Doodling in Math: Connecting Dots Anti-parabola propoganda, plus musing on math class, cardioids, connect the dots, envelopes of lines, even a bit of origami.Extra points to a certain Andrea whose line-enveloped Hilbert curve inspired me to finish this video.
Doodling in Math: Connecting Dots
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- So you're me and you're in math class and
- your teacher's ranting on and on about this article
- about whether algebra should be taught in school
- as if he doesn't realize that what he's teaching
- isn't even algebra which could have been interesting
- but how to manipulate symbols and
- some special cases of elementary algebra which isn't.
- And so, instead of learning about self-consistent systems
- and logical thought, you spent all week
- memorizing how to graph parabolas.
- News flash, no one cares about parabolas.
- Which is why half the class is
- playing angry birds under their desk.
- But, since you don't have a smartphone yet,
- you have to resort to a more noble
- and outdated form of boredom relief.
- That is, doodling.
- And you've invented a game of your own.
- A doodle game that connects the dots
- in ways your math curriculum never will.
- Except instead of connecting to the closest dots
- to discover the mysterious hidden picture
- you've got this precise method of skipping over
- some number of dots and connecting them that way.
- In the past you've characterized how this works
- if your dots are arranged in a circle,
- say 11 dots and connect one to the dot four dots over,
- you get these awesome stars,
- and you can either draw the lines in the order of the dots,
- or you can just keep going around and
- maybe it will hit all the dots, or maybe it won't,
- depending on how many dots are in the circle
- and how many dots you skip.
- But then there's other shapes.
- Circles are good friends with sine waves,
- and sine waves are good friends with square waves.
- And let's admit it, that's pretty cool looking.
- In fact, just two simple straight lines of dots
- connecting the dots from one line to the other in order
- somehow gives you this awesome woven curve shape.
- Another student is asking the teacher when he's ever
- going to need to know how to graph a parabola,
- even as he hides his multi-million dollar enterprise of
- a parabola graphing game under his desk.
- If your teacher thought about it,
- he would probably think shooting birds at things
- is a great reason to learn about parabolas
- because he's come to understand that
- education is about money and prestige and
- not about becoming a better human able to do great things.
- You yourself haven't done anything really great yet
- but you figure the path to your future greatness
- lies more in inventing awesome new connect-the-dot arrangements
- than in graphing parabolas or shooting birds at things.
- And that's when you begin to worry.
- What if this cool liney curvey thing you drew
- approximates a parabola?
- As if your teacher doesn't realize
- everyone has their phones under their desks,
- but he's underpaid and overworked and
- his whole word runs on plausible deniability,
- so he shouts state-mandated, pass the test,
- teach-to-the-middle nonsense at students
- who are not at all fooled
- by his false enthusiasm or false mathematics and
- he pretends he's teaching algebra and
- the students pretend to be taught algebra and
- everyone else involved in the system
- is too invested to do anything
- but pretend to believe them both.
- You think maybe it's a hyperbola,
- which is similar to the parabola
- in that they are both conic sections.
- A hyperbola is a nice vertical slice of cone,
- the cone itself being just like a line swirled around in a circle,
- which is why the cone is like two cones radiating both ways;
- the lovely hyperbola insecting both parts.
- Two perfect curves, looking disconnected when seen alone
- but sharing their common conic heritage.
- While the boring old parabola is a slice taken at an angle
- completely meant to miss the top part of the cone and
- to miss wrapping around the bottom like an ellipse would.
- And it's such a special, specific case of conic section
- that all parabolas are exactly the same,
- just bigger or smaller or moved around.
- Your teacher could just as well hand you parabolas already drawn
- and have you draw coordinate grids on parabolas
- rather than parabolas on coordinate grids.
- And it's stupid, and you hate it,
- and you don't wanna learn to graph them,
- even if it means not making a billion dollars
- from a game about shooting birds at things.
- Meanwhile anyone who actually learns how to think mathematically
- can then learn to graph a parabola or anything else
- they need in like five minutes.
- But teaching how to think is an individualized process
- that gives power and responsibility to individuals
- while teaching what to think can be done
- with one-size-fits-all bullet points and check-boxes
- and our culture of excuses demands that we do the latter,
- keeping ourselves placated in the comforting structure
- of tautology and clear expectations.
- Algebra has become a check-box subject and
- mathematics weeps alone in the top of the ivory tower prison
- to which she has been condemned.
- But you're not interested in check-boxes;
- you're interested in dots, and lines that connect them.
