Discussion on shardulc's question

halirutan

2:38 AM

@shardulc Hi.

shardulc

hi

thanks for taking the time! I just received the USAMTS email today and was wondering what to do with Mathematica

halirutan

@shardulc Yes, it's an expensive gift.

@shardulc One thing I forgot to mention about your 3rd suggestion:

> Find a professional mathematician who would be willing to take a rookie on as an apprentice, give me a research topic, and let me loose for one year. This is the least probable option but in my opinion likely to be the most useful one.

It is not uncommon that research groups use their own tools and stick to it.

shardulc

I thought Mathematica was fairly common in professional mathematics?

halirutan

So when you find a professor that is willing to give you a task and he happens to use Matlab, I'm not sure you would be allowed to use Mathematica.

@shardulc The two major groups Matlab and Mathematica still exist.

Beside that many researchers use free languages like Python.

shardulc

I see

Anyway that would have been a really improbable option.

Like I said in the comments, maybe I can go through a textbook and supplement it with Mathematica. What do you think?

halirutan

2:44 AM

@shardulc For me, doing exercises from a book is probably not the best option either. The reason for that is that your attention will be split between understanding the topic and programming. I assume understanding the book will be the bigger part.

shardulc

But no matter what project I pick up, it will only be interesting if it is challenging.

halirutan

@shardulc Exactly.

shardulc

As in, the project itself should be challenging, not merely the programming part.

halirutan

For me it is fairly simple. Beside work, I like to do things that (a) interest me and (b) are not too easy.

@shardulc Yes, it's exactly how I understood it and it is the same for me.

Let me give you a recent example:

shardulc

@halirutan Yes, I think I will pick up the language quickly and can always look online (Mathematica SE!) for language help.

halirutan

2:47 AM

I stumbled over this question two days ago or so.

I never heard of doyle spirals but I knew of course something about complex computations, logarithmic spirals and their 2d representation.

So there were 2 interesting parts in this question for me:

1. Understand the background of the circle packing and how it is computed

2. how can I divide a disk through the spirals which was from the algorithmic part not directly obvious.

It is exactly the same for the QR Code example I have linked.

shardulc

Yes I think I see what you mean.

halirutan

It started out by explaining my wife how this could be done and I knew the theory pretty well. Then, since she couldn't follow the math, I started making some exmaples and implemented it bit by bit. On the way, I hit some problems that I had to solve.

These problems and their solution are things the bring you forward.

shardulc

So I would have to build up a base level of language knowledge and then tackle a problem.

halirutan

@shardulc So the question is: what are you interested in and what have you done so far.

shardulc

(Not as difficult as the ones you mentioned but something interesting nonetheless.)

I'm interested in a range of things :P

halirutan

2:54 AM

@shardulc Yes, skim over a basic tutorial and just start. The good thing with Mathematica, Python or Julia is that you have an interactive interpreter. You can evaluate every small piece of code and see if it works.

shardulc

Primarily, my interests are in discrete mathematics, artificial intelligence, computational linguistics, etc.

@halirutan I already know Python and yes, interpreters are helpful to learn a language.

I have not done any large mathematical projects. Most of my mathematics has either been 1. competition-oriented or 2. for Math.SE.

halirutan

@shardulc Since you mention discrete mathematics there are plenty of nice problems.

@shardulc I assume you have heard of the Collatz Conjecture?

shardulc

A trend I see on Mathematica.SE is that Mathematica is very good for visualizations?

@halirutan Yes, and I also saw that SE question.

halirutan

Ah.. and I thought I can surprise you :)

shardulc

It is rather artistic :) I like it a lot

halirutan

2:58 AM

@shardulc hehe..

@shardulc @shardulc Yes, due to the structure of Graphics in Mathematica you can basically turn everything into a nice visualization. Beside that there are millioins of plotting functions for parametric survaces, 3d plots, region plots, and many more.

shardulc

Hmm so that is one thing I can do. Art through Mathematica :)

Someone else mentioned going through Project Euler, what about that?

halirutan

@shardulc But note that the visualization capabilities are only one feature that makes Mathematica stand out from many systems.

@shardulc You need to decide what type of person you are. If you like competitions and these problems keep you interested until you tackled them, then please go for it. I did only some of the Euler problems.

My driving force is either a real project that can be used by others or to help people with interesting questions. Thats how I spend my freetime.

