On a Tangent

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You might be thinking, what is that?  And what does it have to do with me?  Or maybe you’re just thinking: Gosh, that looks beautiful – I wish I knew how to make such pretty pictures.

For the latter sort, you could get a math degree, or you could take a shortcut and read my tutoring blog, where I explain other marvels and will soon get into some more technical detail about today’s work of art.  But for the former, read on.

We start off with some really weird curve (read: I mashed a few keys in a graphing program and worked with what came up).  Now, if you were an ant on there looking close up, how would you visualize that curve?

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Maybe you’d think that it was just a line, a line heading off at that same angle that you currently find yourself.  Maybe, as you walk along, you’d constantly revise your view of the world:

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Such an ant could manage themselves just fine.  As long as they keep looking at where they are, they can make it around the curve.  But their conception of the world at large is completely off.  Almost every single judgement they would make about other points on the curve would be wrong.  The ant can live in the “now” as long as they keep it to themselves.

But our ant could be a little more sophisticated:

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This second ant is still using data merely from their local environment; they just estimate a curve of best fit, instead of a line (in mathese: they find a tangent conic, instead of a tangent line.  In worse mathese: they use a truncated Taylor series as calculated at that point, the general technique for all estimations in this post).  This curve still doesn’t match the overall pattern very well, but it does a darn sight better at making decisions about the neighbourhood.

A third ant might be a little more resourceful, and come up with this:

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Or this:

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Now, this ant can start making broader claims about the world.  The ant will still be off, but significantly less so than the previous two.  And finally, we have genius ant here:

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Again, this ant’s knowledge is not perfect – but she absolutely nails entire portions of the world, using just the information that is at her immediate fingertips.  She doesn’t have a larger view of the world, she has a deeper view.  Maybe she still lives in the “here and now”, but she doesn’t rest content with mere appearances.  She doesn’t just calculate how things are, or how they are changing, but how change changes, and so on (technically, up to 8th derivatives).  By assuming that everything is changing, including change itself, she can understand.

And as a bonus (which I’m not sure has any analogical value, but which looks cool), here’s all of the approximations together in one animation:

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