MetaFilter posts tagged with closedcurve
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Posts tagged with 'closedcurve' at MetaFilter.Tue, 22 Feb 2011 19:35:15 -0800Tue, 22 Feb 2011 19:35:15 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60Talking fast and making cool videos does not mean learning is happening
http://www.metafilter.com/100837/Talking%2Dfast%2Dand%2Dmaking%2Dcool%2Dvideos%2Ddoes%2Dnot%2Dmean%2Dlearning%2Dis%2Dhappening
So you're <a href="http://vihart.com/">me</a> and you're in <a href="http://ocw.mit.edu/courses/#mathematics">math class</a> and you're learning about <a href="http://www.utm.edu/departments/math/graph/">graph theory</a>, a <a href="http://oneweb.utc.edu/~Christopher-Mawata/petersen/lesson2.htm">subject</a> too <a href="http://users.encs.concordia.ca/~chvatal/perfect/problems.html">interesting</a> to be included in most <a href="http://users.encs.concordia.ca/~chvatal/perfect/problems.html">grade</a> <a href="http://www.ehow.com/video_4974325_teaching-children-discrete-mathematics.html">school's</a> <a href="http://www.ehow.com/video_4974325_teaching-children-discrete-mathematics.html">curricula</a> so maybe you're in some <a href="http://schools.nyc.gov/community/innovation/SchoolofOne/default.htm">special</a> <a href="https://www3.imsa.edu/about/profile">program</a> or maybe you're <a href="http://www.collegehumor.com/article:1810608">in college</a> and were somehow not <a href="http://books.google.com/books?id=4xRM-JCEm-0C&lpg=PA14&ots=Uje1RaPtKa&dq=%22scarred%20for%20life%20by%20your%20grade%20school%20math%20teachers%22&pg=PA5#v=onepage&q&f=false">scarred for life</a> by your <a href="http://www.youtube.com/watch?v=SXx2VVSWDMo&feature=related">grade</a> <a href="http://www.youtube.com/watch?v=v0XUqliEnPQ&feature=related">school</a> <a href="http://www.youtube.com/watch?v=BbX44YSsQ2I&feature=related">math</a> <a href="http://www.nctm.org/resources/content.aspx?id=530">teachers</a>. I'm not sure why you're not <a href="http://www.hastac.org/blogs/cathy-davidson/why-doesnt-anyone-pay-attention-anymore">paying attention</a>, but maybe you have an in<a href="http://blog.mrmeyer.com/?p=41">competent</a> <a href="http://blog.mrmeyer.com/?p=5368">teacher</a> and it's too <a href="http://www.metafilter.com/100512/A-Darker-Shade-of-Golden#3517841">heartbreaking</a> to <a href="http://www.youtube.com/watch?v=blOrY-nEGaE">watch</a> <a href="http://www.youtube.com/watch?v=XKviYiZhtZY">him</a> <a href="http://ask.metafilter.com/27259/How-much-meat-is-there-on-a-cow">butcher</a> what could have been such a <a href="http://themetapicture.com/why-couldnt-i-have-been-shown-this-in-maths-class/">fun</a> <a href="http://www.youtube.com/watch?v=V1RxTYRKMlY">subject</a> full of <a href="http://www.maa.org/editorial/mathgames/mathgames_08_17_06.html">snakes</a> and <a href="http://vihart.com/balloons/">balloons</a>.
<a href="http://www.boardgamegeek.com/boardgame/5432/snakes-and-ladders">Snakes</a> aren't really all that <a href="http://www.vam.ac.uk/vastatic/microsites/1414_jain/snakesandladders/">relevant</a> to the <a href="http://www.math.niu.edu/~rusin/uses-math/games/chutes/chutes.html">mathematics</a> here but <a href="http://www.dragoart.com/tuts/7235/1/1/how-to-draw-a-king-cobra.htm">being</a> <a href="http://www.dragoart.com/tuts/2035/1/1/how-to-draw-an-anaconda-snake.htm">able</a> to <a href="http://www.dragoart.com/tuts/3953/1/1/how-to-draw-a-black-mamba.htm">draw</a> <a href="http://www.dragoart.com/tuts/6535/1/1/how-to-draw-a-realistic-snake.htm">them</a> will be <a href="http://www.how-to-draw-funny-cartoons.com/cartoon-snakes.html">useful</a> later so you <a href="http://www.youtube.com/watch?v=k9HeP4KtJzY">should</a> <a href="http://www.youtube.com/watch?v=MJjWtx4tRd8">probably</a> <a href="http://www.youtube.com/watch?v=7jq_DhqNZL8">start</a> <a href="http://www.youtube.com/watch?v=qq3uJVZgJEU">practicing</a> <a href="http://www.youtube.com/watch?v=rQ4j-oj13QY">now</a>.