- Or maybe you could connect them with semicircles,
- to give visual structure to lines that would otherwise overlap.
- Or you could say one dot is the center of a circle and
- another defines a radius and draw the entire circle
- and do things that way.
- You could make rules about how every dot
- is the center of a circle with its neighbor being the radius,
- or say one dot stays the center of all circles,
- and all of the others define radii.
- But then you just get concentric circles,
- which I suppose should have been obvious.
- But what if you did it the other way around and
- said one dot always stays on the circle and
- all the other dots are centers, like this.
- Looks more promising.
- So you try putting all the dots in a circle and
- using them as circle centers and
- choose just one dot for the circles to go through and
- you get this awesome shape that looks kind of like a heart.
- So let's call it, oh I don't know, a cardioid.
- Which happens to be the same curve that you get
- when parallel lines like rays of light reflect off a circle
- the same heart of sunshine in a cup.
- Or maybe instead of circle centers
- you could have points all on the curve of a circle,
- which means you need three points to define a circle,
- maybe just a point and its two closest neighbors to start with.
- And of course, any collection of circles is two-colorable,
- which means you can contrast light and dark colors
- for a classy color scheme.
- Or maybe you could throw down some random points
- to make all possible circles.
- Only that would be a lot of circles,
- so you choose just ones you like.
- And then, against your will, you begin to wonder
- how many points it takes to define the boring old parabola.
- Because parabolas are actually a lot like circles in that
- both are like extreme ellipses,
- because a circle is like taking one focus of an ellipse
- and putting the other focus zero distance away.
- And a parabola is like an ellipse where
- one focus is infinity distance away.
- Which is why everyone lies to you and says
- throwing balls or shooting birds is all about parabolas
- when really it's about ellipses
- because the earth is a sphere and
- gravity doesn't actually go straight down.
- And the other focus of the elliptical orbit of
- your thrown object of choice may be very far away,
- but very far away is a great bit closer than infinity.
- So let's not fool ourselves.
- You can't look at everything that seems kind of parabolic
- and call it a parabola.
- Sure, if you connect two dots with a hanging string or chain,
- it looks parabolic, and so do structurally strong arches,
- but they're actually catenary arches and
- maybe you can't tell by sight,
- but if you're an architect you'd better know the difference.
- Though caternarys are quite related to parabolas,
- you get them by rolling around a parabola
- and tracing the focus
- which makes them a cousin of the ellipse and
- even a hyperbola is like an ellipse that got turned inside out
- or whose focus went through infinity and
- came out the other side or something.
- And of course parabolas and hyperbolas and ellipses
- are all conic sections which mean
- they all come from a line that got spun around.
- And a line is just what happens
- when a couple dots get connected.
- Or maybe what happens when your circle is so big that,
- like the extreme ellipse that becomes a parabola,
- the extreme circle is broken at infinity and
- becomes a line before getting larger than infinitely big
- which brings it back to the other side.
- This linear circle, the infinite in-between.
- Or maybe a line is what happens
- when you roll around a circle and trace the focus.
- Or rather the 2 foci, which are zero distance apart, in ellipse terms.
- Which makes you wonder what you get
- when you roll around an ellipse and trace the foci.
- In fact, there's lots of great shapes
- you can get by rolling around shapes on other shapes,
- like if you roll a circle around a circle and trace the focus,
- you get just another circle.
- But, if you trace a point on the edge,
- you get our awesome friend the cardioid again.
- So now it's related to circles in three ways ,
- which means it's a close cousin of the ellipse
- and a second cousin to the infinite ellipse or, parabola.
- Except, not just that, if you take a parabola and
- invert it around the unit circle, reversing inside and outside,
- one half becomes two, one hundred becomes a hundredth,
- one stays one, infinity becomes zero,
- you get once again, the cardioid.
- The cardioid is the anti-parabola which is good
- because parabolas make you sad but you heart cardioids.
- And of course, any time you want to
- connect two dots on a piece of paper,
- instead of drawing the line you could fold the line.
- Here's the thing about connecting dots.
- You can have all the steps laid out for you,
- taking whatever next step is easiest and closest
- and be sure of what you're getting the whole time.
- This way is safe and comfortable.
- Or, you can try new ways of connecting dots
- and not know what you're going to get.
- Maybe it will be something great, maybe it will fail.
- And when it fails it will be your fault.
- You can't blame anyone else, not mathematics
- or the system or the check-boxes.
- But if I am to have faults I would rather they be my own.
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