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

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17
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0answers
388 views

Visualizing the Partition numbers (suggestions for visualization techniques)

So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so ...
11
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0answers
155 views

Is it a good approach to heavily depend on visualization to learn math?

I am a third year undergraduate and I am a beginner on these "real mathematics" (no pun intended). Before contacting the "real math", my math level should be considered to be "good", although I was ...
11
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0answers
455 views

Visualizing a Calabi Yau

I would like to understand how I can visualize the quintic threefold $$ z_1^5 + z_2^5 + z_3^5 + z_4^5 +z_5^5 - 5\psi z_1z_2z_3z_4z_5 = 0$$ For a similar problem, Hanson proposes the following: ...
5
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0answers
66 views

An Auto-Generated Cartography of Mathematical Theories: Has it been done already?

While looking for a way to visualize the logical structure of mathematical theories a graph-like depiction came to my mind, where propositions are represented by vertices. An edge goes from ...
5
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0answers
328 views

Visualizing the domain of the square root

I would like to show someone the domain of the complex square root function (the 2-sheeted riemann surface). Is there a good interactive visualization software for this? I would like some sort of ...
4
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0answers
96 views

How to visualize cotangent spaces.

I was wondering how to intuitively and visually understand dual vector spaces and one-forms. So my question is (1), how to visualize cotangent spaces and (2), how to intuitively understand them? My ...
4
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0answers
89 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} ...
4
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0answers
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 ...
3
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0answers
89 views

Is there a way to visualize a group?

Is there a way to picture a group in ones head? I want to "see" the difference between abelian and non-abelian group. And if f is a group homomorphism, is there a way to see that Ker(f)=1<=>f ...
3
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0answers
123 views

Visualising surface integrals

For a current problem I am working on, I have run into angular surface integrals, i.e. the differential solid angle $\text{d}\Omega$. Specifically the surface integrals are defined by ...
3
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0answers
151 views

Is this a legit way to visualize complex functions?

I am doing laplace transform in a class and I hate how there seems to be no graphical support when things are transformed to laplace domain i.e. nobody cares what they look like in laplace domain But ...
3
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0answers
171 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 $\omega^\omega$: ...
3
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0answers
213 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 ...
3
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0answers
310 views

Imagining four or higher dimensions and the difference to imagining three dimensions

I’m very interested in how people envision four or higher dimensions. And I’m especially interested in how geometers and topologists who actually work in four dimensions do. Now I know of the video ...
3
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0answers
152 views

Visualizing and manipulating 4-dimensional data with 3D technology

It is possible to visualize 3 dimensional data (like a scatter plot) by projecting it on a 2 dimensional screen in a way that allows to interact with it in an intuitive way. Is it possible to ...
3
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0answers
139 views

Illustrations of a line and a curve intersecting for complex field

Are there nice illustrations on the Net of say $y=a·x+b$ and $y=x^2$ intersecting where x and y are complex? I'm thinking of the amplitude of y being depicted as height above the complex plane with ...
2
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0answers
48 views

Ideals in a ring as geometric objects?

I am interested in learing about the possibility of (one-sided) ideals in a ring being repreented geometrically. In other words, about their status as geometric objects (after all, they can be dealt ...
2
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0answers
49 views

What type of diagram could this be?

Some years ago, I saw a diagram somewhat like this somewhere on Wikipedia. I remember that it was supposedly used in some branch of mathematics. Unfortunately, neither Google Image nor trawling ...
2
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0answers
35 views

Hyperplane in a complex vector space

This is my first question on MSE, I'm sorry if there already exists similar questions, I couldn't manage to find it. My friend, who studies Physics, asked me about the meaning of "functional" so I ...
2
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0answers
86 views

Why are inter arrival times in the continuous version of discrete-time Markov chains always exponentially distributed?

I am curious whether there exist continuous time Markov processes for which the times between jumping times (which I call inter arrival times) are not exponentially distributed, but have some other ...
2
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0answers
89 views

Visualising algebraic topology

I'm new to algebraic topology and although I can follow the arguments it would be nice to be able to visualise important concepts like homology and excision. Can anyone recommend a book or other ...
2
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0answers
568 views

Intuition for gradient descent with Nesterov momentum

A clear article on Nesterov’s Accelerated Gradient Descent (S. Bubeck, April 2013) says The intuition behind the algorithm is quite difficult to grasp, and unfortunately the analysis will not be ...
2
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0answers
66 views

Is there a way to graphically show that a solution is the minimum or stationary solution to a functional?

