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5
votes
0answers
188 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. ...
2
votes
1answer
82 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 ...
5
votes
1answer
83 views

Can something like $\text{Hom}(V,K)$ be visualised?

I have no trouble visualising vector spaces like $\Bbb R^3$ and (e.g.) a subspace of dimension $2$, which would just be a plane through the origin of a $3$-D space, but I'm having trouble visualising ...
1
vote
1answer
40 views

Collection of Intuitive / Visual Derivations of Mathematical Concepts and Formulas

I find it difficult to simply memorize mathematical formulas in engineering without understanding what it means and what the result is like, but I realized that many mathematical relationships can be ...
5
votes
0answers
64 views

What group are the group of symmetries of these figures isomorphic to - Fraleigh p. 85 Theorem 8.23, 24, 26

In this section we discussed the group of symmetries of an equilateral triangle and of a square. In Exercises 23 through 26, give a group that we have discussed in the text that is isomorphic to the ...
5
votes
2answers
234 views

How to Visualize Diagonally Opposite Vertices

Consider a cube that exactly fills a certain cubical box. As in Examples 8.7 and 8.10, the ways in which the cube can be placed into the box corresponds to a certain group of permutations of the ...
6
votes
1answer
252 views

Animations or Pictures of Group of Rigid Motions (or Rotations) of the Cube

Consider a cube that exactly fills a certain cubical box. As in Examples 8.7 and 8.10, the ways in which the cube can be placed into the box corresponds to a certain group of permutations of the ...
4
votes
0answers
46 views

Visual Solution - Find All (Cyclic) Subgroups of $D_4$ generated by 1, 2, … elements - Fraleigh p. 84 8.19

Verify that the subgroup diagram for $D_4$ shown in Fig. 8.13 is correct by finding all (cyclic) subgroups generated by one element, then all subgroups generated by two elements, etc. Here, $p_i$ mean ...
4
votes
2answers
71 views

Isomorphism of Group with the Image of the Group - Fraleigh p. 82 Lemma 8.15

I found multifarious duplicates that I listed at http://math.stackexchange.com/a/631364/53934. I edged the purple part because my answer proves it more efficiently. I remember that any function ...
5
votes
2answers
193 views

Visual Group Theory's Intuitive Proof - Cayley's Theorem - Nathan Carter pp. 85, Theorem 5.1

Theorem 5.1. Cayley's Theorem: Every group is isomorphic to a collection of permutations. Figure 5.31. A multiplication table for the group $V_4$, with nodes numbered 1 through 4 to facilitate ...
3
votes
1answer
390 views

If two sets have the same cardinality, then so do their power sets. Converse can't be answered?

For infinite sets $A, B$, $|A| = |B| \Longrightarrow \require{cancel} \cancel{\Longleftarrow} |P(A)| = |P(B)|$. I recast http://ph.answers.yahoo.com/question/index?qid=20100907061641AAE2Vfq : ...
3
votes
2answers
246 views

Easy visualizations of small countable ordinals

The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as ...
1
vote
2answers
80 views

Elliptic Curve and Conjugation

If I consider an elliptic curve $C$ as a Riemann surface cut out in $\mathbb{C}P^2$ by a homogenous cubic, and if that cubic is defined over $\mathbb{R}$, then I think we have a conjugation map ...
2
votes
1answer
59 views

Credit Given - Geometricly Modeling Infinity with 3 planes and 9 circles - Ratio of Circles

Refer to the attached diagram sketch to help visualize the equation. I am requesting help with an interesting math problem. Basically, I am diagraming infinity using three planes. These planes ...
1
vote
2answers
72 views

Visualizing a complex valued function of one real parameter

I'm looking for a way to capture/graph or visualize it in my head, but I can't find how.. a 2-dimensional path won't do, because it doesn't reveal the rate-of-change.. 2 1-dimensional graphs on top ...
9
votes
1answer
123 views

Intuition & Proof of rank(AB) $\le$ min{rank(A), rank(B)} (without inverses or maps) [Poole P217 3.6.59, 60]

I'm aware of analogous threads; I hope that mine is specific enough not to be esteemed one. $\mathbf{a^i}$ is a row vector. $A, B$ are matrices. Prove: $1$. $\mathbf{a^i}B$ is a linear ...
12
votes
1answer
151 views

Handbook of mathematical drawing?

