UserStories.txt

User stories -- Gabe Cunningham

For the past 6 years, many of my projects have been inspired by working with data on abstract polytopes. 
Typically, I will have a question in mind -- ``For which values of p and q is there a regular polyhedron
with q p-gonal faces and p q-valent vertices?'' -- and then I will consult some combination of the Atlas
of Small Regular Polytopes (http://atlas.dr-mikes-maths.com/atlas/index.html) and various lists of data
available on Marston Conder's website (https://www.math.auckland.ac.nz/~conder/). I collect a list of
polytopes that satisfy the desired properties and look for interesting patterns. For example, for the
question mentioned above, I found that whenever p was odd, it seemed that q had to be an even divisor
of 2p. There is typically enough data to make confident conjectures, and then the next step becomes
actually finding a way of proving (or disproving!) these conjectures.

I have several papers related to the above question where I used ad-hoc data analysis: see "Tight 
orientably-regular polytopes", "Classification of tight regular polyhedra", and "Tight chiral polyhedra". 
Eventually I decided that I needed a more robust method of querying the data. I took one of Marston Conder's
freely available datasets (of regular polyhedra with up to 2000 flags), and I put it into a form that I 
could work with interactively in GAP. I pre-computed several properties and created an interface for reading
and writing my list of polytopes to a text file. Now I can easily query the data in GAP -- for example, I 
can ask for all regular polyhedra with 11 vertices and up to 2000 flags. With coauthor Mark Mixer, we found
that for primes p > 3, there seemed to be exactly two regular polyhedra with p vertices, both of which were
polyhedra I had encountered before when working with tight polyhedra. If I had not created my own personal
method for querying the data, I estimate that it would have taken me 8 hours of tedious work to go through
the Atlas of Small Regular Polytopes and determine which regular polyhedra had a prime number of vertices.
I'm not sure if I would have even bothered.

In the community of researchers of abstract polytopes and related structures, there is a deep need for a 
searchable database with polytopes. I have had several conversations that start with someone asking ``What is
the smallest polytope with properties A, B, and C?'' At the moment, you just have to hope you ask the right 
person. Having a facility for exploring the data that we have on regular polytopes would allow rapid testing 
of various hypotheses, and it would help act as a gateway for new researchers entering the field. It would 
also encourage the development of more datasets. There are several fairly straightforward datasets that could 
be developed, and I think the main reason they have not yet been created is that hosting and organizing the 
data represents a significant hurdle.