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33
votes
5answers
6k views

The Stupid Computer Problem : can every polynomial be written with only one $x$?

When I was a child, I wanted to be a mathematician so I asked my parents to buy me a computer to make super complex calculations. Of course, they were not crazy enough to buy an expensive super ...
2
votes
1answer
38 views

How would you draw $(A\setminus B)\times (A\setminus B) = (A\times A)\setminus (B\times B)$?

I know it's useful to prove set equalities to make a quick sketch of the sets described. How can I draw this one? $$(A\setminus B)\times (A\setminus B) = (A\times A)\setminus (B\times B)$$
8
votes
2answers
126 views

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 ...
1
vote
2answers
52 views

What is the fourth dimension of a Tesseract?

Is the fourth dimension of the Tesseract time? That is why it is represented as a moving 3D structure on Wikipedia? I am asking because I have trouble understanding what it is.
4
votes
1answer
100 views

How can I visualize what open sets “look” like in unfamiliar topological spaces?

The question is extremely general, but I do have a specific case I'd like to look at, and I'm hoping that some combination of specific pointers and general advice will help me out. Consider the ...
-1
votes
1answer
35 views

Books on the visual/graphical aspects of geometry

Are there any books providing a general overview of the visual/graphical aspects of geometry? For example, Tilings (e.g. hyperbolic) and tessellations Plane/space filling shapes/objects (e.g. ...
2
votes
1answer
230 views

Is there a geometric projection for every complex function?

I was wondering about the best way to visualize complex functions. As they're $$ {\mathbb R}^2 \rightarrow {\mathbb R}^2\ ,$$ I think best way are complex plane image/grid transforms like they used in ...
2
votes
1answer
41 views

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 ...
2
votes
1answer
586 views

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) ...
1
vote
0answers
35 views

Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
0
votes
2answers
63 views

Looking for intuïtive explanation why contour integral of $\frac{dz}{z} $equals $2\pi i$ in complex analysis

$$\oint \frac{dz}z = 2\pi i$$ I've seen the derivation of it using the parametrisation. Since this result is used all the time in my complex analysis course, i'd like to understand this ...
3
votes
1answer
110 views

Intuition - Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection ...
0
votes
0answers
8 views

Are there tools for presentation and vizualization of deduction?

I read that Kalish and Montague introduced a natural deduction method (http://en.wikipedia.org/wiki/Donald_Kalish), which can be easily implemented in software. Any other tools who can help a logician ...
0
votes
1answer
22 views

How to visualize implicit functions

I have a task of visualizing few implicit functions. Firstly lets say I have the following function of $N$: $$\epsilon = \sqrt{\frac{8}{N}\ln \left( \frac{4(2N)^{50}}{0.05} \right)}$$ Now this is ...
4
votes
0answers
57 views

How to visualise Bollobas' 1965 theorem?

Theorem $[n]=\{1,\ldots,n\}$. Let $\lbrace (R_i, S_i), i \in I \rbrace, R_i, S_i \subset [n]$ be such that $R_i \cap S_i = \emptyset, R_i \cap S_j \ne \emptyset (i \ne j)$. Then $$\sum_{i \in I} ...
1
vote
1answer
55 views

Visualizing second-order Markov chain

You can visualize a first-order Markov chain as a graph with nodes corresponding to states and edges corresponding to transitions. Are there any known strategies to visualize a second-order Markov ...
0
votes
0answers
9 views

Draw an ellipse corresponding to a bivariate normal distribution

Let $$\mu=\left(\begin{array}{c}\mu_1 \\ \mu_2\end{array}\right)$$ and $$\Sigma=\left(\begin{array}{cc}\Sigma_{1,1} & \Sigma_{1,2} \\ \Sigma_{2,1} & \Sigma_{2,2}\end{array}\right)$$ be ...
2
votes
7answers
602 views

Software to display 3D surfaces

What are some examples of software or online services that can display surfaces that are defined implicitly (for example, the sphere $x^2 + y^2 + z^2 = 1$)? Please add an example of usage (if not ...
3
votes
3answers
1k views

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. ...
2
votes
1answer
73 views

Why is it not possible to visualise a 4th dimension object? [duplicate]

By drawing a cube on a paper or by seeing it on a screen (a 2D surface - see Figure below), we can sort of visualise how a 3D cube would look like. I was wondering whether we will be able to ...
9
votes
2answers
884 views

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?
0
votes
1answer
244 views

Visualization of 2-dimensional function spaces

As a follow-up question to what is the norm measuring in function spaces I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...
4
votes
5answers
284 views

Geometric visualization of covector?

How could I geometrically visualize a linear functional?
1
vote
1answer
25 views

How to visualize probability distributions in terms of sets - joint and marginal?

Let there be two sets, $\mathcal{X},\mathcal{Y}$, both finite, and they represent the set of values that the discrete random variables, $X,Y$ can take. $\mathcal{P}_{Y|X}$ be all possible ...
6
votes
3answers
323 views

Proofs without words of some well-known historical values of $\pi$?

Two of the earliest known documented approximations of the value of $\pi$ are $\pi_B=\frac{25}{8}=3.125$ and $\pi_E=\left(\frac{16}{9}\right)^2$, from Babylonian and Egyptian sources respectively. ...
0
votes
1answer
17 views

visualizing a function from the plane into the reals

If $$(x_1,y_1), (x_2,y_2),(x_3,y_3)$$ are points in the plane and if $a,b$ are fixed real numbers, how can I visualize $$f=(ax_1+b-y_1)^2+(ax_2+b-y_2)^2+(ax_3+b-y_3)^2$$ as a function from the plane ...
4
votes
5answers
276 views

How to graph/visualize complicated inequalities

I'm having trouble visualizing areas defined by for example, $$ x^2 + y^2 \leq 2y $$ Or $$ (x^2 +y^2)^2 \leq 2a^2(x^2 - y^2) $$ What is the thought process in picturing these regions?
0
votes
0answers
20 views

Affine Transformation and Continuous Deformation

How do these two concepts relate? Thus far I have a (what I think is a) good intuitive idea of a continuous deformation- the visual basically looks like the boundary being stretched so that it never ...
2
votes
2answers
64 views

How can I visually imagine the area of a circle divided by $\pi$?

