## Picturing a Certain Torus and Klein Bottle

The other day I was explaining orientability to someone and we were walking through some of the statements about orientability on the Wikipedia page on the topic. While I was able to satisfy his curiosity, one statement on that page (which I did not even attempt to delve into with him) has been nagging me … Read more

## How to visualise Bollobas’ 1965 theorem?

Theorem [n]={1,…,n}. Let {(Ri,Si),i∈I},Ri,Si⊂[n] be such that Ri∩Si=∅,Ri∩Sj≠∅(i≠j). Then \sum_{i \in I} \frac{1}{{{r_i+s_i}\choose{s_i}}}\le 1. Question: I am happy with the proof below of Bollobas’ Theorem, but it seems very bashy. Is there a: More elegant way to prove the theorem, and, more importantly Is there a way to visualise the proof? By this I mean … Read more

## How to visualize a category (of “combinatorial” maps)

This is a practical and very soft question, with the combinatorial database http://www.findstat.org in mind. I have a few, around 20, families of combinatorial objects, for example Dyck paths, permutations, perfect matchings, graphs, etc., together with a few, around 200, maps between them. The maps need not be injective or surjective or have any special … Read more

## Is Visualization of Data a Subject of Mathematical Research? [closed]

Closed. This question needs to be more focused. It is not currently accepting answers. Want to improve this question? Update the question so it focuses on one problem only by editing this post. Closed 4 years ago. Improve this question Please excuse my naive question, but what kind of rôle does the visualization of (especially … Read more

## t-Stochastic Neighbor Embedding vs Topological Data Analysis

The shortest form of this question is: How much TDA can be done with tSNE? Specifically, I’m referring to the application of TDA to clustering data, so, think along the lines of Ayasdi’s implementation: My understanding is that TDA constructs simplical complexes on a continuum of scales, to then find persistent components. This is an … Read more

## Explaining patterns in modular multiplication graphs

Let the multiplication graph n/m be the graph with m points distributed evenly on a circle and a line between two points a, b when an≡bmodm. These graphs often look somehow random but by carefully choosing n and m one finds intricate patterns. Let n=20,40,60,80,100,120 and m=n+19 m=2n+19 m=3n+19 Note that it seems essential that … Read more

## When is $2\varphi(n) > n$ – and how to prove it?

When coloring the squares of the Ulam spiral not only by black and white (for being prime or non-prime) but by shades of grey representing the normalized totient function $\varphi(n)/n$ and displaying only those numbers with $\varphi(n)/n > 0.5$, i.e. $2\varphi(n) > n$, one finds that for most $n$ – but not all – it … Read more

## Visualization of an algebraic stack

As the visuallization of an algebraic stack is virtually impossible I warn about this is a soft question. I am interested in thinking visually about algebraic stacks (also higher and derived stacks, but let´s start from the beginning) and I find quite suprising the fact that there isn´t a single “picture” of an algebraic stack … Read more

## Visualizing a Whitehead product: the attaching map S3→S2∨S2S^3\to S^2\vee S^2

There are informative and easily accessible images and videos that illustrate the Hopf fibration S3→S2 by describing what happens to the fibers in the unit cube (0,1)3≈S3∖∗, e.g. those by Niles Johnson. Is there a similar way to visualize the map S3→S2∨S2, which is the attaching map of the 4-cell of S2×S2 and allows one … Read more

## Visualizing functions with a number of independent variables

I need to graph real valued functions (for exposition and analysis). The issue is: there are more independent variables so that the conventional graphing methods can’t be used, and furthermore I don’t want to slice the functions. These functions are like s=f1(x,y,z,t) and s=f2(x,y,z,t,k) I also have vector functions of the same type: v = … Read more