My drawing skills are pretty awful, and although I haven't yet had to teach multivariable calculus, someday I will. (And next semester in calculus II we're already doing some volumes by integrating ...
4
votes
3answers
189 views

Why are orthogonal matrices generalizations of rotations and reflections?

I recently took linear algebra course, all the I learned about orthogonal matrix is that matrices is that Q transposed is Q inverse, and therefore it has a nice computational property. Recently, to my ...
4
votes
1answer
391 views

Intuition for Cayley Table and Cayley Table for identity, inverse but not associativity - Fraleigh p. 47 4.24

$1-2.$ I understand these proofs on pp. 5-6 for Cayley tables but what are the intuitions for Sudoku property : Every element of the group appears only once in each row and each column. Symmetric ...
2
votes
1answer
66 views

Intuition/Picture - Theorems on Linear Independence, Span, Basis, Dimension [Poole, Section 6.2]

I'd like to ask about the intuitions for these theorems, absent in David Poole's Linear Algebra (to which the page numbers refer). Also, are there pictures for these theorems?
3
votes
1answer
90 views

A visual proof of - Curved surface area of a hemisphere = 2(Area of circle)

Suppose we have a circle with radius $r$ . So its area is $\pi r^2$. Now suppose we have a hemisphere of the same radius ie. $r$. Then its curved surface area is $2 \pi r^2$. Which means it is equal ...
3
votes
0answers
138 views

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 $\omega^\omega$: ...
1
vote
0answers
64 views

Complex analysis visualization (Cauchy Theorem, Residue Theorem)?

I usually think of complex functions on the complex plane like vector fields. So basically what I have problems with is visualizing firstly Holomorphic functions. I have also read and successfully ...
2
votes
0answers
36 views

How to Intuit/See Matrix Factorisation [GStrang P250 Ex 5.1A]

I beg leave for your forgiveness over the colours. Please enlighten me if there's a more efficient way. How is the determinant of the checkerboard sign pattern matrix, $ \begin{bmatrix} a(1, ...
2
votes
1answer
145 views

Questions on Prof Gilbert Strang's Picture on the 4 Fundamental Subspaces [Strang P187]

I consulted 1 and 2 but still have questions. What follow are modified editions of Prof Strang's picture from Intro to Lin Alg, 4th Ed: $\Large{{1.}}$ In the given correct version, why is the ...
6
votes
0answers
95 views

Visual Proofs of Series Summations

I'd like to put together a compilation of visually geometric proofs of series summations. I have three famous 2D examples to clarify what I mean below, but other "visually geometric" proofs of an ...
6
votes
3answers
317 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. ...
3
votes
1answer
77 views

Picture for Conditional Version of Law of Total Probability

http://jeremykun.com/2013/03/28/conditional-partitioned-probability-a-primer/ boasts a stupendous picture of the (Law of) Total Probability Theorem: $Pr(A) = \sum_n P(A|E_n) \, P(E_n)$ I'd be ...
1
vote
0answers
92 views

Applet to find least-crossings drawing for an input graph

Is there a convenient online applet that allows me to draw a graph, after which it outputs a plane drawing of an isomorphic graph that has (approximately) the least number of crossings among all ...
3
votes
1answer
61 views

Visualizing $Fct(Op_X, Set)$

I can't seem to wrap my mind around what is going on when I try to visualize $Fct(Op_X, Set)$, as one example. Now I know that a functor is a morphism between categories hence we have a morphism ...
0
votes
1answer
83 views

Visualization of rotation in $\mathbb R^3$

I am trying to visualize the following rotation of $\mathbb R^3$, but it is very difficult. I want to get the answer by intuition, and not by using the Rodrigues rotation formula or conjugation of ...
3
votes
0answers
154 views

Power-set in Hypercube: historical background of indexing each term like Hasse Diagram?

My instructor wants references about the indexation over the hypercube, related question here, he does not believe that I was the first who used it -- [update] thanks to a comment, the name is Hasse ...
1
vote
1answer
53 views

Why it makes sense to think of multivectors as “paralelograms”?

Let $V$ be a vector space over the field $\mathbb{K}$ and let $T(V)$ be it's tensor algebra. We usually define the exterior algebra $\Lambda (V)$ by the following process: we consider the bilateral ...
4
votes
1answer
80 views

Visualizing a homotopy pull back

I am currently taking a course in algebraic topology, which also covers a lot of category theory. My question is pretty straightforward: How do you visualize the (homotopy) pull back of a diagram ...
0
votes
0answers
48 views

Must the “n” in mod(n) always be prime?