I'm looking for the functional analogue to the visual representations of function optimization you most commonly see. To illustrate, if we have some function: $$ f(x) = (x-1)^2+1 $$ We can look at ...
2
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0answers
164 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 $N_G(H)$, ...
2
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0answers
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. $44.$ ...
2
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0answers
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. ...
2
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0answers
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?
2
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0answers
96 views

Help me to vizualise this falling ball on spinning Earth

The earth rotates. The ball falls in an latitude, not equator, let say in Germany. I am trying to understand how to express the ball in terms of the angular velocity on the planet. The constant ...
1
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0answers
53 views

What methods are known to visualize patterns in the set of real roots of quadratic equations?

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations ...
1
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0answers
60 views

How do I evaluate this sum :$\sum_{n=0}^\infty z^{n^3}$ and Is there a visual proof for it?

if $$\sum_{n=0}^\infty z^n = \frac1{1-z}, \quad z \in \mathbb{C},\; |z| < 1 .$$ then is there a way to deduce this sum:$$\sum_{n=0}^\infty z^{n^3}$$ from the above Identitie or any visual proof ...
1
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0answers
31 views

How to visualize orientation of 3d objects

The way I visualize orientations of $1$- and $2$-dimensional objects is by an ant walking along a path. For a $1$d object (like a line/ line segment/ etc), just place the ant on the line and confine ...
1
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0answers
11 views

Visualization of the fact that the integers defining lens spaces must be coprime

This is related to this question I asked: Visualization of Lens Spaces and is also related to this question by @Earthliŋ: Why are the integers appearing in lens spaces coprime? I understand the ...
1
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0answers
81 views

I seek Visual illustrations of Concepts of Mathematics as animated videos for students of Higher Mathematics

There are some very good animated videos explaining concepts of mathematics on youtube, like videos of website " why U" but most of these deal with elementary mathematics. I am searching for videos ...
1
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0answers
32 views

Visualizing Riemannian surface

Given a multi-valued complex function $f: z = x+\mathrm{i}\,y\rightarrow w=u+\mathrm{i}\,v$ with $x,y,u,v\in\mathbb{R}$, we know the image $\{f(z)\,|\,z\in\mathbb{C}\}$ is a Riemannian surface. How to ...
1
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0answers
60 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 ...
1
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0answers
80 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
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0answers
30 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 ...
1
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0answers
122 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 ...
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0answers
111 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 ...
1
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0answers
49 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 ...
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0answers
128 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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0answers
91 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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0answers
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, ...
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0answers
106 views

Which space this space drawn in this picture is homeomorphic?

Based in this question Why this space is homeomorphic to the plane? I would like to know which space this space is homeomorphic, any help or an intuitive idea are welcome. [Context of Image: ...
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0answers
65 views

Visualising the derivative/slope $f'(x_0)$ of $f:\mathbb{R} \rightarrow \mathbb{R}$ as a line segment

A function $f:I\subseteq \mathbb{R} \rightarrow \mathbb{R}$ that is differentiable at $x_0 \in I$ obeys the following equality for all $h\in (I-x_0)$ (i.e. for all $h\in \mathbb{R}$ such that $x_0+h ...
0
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0answers
13 views

Stellating the Octahedron

I am trying to create a very primitive animation/demonstration that shows the stellation of an octahedron to yield the stella octangula. Unfortunately, it seems that the mental image I have for ...
0
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0answers
24 views

How to calculate optimal sizes of rectangles for this type of array visualization?

Given array of positive numbers, I would like to draw this diagram and be able to put descriptions inside: There should be no empty space left, consider that these numbers represent % of total. Do ...
0
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0answers
59 views

How do I visualize this quotient space?

If $V = [0,1] \times [0,1] \subset \mathbb{R}^2$. We define the equivalence relation $\sim$ on $V$ as follows: every element $(x,y) \in V$ is equivalent with itself and besides that the three ...
0
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0answers
7 views

What is angular acuity?

I've been looking for the answer for a while now, but just can't seem to find it. Is it the same as FOV (Field Of View)?
0
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0answers
12 views

How to calculate viewing angles?

Forgive me if this is in the wrong forum, but I was not sure where to ask. Say I have a screen: 1920 x 1080 (27 inch diagonal) How do I work out the viewing angles (horizontal and vertical)? For ...