I've got a <a href="http://www.math.wichita.edu/history/men/bernoulli.html">family</a> of <a href="http://en.wikipedia.org/wiki/3_(number)">three</a> <a href="http://www.google.com/search?hl=en&safe=off&q=related:en.wikipedia.org/wiki/3_(number)+three&tbo=1&sa=X&ei=dEpbTez0Gcqr8AbL9IGCDg&ved=0CC4QHzAA">related</a> <a href="http://vihart.com/doodling/">doodle games</a> to show you all stemming <a href="http://www.perthnow.com.au/entertainment/end-of-the-line-for-mr-squiggle-animator/story-e6frg30c-1225966801098">from</a> <a href="http://squigglepage.blogspot.com/">drawing</a> <a href="http://en.wikipedia.org/wiki/Mr_Squiggle">squiggles</a> all over the page. The <a href="http://mathworld.wolfram.com/OrdinalNumber.html">first</a> <a href="http://en.wikipedia.org/wiki/1_(number)">one</a> goes like this: <a href="http://www.youtube.com/watch?v=_H0dYEjR-jA">Draw</a> a <a href="http://soup2nuts.tv/about.html">squiggle</a>, a <a href="http://www-gap.dcs.st-and.ac.uk/~history/Curves/Lissajous.html">closed</a> <a href="http://www-gap.dcs.st-and.ac.uk/~history/Curves/Curves.html">curve</a> that <a href="http://www.2dcurves.com/">ends</a> where it <a href="http://www-gap.dcs.st-and.ac.uk/~history/Java/index.html">begins</a>.
The only <a href="http://thinkzone.wlonk.com/Numbers/NumberSets.htm">real</a> <a href="http://knowyourmeme.com/memes/rule-34">rule</a> here is to <a href="http://www.makemath.com/">make</a> sure that all the <a href="http://musemath.blogspot.com/2007/06/tricky-crossings.html">crossings</a> are <a href="http://www.maa.org/mathland/mathtrek_10_16_06.html">distinct</a>.
<a href="http://www.nextcomputers.org/">Next</a> <a href="http://blog.makezine.com/archive/2009/12/introducing_math_monday.html">make</a> it <a href="http://www.maa.org/pubs/mm_supplements/farris/frieze.html">start</a> <a href="http://www.maa.org/pubs/mm_supplements/farris/rope.html">weaving</a>.
<a href="http://www.quotegarden.com/math.html">Follow</a> the <a href="http://blog.makezine.com/archive/2010/11/math_monday_3d_hilbert_curve_in_ste.html">curve</a> around and at each <a href="http://www.cut-the-knot.org/do_you_know/CrossingNumber.shtml">crossing</a> alternate going <a href="http://www.cs.arizona.edu/patterns/weaving/webdocs/mo/G/Constraints.pdf">under</a> and <a href="http://www.cs.arizona.edu/patterns/weaving/webdocs/mo/G/Constraints.pdf">over</a> until you've <a href="http://www.cs.arizona.edu/patterns/weaving/webdocs/mo.pdf">assigned all the crossings</a>.
Then put on the <a href="http://local.wasp.uwa.edu.au/~pbourke/texture_colour/periodic/#weave">finishing touches</a>, and <a href="http://www.math.dartmouth.edu/~matc/math5.geometry/syllabus.html">voila</a>!
You <a href="http://www.cut-the-knot.org/ctk/ArtMath.shtml">try it again</a> adding a <a href="http://kennethsnelson.net/category/sculptures/small-sculptures/">little</a> <a href="http://www.kennethsnelson.net/articles/slate.htm">artistic</a> <a href="http://www.ecclesart.co.uk/paintings/mathsandart.html">flair</a> to the lines.
The <a href="http://markdow.deviantart.com/gallery/">cool</a> part is that the <a href="http://fractalartgallery.com/gallery/five/weaved.jpg.html">weaving</a> always works out perfectly. When you're going <a href="http://www.knotplot.com/knot-theory/torus_xing.html">around <a href="https://www.math.lsu.edu/~verrill/origami/tessellations/weave/">alternating</a> <a href="http://lcni.uoregon.edu/~mark/Geek_art/Universal_motifs/Universal_motifs_tilings_knits_weaves_chains.html">over and under</a> and get to a <a href="http://www.c3.lanl.gov/mega-math/gloss/knots/kncross.html">crossing</a> you've <a href="http://www.c3.lanl.gov/mega-math/gloss/knots/kntrcr.html">already assigned</a> it will always be the right one. This is <a href="http://www.vimeo.com/7441291">very</a> <a href="http://lcni.uoregon.edu/~mark/index.html#Geek_art">interesting</a>, and we'll get back to it later, but first I'd like to point out <a href="http://en.wikipedia.org/wiki/2_(number)">two</a> things. One, is that this works for any number of <a href="http://www.webpages.uidaho.edu/~markn/squares/">closed curves</a> <a href="http://local.wasp.uwa.edu.au/~pbourke/texture_colour/tiling/">on the plane</a>, so go ahead and link stuff up or make a <a href="http://jointmathematicsmeetings.org/samplings/feature-column/fcarc-weaving">weaving</a> out of <a href="http://blog.makezine.com/archive/2010/04/math-monday-knitted-cellular-automa.html">two colors of yarn</a>. The other is that this doodle also works for <a href="https://sites.google.com/site/tjgaffneymath/snakes-on-a-projective-plane">snakes on a plane</a> as long as you keep the head and tail on the <a href="http://video.google.com/videoplay?docid=-6626464599825291409#">outside or on the same inside face</a> because mathematically it's the same as if they've linked up.
