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!