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

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