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Origami and Math
TheBoostedBrain writes "I found a nice site that explains a little bit about the math in Origami. Origami is one of my favorite hobbies, but I never thought about it being related to science."
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Sometimes you just have to be creative. Math is everywhere.
Apparently the math goes like this: Origami Website + (/. crowd) = 0
Your paranoia is about as subtle as the alien probe in your neck.
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I've always found that my stress level is directly proportional to the number of times I've tried to fold a goddam pterodactyl or swan or whatever the hell it's supposed to be. I think this guy has the right idea. =)
Man am I sad. When I saw the headline I wasn't thinking about folding paper, and I couldn't figure out what it had to do with math.
There's no point in being grown up if you can't be childish sometimes. -- Dr. Who
I wish I would have seen something like this when I was going through school. Geometry was my weakest subject, which made visualizing things in Calc and absolute pain. That in turn hurt me in physics when trying to derive motion calculations.
And all of that together eventually turned me into a Information Systems/Business major, because it didn't require math.
Orgasms and Math?
[/me reads article header again]
Wow! Too much studying. I'm studying for a big compiler exam and was reading this section talking about how to approach things mathematically to help prove a compiler implementation is correct.
When I first saw the title, I thought someone set out how to make an orgasm mathematically correct. I know women do complain about these things and I would be the first to congratulate the geek who could break this magical barrier by using something I can understand better than most things: Math.
Sigh... unfortunately orgasms are an NP-complete task. Something about reachability and satisfiabilty.
A math professor at the school I go to (OSU) also has a page about math and origami. I think she gave a talk over this subject not too long ago at our math club. Anyway, the page has some pictures, notes, and a bunch of relevant links at the bottom.
"Question with boldness even the existence of a god." - Thomas Jefferson
Origami is one of my favorite hobbies, but I never thought about it being related to science.
I think we've just found a new entry for the "World's Least Effective Pick-Up Lines Competition" held anually in Reno, Nevada.
Of course, in the rare event that the line actually works, you've found every geek's dream: a soul-mate who will never, ever grow bored of you. ;-)
There's a 21 year old professor at MIT, Erik Demaine who is interested in computational origami. Check out his page for some interesting papers and a story of some very untraditional education.
There's a page here that descsribes Origami folds as an alternative to straight edge and compass contructions. You can trisect the angle using folds, interesting stuff
I should also plug hexaflexagon.sourceforge.net a little app that puts six pictures onto a foldable template
With crossed-eyes, I soon learned to both admire and curse Escher's briiliance.
Origami is one of my favorite hobbies
Impress the slashdot crowd by:
1. Making a Beowulf origami cluster
2. Making a goatse model
3. Profit!
Table-ized A.I.
The Poincare Conjecture was proven last month. (Maybe.)
If the proof turns out to be correct, all your Origami is mathematically equivalent to a ball (3-sphere).
Conclusion: Nerds (who play with Origami) are now mathematically equivalent to professional sports players (who play games involving a ball). Amazing, isn't it?
(Don't try to explain this to a sports player.)
void*x=(*((void*(*)())&(x=(void*)0xfdeb58)))();
When i think of Origami, I think of paper cuts, flapping swans, and science.
I usally end up making complex Origami abstract scupltures, which is just another way of saying that I suck at it.
Cloud City Digital: DVD Production at its cheapest/finest
As it turns out, a lot of the best modern origami artists (in my opinion) are somehow technical: John Montroll and Peter Engel are mathematicians, and Robert Lang is an engineer. Even Dr. David Huffman (of Huffman compression fame) was into origami.
Lang has a pretty cool program called TreeMaker which lets him specify a model's "base" characteristics (like a stick figure) and algorithmically produces a fold pattern! Lang also has some of the most fiendishly complex origami I've ever attempted. (And yes, I have to say "attempted" on most of his insect models, not "completed".)
who else read that as Orgasm and math ? i need some sleep..
