## Origami and Math 222

Posted
by
michael

from the finally-a-real-world-use-for-geometry-class dept.

from the finally-a-real-world-use-for-geometry-class dept.

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."*
## Everything can be related to math. (Score:5, Insightful)

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

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

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)

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

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

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

NOT (X) = NAND (x,x)

AND x y = not(nand(x,y))

OR = nand(not(x),not(y))

nor = not or

etc

## 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 +.

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

The 1 [goatse.cx], but I wouldn't say that it's in nature.

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

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

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)

## 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)

## Re: Pi (Score:2)

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

9decimals.I liked the movie, but it ain't exactly a reliable source of mathematical information ;)

## Re: Pi (Score:3, Informative)

## Re: Pi (Score:2)

## Re: Pi (Score:4, Informative)

It's a PDF (obviously), but that's the only good way I've found to express the formula.

## the pattern of pi (Score:3, Informative)

http://fabrice.bellard.free.fr/pi [bellard.free.fr]

And try this one if you can view raw postscript [cecm.sfu.ca].

## Re: Pi (Score:2)

(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)

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?

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

irrational. Pi has been proved irrationallongago. That means there isnorepeating 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)

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: Pi (Score:2)

## Re: Pi (Score:2, Interesting)

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.

## Re: Pi (Score:2)

If that's true (and I think it is) your number is definitely irrational.

What's more, your number is recursively enumerable (it's easy to write a turing machine to compute it).

Ah math. Fascinating stuff. If only there were more mathematicians who were truly gifted at explaining it.

## Re: Pi (Score:3, Interesting)

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.

## Re: Pi (Score:2, Informative)

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

straightline, the circumference isnot. You can constructstraightlines which lengths are algebraic numbers, you cannot construct them with transcendental lengths.## Re: Pi (Score:2)

## Re:Haiku (Score:2)

Math is in origami.

Who would have guessed it?

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

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

Doesn't make sense, but it works.

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

Galois

Julia and Fatou which ledt to

Mandelbrot

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

The other poster's crack about lack of math in the president's budget proposal, on the other hand...

## /.'d after 0 posts (Score:4, Funny)

## 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)

-Jason

## Huh? (Score:4, Funny)

## Re:Huh? (Score:2)

## The two are *definitely* related (Score:5, Funny)

## Re:The two are *definitely* related (Score:2)

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 emotion here folks. I'm the friggin "Pollock" of the wadded paper!!!

-= Stefan

## Not what I read (Score:2)

## 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)

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)

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 during my education was I asked to find the integral of a business function. We keep Finance Majors around for that.

## Re:This would make learning a little more fun... (Score:2)

## Orgasms and Math? (Score:5, Funny)

[/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)

Kama Sutra(translated title: Algorithms) as interpreted by CLRS, exercise 34.4-7]## Another Link (Score:5, Informative)

## Re:Another Link (Score:3, Informative)

## Origami pick-up lines (Score:5, Funny)

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 inReno, 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)

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

However, I would imagine that my fingernails would get in the way of almost anything you could be thinking of.

On a completely unrelated note, good origami paper is almost impossible to find here in Dallas. The KERA Store Of Knowledge over in Fort Worth used to sell it, but then they went out of business.

## Re:Origami pick-up lines (Score:3, Informative)

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)

Regardless, that issue is just me-specific. I thought that that was clear from the way I said it.

How, exactly, do you know that "she" isn't some 80-year-old man (not saying that she is, just what if)? I mean, isn't that one of the wonders of the Internet? Almost total anonymity.

How many women are going to know that you can do origami without knowing you beforehand? It's not like anyone walks around with some folded paper visible on his person.

As far as I know, I was the only person in my entire high school that did origami. I would do it during class sometimes. At first, my teachers would yell at me for not paying attention, but when I could recite the past five minutes of lecture from memory, they started to realize that it doesn't take that much concentration to fold paper.

It never got me any dates.

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

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)

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)

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.

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

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)

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'sGame of Life. In the book, a hexaflexagon was used as a map to show the path through dimensional doorways.Oxinspired 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 friends for their amusement. I'd also doGame of Lifeby hand on graph paper (home computers were not around yet). I sure had a lot of free time back then.## Inorganic chemistry (Score:3, Interesting)

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

## Re:Inorganic chemistry (Score:2)

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)

non-symmetricEscher drawings would make forterribleproblem sets whose purpose was to illustrate symmetry.## Re:Inorganic chemistry (Score:2)

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)

Origami is one of my favorite hobbiesImpress the slashdot crowd by:

1. Making a Beowulf origami cluster

2. Making a goatse model

3. Profit!

## Poincare Conjecture (Score:4, Interesting)

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)

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)

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)

## ok (Score:3, Funny)

## Is Origami just for paper? (Score:2, Interesting)

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)

Rus

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

## origami mathematics (Score:5, Interesting)

## Re: Trisecting the angle (Score:2)

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 slowly, carefully, folding it in to a Z-shape, with each 'segment' having equal length.

However, this sadly does not a geometrical construction construe. Your (any) random angle is transcendental (or the hyperbolic functions of it are).

Finally, you mention a marked ruler: these do not help. When we (mathematicians) say 'construct' we mean 'provably' so. Thus no ruler exists which can measure a line of length pi cm, for example. Having said that, I should again admit something, or at least give a hint to precocious ten-year-olds without their protractor. My Casio (it had a glowing green LED display) calculator would do just fine as long as I had a (marked) ruler. Learning what sines are before you are supposed to CAN help!

## Re: Trisecting the angle (Score:2)

## Re: Trisecting the angle (Score:2)

## Oktaeder out of simply parts (Score:3, Interesting)

## Origami Effect? (Score:4, Funny)

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

## Maths (Score:2, Funny)

"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)

http://web.merrimack.edu/hullt/OrigamiMath.

## Re:Origami + Math = Tom Hull (Score:2, Informative)

Tom is definatly one of the leaders in this field. Those who haven't read his paper The Combinatorics of Flat Folds: a Survey [merrimack.edu] are missing out.

You might also check out Robert Lang's upcoming book Origami Design Secrets: Mathematical Methods for an Ancient Art [amazon.com]

## No Klein bottle ? (Score:3, Funny)

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)

Fun Stuff

## Flexagon (Score:4, Interesting)

Flexagons [cinvestav.mx]. For a real challanager, make a hexaflexagon.

M@

## Re:Flexagon (Score:2)

SourceForge project [slashdot.org]

M@

## Polygons from circles (Score:4, Informative)

## Kawasaki's Theorem (Score:2, Insightful)

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

## Re:come on, michael... (Score:2)

There might be a lot of math in Oragami that impresses 4th graders, but this indeed is not "News for Teacher's Stuff to Assign for Homework".

## Re:its maths damn it (Score:3, Funny)

And we might possibly liberate your oil too.

## Business Card Polyhedra (Score:2)

Instructions are here [hiwaay.net]

Now I have a nice set on my monitor.

## Re:Nifty (Score:4, Interesting)

About 10 years ago, a friend of mine named Joseph Wu [origami.as] 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 [vancouver.bc.ca].

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 [vancourier.com].