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

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96
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20answers
20k views

Visually deceptive “proofs” which are mathematically wrong

Related: Visually stunning math concepts which are easy to explain Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually ...
2
votes
3answers
427 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 disk ...
2
votes
2answers
263 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 ...
4
votes
3answers
441 views

Visualizing mathematics and geometry

Im writing a paper on the role of visualization in mathematics and specifically geometry. I was wondering if it is possible to represent any arbitrary system of relations and manipulable objects ...
18
votes
6answers
2k views

Visual explanation of the following statement:

Can somebody fill me in on a visual explanation for the following: If there exist integers $x, y$ such that $x^2 + y^2 = c$, then there also exist integers $w, z$ such that $w^2 + z^2 = 2c$ I know ...
1
vote
2answers
217 views

Is this a counterexample to “continuous function…can be drawn without lifting” ? (Abbott P111 exm4.3.6)

I'm au courant with http://math.stackexchange.com/a/288133 and http://math.stackexchange.com/a/422001. They're both Abbott P111 exm 4.3.6 which proves "a continuous function is sometimes described, ...
0
votes
1answer
98 views

How to visualize four dimensional tic-tac-toe?

I have played three dimensional tic-tac-toe with three players before, and we had no problem visualizing it. We drew three layers on a sheet of paper and just remembered all the different ways you ...
1
vote
1answer
82 views

Which line is the antiderivative and why?

The graph of a function $f$ is shown. Which graph is an antiderivative of f and why? This should be easy but I keep second guessing myself so I thought I'd check with you magnificent people. I'm ...
1
vote
1answer
69 views

Tools or Resources for pictures and visualizations

The popularity of books like Visual Group Theory and Visual Complex Analysis validates the importance of pictures and visualization for complex subjects. Unfortunately, I'm not aware of similar books ...
0
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3answers
124 views

If $\int^{b}_a f > 0$ then there is some interval and $\delta > 0$ on which $f(x) \ge \delta$ (Abbott pp 199 q7.4.4d)

True or False. If $\int^{b}_a f > 0$, then $\exists \; [c,d] \subseteq [a,b]$ and $\delta > 0$ such that $f(x) \ge \delta$ for all $x \in [c,d]$. 1. We need to determine if true or false. ...
0
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1answer
270 views

Cauchy's Generalized Mean Value Theorem. Required function. (S.A. pp 140 t5.3.5)

Cohen, Henle. Calculus pp 827, (http://www.vias.org/calculus/09_infinite_series_10_06.html) I revised the footnote in pp 14 http://math.uga.edu/~pete/2400calc2.pdf. This theorem can be illustrated ...
-2
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1answer
358 views

If f' exists and f'(c) > 0 then f'(x) > 0 for all |x - c| < d for some d. (S.A. pp 137 question 5.2.8b)

If $f'$ exists on an open interval, and there is some point $c$ where $f'(c) > 0$, then there exists a d-neighborhood $\{x \in \mathbb{R} : |x - c| < d\} = V_d(c)$ around c in which $f'(x) > ...
0
votes
1answer
83 views

logic - how to model or diagram conditional probabilities of multiple related scenarios.

I am interested in modeling questions and specific outcomes so that i can evaluate conditional probabilities and mathematical expectation. I am looking for a way to diagram or otherwise describe the ...
2
votes
3answers
125 views

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 $...
-1
votes
1answer
336 views

Intuition - Theorem - A group homomorphism preserves normal subgroups - Fraleigh p. 149. Theorem 15.16

p. 128, 129. Theorem 13.12. Let $h$ be a homomorphism of groups $G \to G'$. III. If $S \le G$, then $h[S] \le \color{red}{G'}$. IV. If $S' \le G'$, then $h^{-1}[S'] \le G$. p. 149. Theorem ...
0
votes
1answer
93 views

Eigenvalues and Corresponding Eigenspace Bases

Could someone describe the eigenvalues of $ \left( \begin{array}{cc} 2 & 1 \\ -1 & 2 \end{array} \right) $, as well as the bases of the corresponding eigenspaces? I received eigenvalues ...
5
votes
3answers
315 views

Visual explanation of $\pi$ series definition

Can you visually explain why the following is true: $$ \frac{\pi}{4} = \sum\limits_{k=0}^\infty \frac{(-1)^k}{2k + 1} = \frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}\ldots\approx 78.5\% ...
0
votes
0answers
179 views

Intuition of Picture - Collapse, Factor Group, Homomorphism, Normal Subgroup - Fraleigh p. 144 Figure 15.1

