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## Origami and Math222

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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## Origami and Math

Comments Filter:
• #### Everything can be related to math. (Score:5, Insightful)

<dleblanc@gmail.com> on Wednesday May 07, 2003 @12:28AM (#5898757)
Sometimes you just have to be creative. Math is everywhere.
• #### Re:Everything can be related to math. (Score:2, Interesting)

In the end... you can reduce everything to 0's and 1's.... and logic operators...
• #### Re:Everything can be related to math. (Score:2, Funny)

Yes, but you can break the logic operators into even smaller 1's and 0's...

The plus sign is simply two 1's criss-crossing each other.

The multiplication sign is the same thing as the plus sign, but at a 45-degree angle.

The division sign is a sideways '1' with very small 0's above and below it...
• #### Re:Everything can be related to math. (Score:2)

Those aren't logic operators, those are arithmetic operators. Logic operators are AND (&), OR (|), NOT (!), and XOR (^). There are a few other less common ones too, but they don't have any symbols I'm familiar with.
• #### Re:Everything can be related to math. (Score:2, Informative)

You don't have to include XOR. You can create it out of ((x OR y) AND (NOT (x AND y)))

• #### Re:Everything can be related to math. (Score:2, Insightful)

u dont need and, not, or, just use NOR xor NAND like the hole IC industry does
• #### Re:Everything can be related to math. (Score:2, Informative)

you need only NANDs to make all other gates, y'know ?
NOT (X) = NAND (x,x)
AND x y = not(nand(x,y))
OR = nand(not(x),not(y))
nor = not or
etc ... (even a=>b, the logical implication, which is not a or b)
• #### Re:Everything can be related to math. (picky) (Score:2)

Those aren't logic operators, those are arithmetic operators. Logic operators are AND (&), OR (|), NOT (!), and XOR (^).

If you insist:
The '|' is just a one in another form.
! is a one with a small 0 beneath it.
^ is two ones
& is a bit more difficult, but it can be reasonably done as two 0's one small '1' and two even smaller '1's side by side (for the up=pointing stub.

btw: AND and OR are often designated as: * and +.

• #### Even the c0ffee you like (Score:2)

110000001111111111101110
• #### Re:Everything can be related to math. (Score:5, Funny)

by Anonymous Coward on Wednesday May 07, 2003 @12:52AM (#5898865)
Math is everywhere.

Well, not everywhere.

Math doesn't exist in our President's budget proposal, for example...
• #### Re:Everything can be related to math. (Score:5, Funny)

by Anonymous Coward on Wednesday May 07, 2003 @01:31AM (#5898998)
Imaginary numbers are part of math.
• #### Re:Everything can be related to math. (Score:2)

Imaginary numbers are part of math.
what like eleventeen and thirtytwelve?
• #### Re: Pi (Score:4, Interesting)

on Wednesday May 07, 2003 @01:29AM (#5898992) Homepage
"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. [pithemovie.com]
• #### Re: Pi (Score:2)

Oh, yeah, the movie that fucks up Pi [imdb.com] after 9 decimals.

I liked the movie, but it ain't exactly a reliable source of mathematical information ;)

• #### Re: Pi (Score:3, Informative)

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.
• #### Re: Pi (Score:2)

What is it? do you have a web link?
• #### Re: Pi (Score:2)

How can this be - how can there exist a formula to get a hexadecimal digit but not a decimal digit? What is so special about base 16?

(Unless the origianl formula is to get a binary digit, and you clump four of them together to get hex... but then why is binary special?)
• #### Re: Pi (Score:2)

How can this be - how can there exist a formula to get a hexadecimal digit but not a decimal digit?

Easy. There can't. Pi is irrational. By the definition of an irrational number there is no repeating pattern that defines the number, hence no formulas. And for the second impossibilty, how can there possibly be a formula for aribtrary hex digits and not decimal? All you have to do is find at most two hex digits and convert to find the decimal digit.

• #### Re: Pi (Score:3, Informative)

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 g
• #### Re: Pi (Score:5, Informative)

on Wednesday May 07, 2003 @06:44AM (#5899874) Homepage Journal
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...
• #### Re: Pi (Score:2)

If a formula isn't a pattern, then what is?

Patterns don't need to repeat. We have trig functions that do, but if you give them a little bias, they follow a line instead of an axis. Surely no one denies that y(x)=x+Sin(x) is a pattern, and yet, it doesn't repeat.