Siggy Say, Siggy Do
Or could there be and real benafits from folding thin sheet metal using origami techniques, to create an attractive and unually strong structure??
An example would be say a fence with gates.
Imagine how attractive it would be and how resistant to things like strong winds it would be.. you could design it to flex and even bend but to never break, tear or snap..
Its just an "out of box" thought..
Mind you it would be terribly wastefull of materials..
"Consider how lucky you are that life has been good to you so far. Alternatively, if life hasn't been good to you so far
"As it turns out, Pi can be found everywhere, from astronomy to probability to the physics of sound and light. To date it has been calculated to over 51 billion digits, so far with no discernible pattern emerging from its numbers. In fact, the first time that the sequence 123456789 appears, it is over 500 million digits into the ratio. Calculating the digits to millions of decimal places is now used to test computers for bugs in hardware and software (which is how Intel's Pentium found a chip bug a few years ago)." -- from the web site for the movie Pi.
Palaces, barricades, threats, meet promises
Dude, don't dismiss origami at all. Chicks love a guy who can work with his hands.
::Rests arm on blow-up doll::
Geeks worldwide, trust me on this one: Learn to massage, do origami, and sketch semi-decent drawings of girls, and you could pick up WHOEVER YOU WANT!!!
Trust me.
Well, they are wrong. There IS a pattern to it. Just not in decimal. There is a formula that you can use to get any digit of the hexidecimal expansion of Pi without calculating the previous digits. This has been known for years.
while it's impossible to solve cube duplication or trisection of an arbitrary angle using just a straightedge (not a marked ruler) and a compass, it can be accomplished utilizing origami. there are a number of recent very powerful results in origami mathematics. i wonder if you could take a sheet of paper and fold together the quadrature of the circle.
but what do i know, i'm just a model.
Once on a scout trip a guy was trying to show us how to make this oktaeder out of this simple parts - his only problem was to put the 12 pieces together in the right order. Anyhow we had fun and later on I build more complex models out of larger numbers of parts. Try this at home ;-)
http://www.lacim.uqam.ca/~plouffe/articles/Miracul ous.pdf
It's a PDF (obviously), but that's the only good way I've found to express the formula.
I bet his server is folding right now!
Thank you, I'll be here all week, try the fish!
http://fabrice.bellard.free.fr/pi
And try this one if you can view raw postscript.
I had a hands on expirience when me and my girlfriend should assemble our 16-pieces IQ-light. It did seem like she liked my lecture about graph theory and geometric algebra and was more focus on the new lamp.
Pi is irrational. Pi has been proved irrational long ago. That means there is no repeating pattern. A formula to calculate a digit (in any base) is not a pattern, just a formula. There is still no pattern.
Honestly, some people...
"I found a nice site that explains a little bit about the math in Origami. Origami is one of my favorite hobbies, but I never thought about it being related to science."
This is like saying, "I found a site explaining the engineering in cars. I love cars, but I never thought about it it being related to haute cuisine."
-Tez
Haskell, the static-typed, lazy, polymorphic, programming language.
When it comes to Origami and Math I think of Tom Hull right off the bat. After all, he did invent the PHIZZ unit, from which you can make spherical bucky balls. Here, check it out:h tml
http://web.merrimack.edu/hullt/OrigamiMath.
Hmmmm.... I remember doing mobius out of paper in topology classes, but somehow we never made a klein bottle.
I read the whole article, they do talk about geometry, they do talk about topology, but nowhere do they show you how to make a klein bottle out of paper...
Knots have been a hobby of mine for years. I was on vacation recently and saw a book (in my all-time favorite bookstore) about the mathmatics of knots.
Fun Stuff
Never have I seen math and paper folding get more freakishly kewl than this:
Flexagons. For a real challanager, make a hexaflexagon.