Or just actually link up the head and tail into a <a href="http://en.wikipedia.org/wiki/Ouroboros">Ouroboros</a>. For example here's three ouroborii in a configuration known as the <a href="http://www.liv.ac.uk/~spmr02/rings/index.html">borromean</a> <a href="http://vismath5.tripod.com/bor/bor1.htm">rings</a> <a href="http://katlas.math.toronto.edu/wiki/L6a4">which</a> has the <a href="http://www.brainbashers.com/showillusion.asp?126">neat property that</a> no two snakes are actually linked with each other. Also because I like naming things this design shall henceforth be known as the <a href="http://www.google.com/search?sourceid=chrome&ie=UTF-8&q=Ouroborromean+Rings">Ouroborromean Rings</a>.
But you are me, after all, so you're finding a lot to think about even with just drawing one line that isn't a snake, such as what <a href="http://katlas.math.toronto.edu/wiki/Main_Page">kind of knots</a> are you drawing? And, can you <a href="http://knotplot.com/zoo/">classify them</a>? For example, these knots all have <a href="http://newweb.cecm.sfu.ca/cgi-bin/KnotPlot/KnotServer/kserver?knot_type=4">five</a> <a href="http://newweb.cecm.sfu.ca/cgi-bin/KnotPlot/KnotServer/kserver?knot_type=5">crossings</a> but two are essentially the same <a href="http://www.cut-the-knot.org/ctk/August2001.shtml">knot</a> and one is different. <a href="http://www.oglethorpe.edu/faculty/~j_nardo/knots/intro.htm">Knot theory</a> <a href="http://infohost.nmt.edu/~jstarret/sierpinski.html">questions</a> are actually really <a href="http://web.williams.edu/go/math/cadams/knotproblems.html">difficult</a> and interesting but <a href="http://www.math.cornell.edu/~mec/2008-2009/HoHonLeung/intro_knots.htm">you're going to have to look that one up yourself</a>.
Oh, and you should actually learn how to <a href="http://www.flickr.com/photos/24774692@N00/4686898230/in/pool-35241465@N00">draw a rope</a>, because it's an <a href="http://www.youtube.com/view_play_list?p=2B059F10DDE1E5E9">integral</a> <a href="http://www.sosmath.com/calculus/integration/byparts/byparts.html">part</a> of <a href="http://www.sciencenews.org/view/generic/id/38237/title/Unknotting_knot_theory">knot theory</a>. So <a href="http://archives.math.utk.edu/visual.calculus/4/index.html">integral</a>, in fact, that if you draw a bunch of <a href="http://www.math.usma.edu/people/rickey/hm/CalcNotes/Integral-Sign.pdf">integral signs</a> in a row, a sight which is often quite daunting to a mathematician, you can just shade it in, and ta-da. But, being able to draw snakes is also super useful, especially as this doodle game is excellent for <a href="http://www.the-leaky-cauldron.org/features/crafts/othercrafts/darkmarktattoo">producing dark mark tattoo designs</a>.
Also this doodle game can be combined with <a href="http://vihart.com/doodling/stars.mp4">the stars doodle game</a>. For example, if this <a href="http://www.jimloy.com/geometry/pentagon.htm">pentagram</a> gets knighted, it will henceforth be known as <a href="http://www.btinternet.com/~connectionsinspace/Form_and_Structure/Sierpinski_Gaskets/sierpinski_gaskets.html">SerPentagram</a>. Also notice that this snake is a <a href="http://www.math.uiowa.edu/~roseman/moebTestN.html">five twist mobius strip</a> so you can also call it a <a href="http://www.google.com/search?num=100&hl=en&safe=off&q=mobiaboros">mobiaboros</a> but we'll get back to <a href="http://plus.maths.org/issue26/features/mathart/index-gifd.html">one-sidedness</a> later.
