A possible meaning of “meta” – Mathematical writing

I posted one of my exponent fractals on Dan Meyer’s 101qs.com, with the question, “how many dots is that?”  I was sort of disappointed that I had to pose the first question.  That seems to defeat the point, if I’m interested in what questions the photo prompts on its own.  I did get a few interesting questions though…

The question I’m a little more interested in, the one I posed in my first post on this stuff, is this one; If 9 is a meta-triangle (a triangle of triangles), is 27 a meta-meta-triangle?  I’ve often called it that, but I can see a different interpretation, so I asked the meaning.

I got a bite and this comment from Max Goldstein, but I wanted to share what I thought was in interesting answer, and once I started writing, the math started to push back, and I noticed new things, and on and on.   This is me doing math, so I thought I’d publish some of my work here.

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Question: What’s a meta-meta-triangle?

Like most math, this depends on the meaning of the terms. Let’s take “triangle” as understood and consider “meta.”  A meta-movie is a movie about a movie.  When faced with the challenge of writing, young poets often write about writing.  That’s meta-poetry.  9, then, is a meta-triangle, because it’s a triangle of triangles.

Then what’s a meta-meta-triangle? It’s a meta-triangle of meta-triangles!  So in the powers of 3, that would be a 9 of 9’s.  That’s 81, not 27!

[Do you buy that?  Is it clear?]

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Here’s an argument by notation:

Definition: meta(x)= x(x).
Examples: meta(play)=play(play)     —–>  Hamlet
meta(triangle)=triangle(triangle)     —–>  9

Let’s use m instead of meta.  It starts to sound weird if you say it too much.  Now m(string)=string(string), and m(asdf!3!)=asdf!3!(asdf!3!).

Note: The definition of meta(x) relies on x(x) making sense. If x is in meta’s domain, then x must be in its own domain! This is just spooky to me…

Then if “meta” is in its own domain, we know m(m)=m(m), by definition.  So meta(meta) is itself!  (I’m not making this up.)  As above, m(triangle)=triangle(triangle)=9.  In short, m(t)=9.  Get ready…

meta-meta-triangle is m((m(t))=m(9)=9(9)…  81?

[Does the notation make this clearer or obfuscate the ideas?]

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Well that’s it.  It’s a little piece of mathematics that I spent time carefully wording for clarity and communication’s sake.  Choosing a single notation, while proper, can be a little austere and hard for the reader, so I picked and chose which representations to use on each line, in the same way I choose punctuation.  I hope that comes across.

There’s so much unreadable mathematics in this world.  It breaks my heart.  I’m convinced it’s helping to kill mathematics.  This means I have a responsibility to try and work on quality writing – concise, elegant, clear, and convincing mathematical arguments that can be read widely.  We’ve been putting extra effort towards this in my department.  In short, this is the essence of proof – “convincing” argument.  (note: unreadable symbology and jargon is often NOT convincing at all.)

Maybe you’ll share your own mathematics with the world.  The goal is the clear and simple communication of ideas.  Two-column proof is NOT the only way.

Anyhow, with MArTH Madness and everything else going on, life is really rushing straight at me right now.  Somehow it feels amazing!

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Problems vs. exercises – Pointilism and more exponent fractals

In my last post, I showed the powers of 3  as a series of fractal drawings.  I had lots of other doodles in my math notebook, so I spent the morning dotting little pieces of paper to share them with you.

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These could serve as exercises for a student wanting to practice using exponent notation, but they could also serve as decent stand alone problems.  Something as simple as “How many dots?” maybe.  Perhaps I’ll post some on 101qs.com and see what kind of questions they get.  (I did.)

Problems and exercises can both be valuable, but they aren’t the same thing.  Math students should know the difference between doing math in its most free and open form, and completing exercises to build familiarity, skill, and speed.  I need to do a better job of including this in my teaching.

What do you make of these?

Perhaps you’re wondering “How much time did he spend drawing all those dots?

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Through school and work, I’ve been able to download two copies of Mathematica, and I LOVE it!  It’s especially good for all of my mathematical art and imaging needs.  It also has an exceptional documentation system with hyperlinks and examples that make it very easy to teach yourself and play.  I’m amazed how much I’ve learned in two weeks.

Mathematica also handles recursive programming nicely, which is perfect for fractals.  I wrote some code this afternoon and generated these images.  Enjoy.

Now how long would THAT have taken me to draw?