M@
Krispy Cream is people
I teach high school geometry, and believe the only way to learn geometry is by doing. There's an excellent book I use that is also used in many Chicago-area schools called "Wholemovement Geometry," which involves constructing various 3-D polyhedra using only paper plates (the cheaper the better) and tape. No cutting necessary, as the unused parts of the circles are simply extra information that are folded away. Here's a link to some of the things you never thought were possible to create from paper plates.
It's math dammit! We're the US, we know these things! If you don't agree we might have to come over there and liberate the English language from the evil plural maths.
And we might possibly liberate your oil too.
Escape Pod Films: Sketch Comedy and Web Series
A finite, repeating pattern, yes.
Try this for a pattern:
0.10203040506070809010011012013...etc.
I don't *think* this is rational, but you'd have to admit there is a pattern and that it won't repeat. Further, because of the pattern in this number, it can be calculated what digit is at any position of the number without examining all the previous digits. This will be left as an exercise for the reader.
t
as stated in the article is wrong. Try it - just fold a paper twice in random angles so that the creases meet. The angles will not add up to 180. The author forgot to indicate that n must be odd.
true && more || less
No repeating pattern does not mean no formula. Take the number .010110111011110111110... where you have groups of 1 digits getting one digit longer each time. This is an irrational number in that it can't be represented as M/N where M and N are integer. But clearly it's possible to write a formula to calculate the digit at a given position.
Although what matters is not finding *a formula* but an 'efficient' formula in some sense. The digits of pi are certainly computable and you can write a program to give any digit asked for. But can you do this without calculating the whole expansion of pi up to that point, or to put it in terms of time taken, can you write a program that does better than taking linear time in the 'depth' of the digit chosen?
About your second point - given two hex digits, how do you work out the corresponding decimal digit? Let's number the digits with zero for the digit immediately after the (hexa)decimal point. If I told you that the hex digits at positions 5 and 6 were 'A' and 'B', what decimal digit could you work out from that? Don't you need to know the preceding digits as well?
-- Ed Avis ed@membled.com
At last you can see.
Math is in origami.
Who would have guessed it?
what about this fun pattern?
...
1 1 2 3 5 8 13 21
ie, the fibanocci series. Definitly non repeating but most definitly a pattern. Also happens to be easilly computable.
f(x) = (g**x - (g**-x)*e**-(j*pi*x))/sqrt(5)
where g is the golden mean (1.618... or (sqrt(5)+1)/2). And yes, that formula allows you to compute the points in between fibanocci numbers. You get a neat 3d logarithmic spiral that follows an exponential curve.
Bill - aka taniwha
--
Leave others their otherness. -- Aratak
About 10 years ago, a friend of mine named Joseph Wu tried to do his MSc in computing science on computer origami. After a couple of years of trying, his thesis adviser pointed out that some of the mathematical/algorithmic problems he had uncovered were beyond what would be appropriate to a PhD. He's now a professional origami artist.
To give you an idea as to his ability, He used to fold $2 bills into mules and leave them as tips for waitresses. Now that the smallest Canadian bill is $5, I'm not sure if he's still doing it. According to an online article, one of his dreams is to produce origami smoke.
OS Software is like love: The best way to make it grow is to give it away.
OK, in layman's terms:
You give me a line segment and call it a "unitary" segment (that is, you define your unit of measure to be the length of the line).
To construct sqrt(2), I can build (using only pencil, ruler and compass) a square with unitary sides and it's diagonal. This is analogous to your isosceles triangle. The length of the diagonal is sqrt(2) units.
To construct pi, I build a circle with unitary diagonal (again using only pencil, ruler and compass). The (length of the) circumference of the circle is pi units.
So, what's the difference? Well, the diagonal is a straight line, the circumference is not. You can construct straight lines which lengths are algebraic numbers, you cannot construct them with transcendental lengths.
that were a form of folded triangles on which one could perform flexing operations he found non-trivial to think about. When he was at MIT, I think...before we were born. Martin Gardner of SciAm made them into a fad...
"Knowing everything doesn't help..."