Or, if you want to draw something super complicated like the eighth <a href="http://mathworld.wolfram.com/StarPolygon.html">square star</a>, combining snakes and stars is a great technique for that, too. Here's a boa that <a href="http://www.homophone.com/">ate</a> <a href="http://en.wikipedia.org/wiki/8_(number)">eight</a> <a href="http://mathforum.org/dr.math/faq/faq.polygon.names.html">eighth-gons</a>.
The <a href="http://nuwen.net/poly.html">creativity</a> that your mind is forced into during these boring classes is both a <a href="http://www.thinkgeek.com/brain/whereisit.cgi?t=math&icpg=WhatyoulikeSearch">gift</a> and a burden.
But, here's a few authentic doodles using these techniques that I did when I was in college just to show you I'm not making all this up. These are from a freshman <a href="http://plus.maths.org/content/magical-mathematics-music">music</a> history class because I happened to be able to find this notebook, but this is a doodle I actually did most often during my ninth grade <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibBio.html">Italian</a> class, language being another subject usually taught by unfathomably stupid methods.
For example these snakes are having trouble communicating because one speaks in <a href="http://harrypotter.wikia.com/wiki/Parseltongue">Parseltounge</a> and the other speaks in <a href="http://www.python.org/">Python</a> and their language <a href="http://docs.python.org/tutorial/classes.html">classes</a>, much like math classes focused too much on memorization and not enough on immersion.
But just pretend you're in math class learning about <a href="http://graph-theory.blogspot.com/2008/11/good-will-hunting-problem.html">graph theory</a> so I can draw the <a href="http://www.math.tamu.edu/outreach/mam/illusions/">parallels</a>.
Because here's the second doodle game which is very much <a href="http://momath.org/">mathematically related</a>: Draw a <a href="http://en.wikipedia.org/wiki/Tilde#Other_uses_2">squiggle</a> all over the page and make sure it closes up. Pick an outside section and color it in. Now you want to alternate coloring so that <a href="http://www.mathpages.com/home/kmath266/kmath266.htm">no two faces of the same color touch</a>.
Curiously enough, much like the <a href="http://tensegrity.wikispaces.com/Weaving">weaving</a> game this game also always mathematically works out. It also works really well if you make the lines spiky instead of a smooth curve. And once again it works with multiple lines, too.
It probably has something to do with the two-color-ability of <a href="http://web.mit.edu/newsoffice/2004/origami-lang.html">graphs of even degree</a>, which <a href="http://www.personal.kent.edu/~rmuhamma/GraphTheory/MyGraphTheory/eulerGraph.htm">might even be what your teacher is trying to teach</a> you about at this very moment, for all your paying attention.
But maybe you can chat with him after class about snakes and he'll explain it to you, because I'd rather move on to the next doodle game. This is a <a href="http://malini-math.blogspot.com/2010/06/permutations-and-combinations.html">combination</a> of the last two. Step one: draw a smooth closed curve. Step two: assign overs and unders. Step three: shade in every other face. After that, it takes a little <a href="http://www.yehrintong.com/">artistic finesse</a> to get the shading right, but you end up with some sort of <a href="http://newweb.cecm.sfu.ca/cgi-bin/KnotPlot/fgetknot?crazy=6">really neat</a> <a href="http://www.math.osu.edu/~fiedorow/math655/classification.html">surface</a>.
For example <a href="http://knotplot.com/moebius/t1/PretzelA.html">this one only has one edge and only one side</a>. But if you're <a href="http://www.ams.org/mathimagery/">interested in this</a> you should really be talking to your resident topology professor and not me.
But here's the thing: If someone asked you five minutes ago what <a href="http://nrich.maths.org/5681">tangled up</a> snakes, <a href="http://www.youtube.com/watch?v=QKCSBkdEUXQ">demented checker boards</a> and <a href="http://www.youtube.com/watch?v=_0nX-El-ySo">crazy twisty surfaces</a> <a href="http://www.btinternet.com/~connectionsinspace/index.html">have in common</a>, what would you have answered?
This is why I love mathematics.
The moment when you realize that something seemingly <a href="http://random.org">arbitrary</a> and confusing is actually <a href="http://www.youtube.com/watch?v=wm_T-FiXkmY">part</a> of something.
It's better than the cleverest possible ending to any <a href="http://numb3rs.wolfram.com/616/">crime show</a> or <a href="http://gutenberg.net.au/ebooks02/0200241.txt">mystery novel</a> because that's only the beginning.
Anyway, have fun with <a href="http://www.youtube.com/watch?v=heKK95DAKms">that</a>.
(<a href="http://www.metafilter.com/98150/Explorations-of-a-Recreational-Mathematician">Previously</a>.)</a> tag:metafilter.com,2011:site.100837Tue, 22 Feb 2011 19:35:15 -0800achmorrison