I'm experimenting with mod(n) and have the following questions even after reading the Wiki page and numerous articles about the subject. Must mod(n) always be prime for cryptographic purposes? Is ...
4
votes
1answer
230 views

Alternate bases and visualized counting and arithmetic: See images

I've wondered how math would be different if we used a different base for counting (pi? e? most equations would be different). Attached are 2 images that I created to illustrate the concept. The ...
1
vote
1answer
74 views

Software to produce graphics of triangulated surfaces

I would like to find a software that lets me create graphics of a surface with a triangulation on it. It doesn't need to be very fancy; I just need to explain to a bunch of high schoolers what a ...
2
votes
1answer
82 views

Visual proof ot the distributive property in $\mathbb{Z}$

Is there a intuitive/visual (not formal) "proof" that the distributive property holds in $\mathbb{Z}$? For the natural numbers $\mathbb{N}$ I know something like this: There are two ways to get ...
3
votes
2answers
94 views

Visualizing a projective variety

What does the variety $V(x_0^2+x_1^2+x_2^2)\subset \mathbb{P}^2$ look like? It seems to me like a single point... In general, are there any good ways/tips/tricks to visualize projective varieties?
7
votes
4answers
404 views

Visualising finite fields

I'm interested in finding visual and/or physical approaches to understanding finite fields. I know of a few: V. I. Arnold has a few pictures of 'finite circles' and 'finite tori' in his book Dynamics, ...
4
votes
2answers
139 views

Geometric/Visual Solution - Shortest Vector for which Dot Product = x + 2y = 5. (Strang P21 1.2.26)

Much as I referenced this same exercise, I'm questing after an exclusively geometric solution. Question: If $\color{#0070FF}{\vec{v} = (1,2)}$ draw all vectors $\vec{w} = (x,y)$ in the plane ...
1
vote
0answers
50 views

Visually apealing holologous transformation of a given contour

There is this problem which roughly says: You want to put a framed picture onto the wall with a cord to the picture frame. The cord is a single one, and both ends are attached to the frame. ...
0
votes
2answers
262 views

Plotting complex maps as z-plane and w-plane

I have seen many plots of complex maps as colors, such as $w = sin(z) = 0$: However, I am looking for more involved plotting capabilities. For example I would like the ability to see the z-plane ...
42
votes
13answers
7k views

Interesting math-facts that are visually attractive

To give a talk to 17-18 years old (who have a knack for mathematics) about how interesting mathematics (and more specifically pure mathematics) can be, I wanted to use nice facts accompanied by nice ...
1
vote
0answers
218 views

Help to understand the concept of diminishing returns

Suppose I have this function: $q=f(k,l)=600k^2l^2-k^3l^3$ Then, $f_l=1200k^2l-3k^3l^2$ $f_k=1200kl^2-3k^2l^3$ $f_{ll}=1200k^2-6k^3l$ $f_{kk}=1200l^2-6kl^3$ $f_{kl}=f_{lk}=2400kl-9k^2l^2$ Now, ...
3
votes
1answer
309 views

Mental Math Visual Retention

I have always been good at math but even I struggle with visualizing numbers in my head. I am seeking help on this forum to see if any mathematicians here have experienced the same issue I currently ...
4
votes
1answer
170 views

Is 4D visualization necessary? [closed]

Is 4D visualization necessary in order to be successful at math (complex analysis for example)?
2
votes
1answer
191 views

elements of $SL(2,\mathbb{Z})$ which fix roots of Klein's absolute invariant $j(\tau)$

As a followup to this question (resulting video here), I'd like to make a video showing elements of $\mathbf{SL}(2,\mathbb{R})$ which fix roots of Klein's absolute invariant $j(\tau)$, stylized before ...
1
vote
1answer
118 views

Does a Möbius strip have only one shape? Or may it have different shapes?

I'm reading a book about geometry, and after thinking and viewing the Möbius strip, I want to know whether the book is right or not. The book says with a little description (that I can't write here ...
3
votes
1answer
178 views

Visualization of the diffeomorphism!

Basic to all mathematics is the notion-here used quite informally-of a set with structure. For every type of structure there is a notion of equivalence (or isomorphism)-a one-to-one onto ...