Tagged Questions

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

187 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 ...
369 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. ...
119 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 ...
123 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 ...
68 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 ...
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 ...
151 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 ...
214 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 ...
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 ...
102 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 ...
357 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 ...
104 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 ...
99 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 ...
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?
711 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, ...
156 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 ...
59 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. ...
381 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 ...
16k 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 ...
307 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, ...
587 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 ...
210 views

Is 4D visualization necessary? [closed]

Is 4D visualization necessary in order to be successful at math (complex analysis for example)?
201 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 ...
151 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 ...
235 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 ...
149 views

Visualization of a set

How can I imagine the set $$M:=\left\{(x,y,z)\in\mathbb{R}^3:z=xy\right\}?$$ Is there a program that can visualize that?
141 views

How can I “move through a hypersphere?”

A man walking along a 2 dimensional circle will take a periodic path that begins and ends at the same point. Since he can travel in only a single direction, let's say how far along he is in his ...
60 views

Definition of the punctured $\mathbb{R}P^3$

I faced "punctured $\mathbb{R}P^3$" denoted by $\mathbb{R}P^3-\{{pt}\}$ in my studies. I dont know its definition and also my searches in the web are failed. Can anyone help me? What is the ...
382 views

Kakeya Needle problem video

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid: What I'm searching for is a visualisation of the ...
186 views

Software for visualizing partial derivatives?

I'm whipping up a set of notes, and I want to include a diagram or two showing some partial derivatives. Specifically, a diagram would include: a 3D surface of the form z=f(x,y), a plane of the form ...
388 views

Geometric visualization of covector?

How could I geometrically visualize a linear functional?
222 views

Visualization of immersed submanifold

I am trying to visualize the difference between immersed submanifold and embedded submanifold. At first, I thought that, for example, if I can embed manifold $M$ in $\mathbb{R}^4$ and if my friend can ...
199 views

Looking for proof-without-words of Bezout's identity

I'm looking for a "proof-without-words" of Bezout's identity (for integers). Does anyone know of one?
179 views

Visualise 3 simultaneous cubic equations

I have three equations of the form: $$\frac{i_1^3}{P_1}+i_1(Z_1+Z_2)+(i_2+i_3)Z_2-U_1=0$$ $$\frac{i_2^3}{P_2}+i_2(Z_1+Z_2)+(i_1+i_3)Z_2-U_2=0$$ $$\frac{i_3^3}{P_3}+i_3(Z_1+Z_2)+(i_1+i_2)Z_2-U_3=0$$ ...
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visualizing functions invariant (or almost) under modular transformation

In the spirit of Möbius Transformations Revealed, I'd like to make a pair of movies depicting how Klein's absolute invariant $j(\tau)$ and the Dedekind eta function $\eta(\tau)$ transform when ...
272 views

The complement of a torus is a torus.

Take $S^3$ to be the three-sphere, that is, $S^3=\lbrace (x_1,x_2,x_3,x_4):x_1^2+x_2^2+x_3^2+x_4^4=1\rbrace$. Using the stereographic projection, $S^3=\mathbb{R}^3\cup \lbrace \infty \rbrace.$ Can ...
2k views

Algebra: Best mental images

I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on ...
44 views

How to represent a bounded function

I am not completely sure whether this question belongs on mathematics.SE but I figured to give it a shot: I have a function which mathematically looks like this: $f(x)=\max(A,B\cos x)$ This will ...
357 views

Fuzzy Venn diagram regions labeled in ternary

I have a couple of questions about the Venn diagrams object : Words from the binary alphabet with n letters label each region of an order-n Venn diagram. Is there any more profound connection ...
119 views

Visualizing diffeomorphisms

This is probably a really basic question (hence my asking it here as opposed to MO). In a comment to a question on mathoverflow ...
I have a trouble learning Lie groups --- I have no canonical example to imagine while thinking of a Lie group. When I imagine a manifold it is usually some kind of a $2$D blanket or a circle/curve or ...