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

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Visualizing a complex curve

This may be as much a question about computers as about math. Let $C=\{f(r,s,t)=0\}$ be a curve in $\mathbb{CP}^2.$ By forgetting about the points at infinity we can view $C$ as a surface in ...
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2answers
43 views

Graphic of function similar to Sin(x) but with scaled size

I wont to build a graphical representation of numbers, as "waves": Similar as the graphic of sin x but for each number N it gets bigger and bigger crosses the ...
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45 views

Motivation behind Definition of Projection [Poole P27]

In the long paragraph above equation $(2)$, http://mathinsight.org/dot_product avers: This leads to the definition that the dot product $\mathbf{a⋅b}$, divided by $∥\mathbf{b}∥$ (= magntitude of ...
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2answers
455 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 ...
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1answer
211 views

Visualize left, right cosets and conjugation

I drew everything that's in orange. Figure 6.8. Left illustration - Each left coset gH is where H arrows can reach from g, which looks like a copy of H based at g, as in the left illustration. ...
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1answer
64 views

Visualize cosets of $\left<(0,1)\right>$ partition $C_3 \times C_3$

Page 105 says - A careful look at Figure 6.9 reveals that the cosets of $\left< \, (0,1) \,\right>$ partition $C_3 \times C_3$. How is this true? The picture shows $gH = left picture = ...
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1answer
63 views

Visualize $C_2 \times C_4$ is normal subgroup

Page 120 says: Given our recent work with subgroups, you may have noticed that $C_2$ is a subgroup of $C_2 \times C_4$; specifically, it is the subgroup $<(1,0)>$. Furthermore, the cosets of ...
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1answer
132 views

Visualize $A_4$ and $\langle x, z\rangle$ isomorphic to the Klein 4 group

Page 136 says Following Step 1 of Definition 7.5, the top of Figure 7.23 shows $A_4$ organized by the subgroup $\langle x, z\rangle$ (which is isomorphic to the Klein $4$ group. This reorganization ...
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120 views

Infer distance from a point to a line, from the distance from a point to a plane [Stewart P793 12.4.44]

I'm able to prove $44$, but how would one deduce $43$ from it without further industry, forthwith? $43$ seems like a reduced, 2D version of $44$? I'm not enquiring about individual proofs. $44.$ ...
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2answers
105 views

Visualise all vectors perpendicular to one vector and two vectors in R^3 [Strang P19 1.2.6]

I'm only asking about visual/geometric solutions herein. (b) The vectors perpendicular to any vector in $\mathbb{R^3}$ lie on what?. (c) The vectors perpendicular to any two vectors in $\mathbb{R^3}$ ...
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1answer
151 views

Intuition - Homomorphic Image of Group Element is Coset - Fraleigh p. 135 13.52, p.130 Theorem 13.15

Theorem 13.15: Let $\phi: G \rightarrow G'$ be a group homomorphism, $g \in G$. Then $g\ker\phi = (\ker\phi)g = \operatorname{Im}^{-1} \left[ \; \{ \; \phi(g) \; \} \; \right] = \phi^{-1}[ \; \{ ...
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2answers
48 views

3-space viewer?

Is there a software package that would allow visulaizing/rendering some example structures in 3-space? Specifically, I'm thinking of something that would provide a 3-D rendering of, say, 3-vectors ...
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1answer
295 views

A subgroup has the same number of left and right cosets - Tricks - Fraleigh p. 103 10.32, 35

(32.) Let $H \le$ group G and let $a, b \in G.$ Prove or disprove. If ${aH= bH},$ then $Ha^{-1} = Hb^{-1}.$ $\color{blue}{Ha^{−1}} = \{\color{magenta}ha^{−1} | h ∈ H\} = ...
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1answer
109 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 ...
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1answer
89 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 ...
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1answer
45 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 ...
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126 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 ...
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2answers
348 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 ...
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1answer
390 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 ...
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2answers
80 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 ...
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2answers
378 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 ...
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1answer
820 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 : ...
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282 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 ...
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2answers
91 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 ...
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1answer
73 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 ...
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77 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 ...
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1answer
202 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 ...
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172 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 ...
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3answers
518 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 ...
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1answer
619 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 ...
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1answer
78 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?
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1answer
141 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 ...
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158 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 $\omega^\omega$: ...
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87 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 ...
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1answer
217 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 ...
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174 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 ...
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3answers
365 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. ...
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1answer
109 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 ...
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118 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 ...
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1answer
66 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 ...
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1answer
95 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 ...
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1answer
135 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 ...
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0answers
207 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 ...
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1answer
58 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 ...
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1answer
98 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 ...
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1answer
335 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 ...
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1answer
98 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 ...
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1answer
98 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 ...
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2answers
103 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?
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688 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, ...