@shardulc Are you still a student with regular lectures?

shardulc

I don't particularly like competitions but from previous experience, I have found Project Euler to be a good language-teaching tool. (This is how I learned LISP by the way.)

@halirutan just finishing 11th grade

halirutan

@shardulc Oh, school then? That is nice.

shardulc

Yes

halirutan

3:10 AM

@shardulc What about home-work assignments or deeper insight in the topics of your lectures? That would be another thing were you work on something that is really useful to you and you don't have to work on an artificial problem.

shardulc

@halirutan That is a very interesting perspective.

@halirutan They aren't particularly deep, and anyway I have exhausted my school's math curriculum.

I am not taking any mathematics or computer science classes in my next (and final) year.

Most of my mathematics education is outside school, from a really nice math circle I used to attend and from the Internet.

halirutan

@shardulc Another question: Let's see you have free-time and you have to choose something interesting you want to learn. What would that be?

Something specific and not as general as discrete math.

shardulc

Recently I have been interested by generating functions, so I would like to learn more about them and how they are applied to mathematical problems.

On a more 'applied' level I would like to learn about natural language processing, but again this is somewhat general as I know little about the field.

Of course, I would always like to learn something totally new and unexpected :)

halirutan

@shardulc Somewhat related, highly visual and theoretically has deep as you want to dive is the book "indra's pearls".

Indra's Pearls (book)

Indra's Pearls: The Vision of Felix Klein is a geometry book written by David Mumford, Caroline Series and David Wright, and published by Cambridge University Press in 2002. The book explores the patterns created by iterating conformal maps of the complex plane called Möbius transformations, and their connections with symmetry and self-similarity. These patterns were glimpsed by German mathematician Felix Klein, but modern computer graphics allows them to be fully visualised and explored in detail. == Title == The book's title refers to Indra's net, a metaphorical object described in the Buddhist...

@shardulc I'm not sure how frustrating this topic might be if you start with that. I have only a glimpse of what is currently possible but the algorithms might be too deep for a start. But that is only a guess.

shardulc

3:46 AM

Oh, this seems interesting.

Do you know of any other books on similar topics? Pertaining to visualizing mathematical structures?

Or what general mathematical field does this fall under?

halirutan

@shardulc This book is indeed very popular. The reason is simple and pointed out at the end of the wiki article:

> Indra's Pearls is unusual because it aims to give the reader a sense of the development of a real-life mathematical investigation, rather than just a formal presentation of the final results. It covers a broad span of topics, showing interconnections among geometry, number theory, abstract algebra and computer graphics. It shows how computers are used by contemporary mathematicians. It uses computer graphics, diagrams and cartoons to enhance its written explanations.

I have another one, quite old from a professor of mine:

But I guess you wouldn't need that. If you are interested in chaos and fractals there is so much information online available. One starting point would be

L-system

An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into some larger string of symbols, an initial "axiom" string from which to begin construction, and a mechanism for translating the generated strings into geometric structures. L-systems were introduced and developed in 1968 by Aristid Lindenmayer, a Hungarian theoretical biologist and botanist at the University of Utrecht. Lindenmayer used L-systems to describe the...

shardulc

Oh this last one seems nice.

halirutan

I lied, I have two others:

shardulc

I just thought of the visualizations in Hofstadter's book "Godel Escher Bach" which are along similar lines

halirutan

@shardulc Yes. And if you like more pde we have this here

those patterns base of reaction-diffusion systems and are quite easy to implement.

And this is the last one I have here

shardulc

4:01 AM

I think I found their webpage: algorithmicbotany.org/papers

halirutan

So there are 3 topics: Indras pearls is probably the most interesting one for you as it concerns complex functions, maps and symmetry and contains more mathematics. The fractals and the beauty of plants base on rewriting system where you make iterative replacements of the same structure. Easy to understand and visually very appealing. The seashells are 1d PDEs that develop over the time (at the front of the seashell) creating various patterns.

shardulc

I think I have an idea now of what direction to explore in.

Thank you for all your help and for spending your time with me!

halirutan

@shardulc That is nice and absolutely no problem. That was time well spent!

shardulc

If not for SE it would have been very hard to find a Mathematica expert and enthusiast like you

halirutan

@shardulc Ahh, one final piece you should definitely check out is Paul Nylanders page!

bugman123.com

There, you find everything from minimal surfaces to fluid motion. I never have seen a site like this.

shardulc