ |\sqrt{x}-\sqrt{y}|^{2}\leq|\sqrt{x}-\sqrt{y}||\sqrt{x}+\sqrt{y}|=|x-y|<\epsilon^{2}. $ Hence $|\sqrt{x}-\sqrt{y}|<\epsilon$. 1. Where does $|\sqrt{x}-\sqrt{y}|\leq|\sqrt{x}+\sqrt{y}|$ issue from? How […]

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, see it in 2-space or 3-space, enter a transformation vector or matrix, and see the result. For example, enter a […]

Visualizations of ordinal numbers

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 $\omega^\omega$: My questions are: (Where) has this way of visualization been suggested before? How far can it been generalized by repeating the indicated […]

Geometric visualization of covector?

How could I geometrically visualize a linear functional?

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 obscured the data too much. So I was wondering, is there […]

On visualizing the spaces $\Bbb{S}_{++}^n$ and $\Bbb{R}^n\times\Bbb{S}_{++}^n$ for $n=1,2,\ldots$

Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive-definite $n\times n$ real matrices. I am looking for hints concerning the visualization of such spaces for $n=1,2,\ldots$. I know that $\Bbb{S}_{++}^n$ is a convex cone, but I am not sure how it does “look like”. How could I compute the equation of the cone analytically? My idea […]