So, how does the BBP formula not show a pattern? Without one, the formula wouldn't work, because it can calculate the nth digit without calculating any of the previous digits.
• #### Re:Haiku (Score:2)

At last you can see.
Math is in origami.
Who would have guessed it?
• #### /.'d after 0 posts (Score:4, Funny)

on Wednesday May 07, 2003 @12:31AM (#5898768)
Apparently the math goes like this: Origami Website + (/. crowd) = 0
• #### Re:/.'d after 0 posts (Score:3, Funny)

That would imply that either both the website and slashdot are zero, or that they are opposites. Perhaps this:

lim responsiveness = 0
hits->slashdot_users
• #### Re:/.'d after 0 posts (Score:2, Funny)

So then the server "folded" under the pressure? Who would have thought :P
-Jason
• #### Huh? (Score:4, Funny)

<<jessica> <at> <supjessica.com>> on Wednesday May 07, 2003 @12:31AM (#5898772)
How does math relate to the Origami Boulder? [origamiboulder.com] :)
• #### Re:Huh? (Score:2)

You mean besides this [ohio-state.edu] and these [uchicago.edu]? And you thought you were being funny...
• #### The two are *definitely* related (Score:5, Funny)

<slashdot@leSLACKWARExal.net minus distro> on Wednesday May 07, 2003 @12:31AM (#5898774) Homepage
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 [origamiboulder.com] has the right idea. =)
• #### Re:The two are *definitely* related (Score:2)

Fuck, and I've been recycling all of my wadded up paper.

Hell, instead of wadding up those "Thank you for submitting your resume. You will be contacted if your skills match the job requirements." type of letters in anger and frustration, I could be selling them for \$10!!!

And I'll offer more then just wadded oragami like that cheapo. I mean the real stuff: paper wads, shredded paper, paper that I ripped into a million pieces, dipped in whiskey, set on fire and spit on the dead, charred remains.

Real emotio
• #### Not what I read (Score:2)

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.
• #### Re:Not what I read (Score:2)

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.

I dunno, maybe graph the projectile of a fluid?
• #### This would make learning a little more fun... (Score:4, Insightful)

on Wednesday May 07, 2003 @12:32AM (#5898781) Homepage Journal
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.
• #### Re:This would make learning a little more fun... (Score:2)

And all of that together eventually turned me into a Information Systems/Business major, because it didn't require math.

Sorry for the jab, but...

As a Business Major, of course you don't need math! If things don't add up right (taxes, extra losses you don't want people to see, bonuses for the heck of it, etc.) you can always use the origami paper shredder, ala Enron. :)
• #### Re:This would make learning a little more fun... (Score:2)

It required doing math, it just didn't require taking any more math courses. I'd survived through Calc 1, they seemed to think that was enough.

At least I didn't say that I learned how to code all of my business applications in Visual Basic, and I just need to trust Microsoft to get my math right.

Actually, I did learn VB, JAVA, and ASP, but I never trusted any figures unless I could work the same thing out on my trusty TI-30 (or whatever model I had then) calculator and get the same answer.

But at no time
• #### Orgasms and Math? (Score:5, Funny)

by Anonymous Coward on Wednesday May 07, 2003 @12:34AM (#5898785)
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.
• #### Re:Orgasms and Math? (Score:2, Funny)

Orgasms are only NP-complete in a threesome (3-CNF-SAT). It has been shown that the task can be completed in polynomial time when the conjunction is only between 2 entities (2-CNF-SAT). [see Kama Sutra (translated title: Algorithms) as interpreted by CLRS, exercise 34.4-7]
• #### Another Link (Score:5, Informative)

on Wednesday May 07, 2003 @12:34AM (#5898788)
A math professor [ohio-state.edu] 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.
• #### Re:Another Link (Score:3, Informative)

Ahh, her origami models for her undergrad math thesis still floats gloriously in the Amherst College math building. Here's another link: http://web.merrimack.edu/hullt/OrigamiMath.html Tom's a graph theorist who's been studying this subject basically for as long as mathematics and origami were linked. There are some very interesting stuff there, like curricula to courses involving origami that he's taught.
• #### Origami pick-up lines (Score:5, Funny)

on Wednesday May 07, 2003 @12:38AM (#5898803)

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. ;-)

• #### Of Course It's in NV (Score:2, Funny)

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 it's held in Nevada. If the line fails, you hit up the whore-house down the road.

Repeat to yourself: "Location, location, location."

• #### Re:Origami pick-up lines (Score:2, Funny)

it would work for me -- a guy that is that capable with his fingers, he would be worth dating! :)
• #### Re:Origami pick-up lines (Score:2)

Well, some of us that do origami don't necessarily have great fingers. For instance, I have a surgeon's fingers (not literally, of course), but I tend to grow my fingernails inordinately long. This is actually helpful in certain ways. It makes creasing the paper much easier. They also let me easily flatten foil Reese's wrappers and make some really cool models out of real metal as opposed to metallic paper.