Let $N \unlhd G$. In the factor group $G/N$, the subgroup $N$ acts as identity element. Regard N as being collapsed to a single element, to the identity element. This collapsing of N together with ...
1
vote
0answers
410 views

Intution, Figure. Negation of Continuity and Uniform Continuity (S.A. pp 117 T4.4.6)

Every time I need negation, I have to write out all the logical symbols to negate manually. I know how to determine these negations myself. But I want to compehend intuition or figure like http://...
1
vote
0answers
174 views

Intuition on Axiom of Completeness

♪ (J. Stewart. Calculus 6th ed. pp 682) Axiom of Completeness = AoC = A nonempty set of real numbers that has an upper bound has a least upper bound. AoC is an expression of the fact that there is ...
3
votes
3answers
2k 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
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1answer
345 views

Characterize normal subgroups - Find all subgroups of $S_3$ conjugate to $\{id, (1,3) \}$ - Fraleigh p. 143 14.29

(27.) A subgroup H is conjugate to a subgroup K of a group G (viz. p. 141 $K \le G$ is a conjugate subgroup of $H$), if $i_g[H] = gHg^{-1} =K$ for some $g \in G$. Show that conjugacy is an ...
3
votes
2answers
339 views

Why is this a good picture of a covector?

I'm reading a book about applied differential geometry and the author says: "suppose $V$ is a finite dimensional vector space. For a given covector $\omega \in V^\ast$, the set $\hat{\omega}$, of ...
1
vote
1answer
42 views

Draw following set in $\Bbb{R}^3$

How can I draw following set in $\Bbb{R}^3$? $$ M = \left\{(x,y,z) : \sqrt{x^2 + y^2} \le z \le 1 \right\} $$ I have the answer in the book but I don't want to check it before i try to solve it ...
-1
votes
1answer
71 views

If $\phi[H] \subseteq H'$, homomorphism from G to G' induces homomorphism from G/H to G'/H' - Fraleigh p. 143 14.39

Let $H \trianglelefteq \text{ group } G$ and let $H' \trianglelefteq \text{ group } G'$. Let $\phi$ be a homomorphism of G into G'. Show that if $\phi[H] \subseteq H'$, then $\phi$ induces a natural ...
2
votes
2answers
135 views

Intuition - Fundamental Homomorphism Theorem - Fraleigh p. 139, 136

Let $\phi: G \to H$ be a group homomorphism with $K = \ker\phi$. Then $G/K \simeq \phi[G]. $ The hinge to the proof is to define $\Phi: G/K \to \phi[G]$ given by $\Phi(gK) = \phi(g)$. Then we must ...
2
votes
1answer
148 views

Visualize meaning of quotient in quotient map, group - etc?

What are the reasons for the name "Quotient" in Quotient map, group - etc? Overhead picture shows each of the three cosets in $A_4$ is mapped to a single - gray - node. But this isn't division? ...
3
votes
0answers
255 views

Visualize normal subgroup, normalizer, cosets.

A few important aspects of the relationship $H \lhd N_G(H) \le G$ are highlighted in Figure 7.31. First, the size of $N_G(H)$ is some multiple of |H|, and the size of G is some multiple of $N_G(H)$, ...
4
votes
2answers
240 views

Visualize Fundamental Homomorphism Theorem for $\phi: A_4 \rightarrow C_3$

Question 1. How do you see $\ker\phi = V_4 $ = Klein 4 group ? Book doesn't give formula for $\phi$? Question 2. What's $H$ in $i(aH) = \phi(a)$? I think $H = \ker\phi$ ? Question 3. Why is $i: \...
1
vote
0answers
71 views

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 $\mathbb{...
-1
votes
2answers
63 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 <...
0
votes
0answers
65 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 $\...
2
votes
2answers
767 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 ...
1
vote
1answer
326 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. ...
2
votes
1answer
85 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 = (1,0)\...
0
votes
1answer
69 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 $&...
0
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1answer
202 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 ...
3
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1answer
141 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. $44.$ ...
0
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2answers
122 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 $\mathbb{R^3}$ ...
1
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1answer
184 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}[ \; \{ \; \...
0
votes
2answers
68 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 (...
2
votes
1answer
432 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\} = \{\color{magenta}{h^{−1}...
1
vote
1answer
156 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 the ...
5
votes
1answer
103 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 ...
2
votes
1answer
56 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 ...
1
vote
0answers
177 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 ...
1
vote
2answers
546 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 ...
2
votes
1answer
582 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 ...
0
votes
2answers
88 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 $...
4
votes
2answers
584 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 ...