If I have a circle with an area of 100 units^2, and I divide it by $\pi$, how can I imagine that visually in my mind? Since 100 / $\pi$ =~ 31.83, and the square of that is =~ 5.64, I currently ...
5
votes
1answer
227 views

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, ...
4
votes
2answers
70 views

Visualizing Lie algebra of SO(3)

Let $SO(3)$ be the Lie group of 3D rotations. Rotation about z-axis by an angle $\phi$ is represented in standard basis by this matrix: $$ \begin{pmatrix} \cos \phi & -\sin\phi & 0 \\ ...
1
vote
2answers
40 views

Parallel Lines Intersecting in the Projective Plane

My question is about visualizing projective space, in particular the real projective plane $\mathbb{P}^2(\mathbb{R})$. I know there are different ways to define this space, but in each we can say that ...
3
votes
3answers
71 views

Viewing an abelian group using cayley diagram

I cannot understand this way of viewing whether a group is abelian using cayley's diagram: (from Visual group theory book) What I can't understand is that while checking being abelian we check ...
1
vote
1answer
35 views

Order of an element in direct product using cayley's diagram

How can I find the order of element (1,1) of the group $C_4\times C_3$ visually in the diagram below :
0
votes
1answer
35 views

Analysing/Visualising shape of multi-variate function.

I have an unknown function $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ for which I'm determining a first order Taylor approximation through a non-linear optimization process in six variables (the ...
2
votes
1answer
38 views

Criteria of regularity that a Cayley's Diagram should meet .

As referred in the Visual group theory Book by Nathan Carter- The unofficial definition of a group says that : A group is a collection of actions satisfying the rules: 1. there is a predefined list ...
5
votes
0answers
194 views

Colored Picture for Equivalence Classes, Relations, Partitions, ..

Origin — A Book of Abstract Algebra — Charles Pinter — p120. I'm trying to sketch a colored picture for the ideas from equivalence classes, equivalence relations, partitions, etc... underneath. ...
0
votes
0answers
23 views

Nyquist limit explanation

Kindly explain Nyquist in easy words. The actual question is as follows. We can attempt to display sampled data by simply plotting the points and letting the human visual system merge the points into ...
1
vote
2answers
262 views

Visualization of Eratosthenes’ sieve

In otherwise great paper on prime numbers, I found following visualization of Eratosthenes’ sieve: I found it somewhat scary and confusing. Is there any better visualization of Eratosthenes’ sieve ...
0
votes
1answer
63 views

Line integrals of vector fields-positive, negative, or zero

I have a question about line integrals of vector fields being positive, negative, or zero. If you are measuring the work it takes to "push" a point on the curve through the vector field, does this ...
1
vote
0answers
75 views

Geometrical application of generation function for permutation

It is quite well known that the generation function for permutations is represented as $$(1+x)(1+x+x^2)\dots(1+x+x^2+x^3...+x^{n−1})$$ (See, e.g., question The generating function for permutations ...
1
vote
1answer
53 views

Plotting the intersection of multiple surfaces with WolframAlpha

I want to plot the intersection of two surfaces like in this post. But if I enter the much simplified expression ContourPlot3D[{x^2 + y^2 + z^2 - 4=0, xy=1}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}] ...
86
votes
5answers
10k views

How were 'old-school' mathematics graphics created?

I really enjoy the style of technical diagrams in many mathematics books published in the mid-to-late 20th century. For example, and as a starting point, here is a picture that I just saw today: ...
1
vote
2answers
43 views

$2\times$ distance to a line${} = {}$distance to a point, in geogebra

While preparing for my math exams, I got this question: give the locus (if thats the right word) where $2$ times the distance to the line l $x=8$ equals $1$ time the distance to point F $2,0$. I was ...
1
vote
0answers
20 views

Different ways of visualizing a certain classes of single variable complex functions

Single variable complex functions of a real variable are ubiquitous in engineering contexts such as control engineering and signal processing, and visualizing them is of utmost importance in designing ...
16
votes
10answers
2k views

Visualizing the square root of 2

A junior high school student I am tutoring asked me a question that stumped me - I was wondering if anyone could shed some light on it here. We were talking about how the square root of 2 is an ...
2
votes
3answers
292 views

Polar coordinates for $xz$-plane: $z = r\sin\theta$ ? [Stewart P1091 16.7.25]

$1.$ The unit disk is projected onto the xz-plane, so shouldn’t $x = 1\cos \theta$ and $\color{red}{z = 1 \sin \theta} $? User Semsem below kindly identified the problem: The normal to the ...
2
votes
2answers
170 views

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 ...
2
votes
1answer
84 views

What are all the boundary curves for this combined cone and cylinder? [2013 10C]

Consider the bounded surface S that is the union of $x^2 + y^2 = 4$ for $−2 \le z \le 2$ and $(4 − z)^2 = x^2 + y^2 $ for $2 \le z \le 4.$ Sketch the surface. Use suitable parametrisations for ...
0
votes
2answers
305 views

Parametric equations and specifications of a triskelion (triple spiral)

I haven't been able to find the parametric equations and specifications to form a triskelion, a triple spiral (this is made of three interlocked couples of spirals). Using the parametric equation of ...