However, I would imagine that my fingernails would get in the way of almost anything you could be
• #### Re:Origami pick-up lines (Score:3, Informative)

I picked up some decent quality stuff from AC Moore. Don't know if there are any in your area...

If you have access to a decent paper cutter, some wrapping paper makes good folding paper, as well.

And be really careful... I thought that was handy, too, until I started doing complex models. My first try on a rhino tore about 1/2 way through because of too-strong creasing. Not that I've gotten it right yet, but still.

• #### Re:Origami pick-up lines (Score:2)

(obligatory) Would you like to see my SR-71?

Sadly enough, I'm serious... It's right above my monitor. Along with a crane, chrysanthemum, antelope, giraffe, frog, kangaroo, and an eagle. I also have another (actually, the standard) style crane folded out of aluminum foil bonded to tissue paper, which is a really neat material.

• #### Computational Origami and protein folding (Score:5, Informative)

on Wednesday May 07, 2003 @12:48AM (#5898847)
Don't dismiss origami immediately - it could have implications for things like protein folding. As it stands, computing and examining the number of ways a protein can fold is an NP-complete problem. Imagine the insights into molecular biology we might get with further research into the computational complexity of origami.

There's a 21 year old professor at MIT, Erik Demaine [mit.edu] who is interested in computational origami. Check out his page for some interesting papers and a story of some very untraditional education.

• #### Chick magnet, dude... (Score:4, Funny)

on Wednesday May 07, 2003 @01:40AM (#5899024)
Dude, don't dismiss origami at all. Chicks love a guy who can work with his hands.

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!!!

::Rests arm on blow-up doll::

Trust me.
• #### Re:Chick magnet, dude... (Score:2)

Ha, ha, ha. Good one. I can do all three quite well and still everyone hates me.

Maybe I'm just anti-social.
• #### Re:Computational Origami and protein folding (Score:2)

Don't dismiss origami immediately - it could have implications for things like protein folding.

And download folding@home [stanford.edu] while you're attention's in this vicinity.
• #### Re:Computational Origami and protein folding (Score:2)

Check out his page for some interesting papers...

... folded into a swan, a viking longboat, and a scale model of the moon.

• #### Origami for geometrical constructions and a plug. (Score:5, Interesting)

on Wednesday May 07, 2003 @12:49AM (#5898853) Homepage Journal

There's a page here [merrimack.edu] 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 [sourceforge.net] a little app that puts six pictures onto a foldable template

• #### Re:Origami for geometrical constructions and a plu (Score:2)

Wow. I haven't thought about hexaflexagons in a long long time. When I was in middle school (in the early 1970's), I read the Piers Anthony science fiction book, Ox, which featured not only a hexaflexagon, but also a sentient being based on Conway's Game of Life. In the book, a hexaflexagon was used as a map to show the path through dimensional doorways.

Ox inspired me to dig deeper into the mathematics presented in the book. I made hexaflexagons when I was bored in class, and would give them out to fr

• #### Inorganic chemistry (Score:3, Interesting)

<ken,lin&gmail,com> on Wednesday May 07, 2003 @12:49AM (#5898854)
I remeber many homework assignments/problem sets in my inorganic chemistry class (Cornell '96) that ask ones to find and name all the symmetry in Escher drawings. (It's harder than you think.)

With crossed-eyes, I soon learned to both admire and curse Escher's briiliance.

• #### Re:Inorganic chemistry (Score:2)

Well, not all of Escher's stuff is symmetric. Try the one of the sphere in his hand. It has very little symmetry at all.

Of course, then you try to do it with the one with the Sausage Rolls going up and down the stairs.

Really, I quite like Escher's art. It's right up there with Salvador Dali on my scale of great art.
• #### Re:Inorganic chemistry (Score:2)

Of course, but the non-symmetric Escher drawings would make for terrible problem sets whose purpose was to illustrate symmetry.
• #### Re:Inorganic chemistry (Score:2)

Well, there is that.

Why was your inorganic chem class even assigning problems like that? Most interesting symetries that I've found are organic. Stuff like Hemoglobin. The structure of that molecule just facinates me.
• #### "for my next trick...." (Score:5, Funny)

on Wednesday May 07, 2003 @12:52AM (#5898868) Journal
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!
• #### Poincare Conjecture (Score:4, Interesting)

<yoni.vl@gmail.com> on Wednesday May 07, 2003 @12:54AM (#5898879) Journal

The Poincare Conjecture [wolfram.com] was proven [wolfram.com] 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.)

• #### Never thought of science!?!?! (Score:3, Funny)

on Wednesday May 07, 2003 @12:57AM (#5898893) Homepage
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.
• #### Modern origami artists familiar with math (Score:5, Interesting)

on Wednesday May 07, 2003 @01:20AM (#5898959)

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 [sgi.com] (of Huffman compression fame) was into origami.

Lang has a pretty cool program called TreeMaker [origami.kvi.nl] which lets him specify a model's "base" characteristics (like a stick figure) and algorithmically produces a fold pattern! [siam.org] Lang also has some of the most fiendishly complex origami [origami.kvi.nl] I've ever attempted. (And yes, I have to say "attempted" on most of his insect models, not "completed".)

• #### Re:Modern origami artists familiar with math (Score:3, Informative)

Interesting that you should mention Engel. The introduction to his book Origami From Angelfish To Zen [bestwebbuys.com] deals with the mathematical aspects of origami, including its fractal aspects.
• #### ok (Score:3, Funny)

on Wednesday May 07, 2003 @01:22AM (#5898965)
who else read that as Orgasm and math ? i need some sleep..
• #### Is Origami just for paper? (Score:2, Interesting)

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..
• #### Paper folding (Score:2)

This is all cool to know but it doesn't help me with my basic problem of not being able to fold paper in a straight line. Prehaps I'm using the wrong type of paper

Rus
• #### This is funny (Score:2, Funny)

I like Origami. Cranes are cool, but what I really like are boulders [origamiboulder.com] and rocks [origamirock.com].
• #### origami mathematics (Score:5, Interesting)

on Wednesday May 07, 2003 @02:14AM (#5899130) Journal
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.
• #### Re: Trisecting the angle (Score:2)

What a load of codswollop.

In fairness, I was deeply impressed with myself when _I_ managed to trisect an angle at around a third of my current age (which is around a third of a century). However, I discovered (when junior school arithmetic became senior school mathematics) that trisecting a right angle using origami is not particularly difficult (no harder than creating an equilateral triangle).

Now presumably your 'method' involves taking a corner of a piece of paper (perhaps not 90 degrees even) and slow
• #### Re: Trisecting the angle (Score:2)

In answer to myself: of course I already had the 'trisecting the angle' page open in another tab! Although it does admit that the vital fold is a matter of trial and error, it is rather more elegant than folding a Z. As such, it is not a mathematical construction, but certainly could be handy if stuck without a protractor AND a calculator!
• #### Re: Trisecting the angle (Score:2)

I'm still wrong and blathering. I'm too old. Extra axioms are being added which if not easy to fold physically are at least mathematically sound. My apologies for my witterings, and my thanks for having learned something after all that (no, not to not post before reading, about origami and third order equations, grin).
• #### Oktaeder out of simply parts (Score:3, Interesting)

on Wednesday May 07, 2003 @02:58AM (#5899273)
Once on a scout trip a guy was trying to show us how to make this oktaeder [origamiseiten.de] out of this simple parts [12testing.net] - 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 ;-)
• #### Origami Effect? (Score:4, Funny)

by Anonymous Coward on Wednesday May 07, 2003 @03:12AM (#5899332)
I bet his server is folding right now!

Thank you, I'll be here all week, try the fish!
• #### IQ Light (Score:3, Interesting)

on Wednesday May 07, 2003 @04:35AM (#5899584)
I had a hands on expirience when me and my girlfriend should assemble our 16-pieces IQ-light [iqlight.com]. It did seem like she liked my lecture about graph theory and geometric algebra and was more focus on the new lamp.
• #### Maths (Score:2, Funny)

Yes, very interesting article. But to quote the post:

"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

• #### Origami + Math = Tom Hull (Score:4, Interesting)

on Wednesday May 07, 2003 @07:18AM (#5899998)
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:
http://web.merrimack.edu/hullt/OrigamiMath.h tml
• #### No Klein bottle ? (Score:3, Funny)

on Wednesday May 07, 2003 @07:50AM (#5900156) Homepage
Hmmmm.... I remember doing mobius [umn.edu] out of paper in topology classes, but somehow we never made a klein bottle [umn.edu].

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 are great fun too (Score:2, Interesting)

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 [tatteredcover.com].

Fun Stuff
• #### Flexagon (Score:4, Interesting)

on Wednesday May 07, 2003 @08:20AM (#5900298) Homepage Journal
Never have I seen math and paper folding get more freakishly kewl than this:
Flexagons [cinvestav.mx]. For a real challanager, make a hexaflexagon.

M@
• #### Re:Flexagon (Score:2)

A much better flexagon link has been posted in another reply:
SourceForge project [slashdot.org]

M@
• #### Polygons from circles (Score:4, Informative)

on Wednesday May 07, 2003 @08:42AM (#5900471)
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 [depaul.edu] to some of the things you never thought were possible to create from paper plates.
• #### Kawasaki's Theorem (Score:2, Insightful)

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.
• #### Feynman invented Flexagons... (Score:2, Informative)

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...

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