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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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And everything old is new again. Fascinating art
"Sanity is not statistical", George Orwell, "1984"
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
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Once Wiles proves the Goldbach Conjecture (Journal of the American Mathematical Society, Spring 2019), the entire art of origami ends up reducible to polynomial-time modelling.
Interestingly, Wiles publishes the proof at the age of 68, while residing at the Shady Acres Convalescent Center in Far Rockaway, New Jersey. Perhaps the most important aspect of his discovery is that no, as a matter of fact, mathematicians are not all washed up by age 40. At the time I came back (no pun intended), there was talk of a third Fields Medal for him.
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
I agree, this sum total of the interesting bits of this "Math in Oragami" page is a single proof regarding coloring that is not any more profound that what you would find in a u-grad course on graph theory.
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".
Have you ever seen FRIED GREEN TOMATOS? Well thats the type of oragami I practice! So come on up to chicago if you want your cock in my ass!!!!
L,
Scott Lockwood
We must return to the Tribe. It is our only hope of survival! Just because something is better (as IE is), doesn't mean you must use it. When you do, you lose your identity. Kill your TV, your SUV and your IE. (-;
110000001111111111101110
Help fight continental drift.
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."
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
what would "it" be called if one used no creases and instead used only curvaceous bends to get around making things outa paper?
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
Boy am I glad I hadn't taken out the recycling and gotten rid of more than 1000 old business cards yet.
:)
I have a new-found idle-time project thanks to finding out how to build business card cubes via this story
"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
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
Cheap UK and US VPS
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.
I like Origami. Cranes are cool, but what I really like are boulders and rocks.
Oh, yeah, the movie that fucks up Pi after 9 decimals.
I liked the movie, but it ain't exactly a reliable source of mathematical information ;)
Karma: Could be worse (could be raining)
...all this research about "folding protiens" and such...
Ok, that was awful, I'm going to bed now.
CAn'T CompreHend SARcaSm?
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.
And so obviously God himself, something so beautiful and ornate, must have come into existence from another God, and on and on...
Is a universe who's existence is contingent on infinite recursion that much weirder than one that popped into existence from nothing? Is it stranger than having existed forever?
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.
What is it? do you have a web link?
All I want is a secure system where it's easy to do anything I want. Is that too much to ask ~~ Randall Munroe
You mean besides this and these? And you thought you were being funny...
Sure I'm paranoid, but am I paranoid enough?
Bit racist isn't it?
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.
This discussion is getting a little off topic, however you are incorrect in saying "maths" is not a word. In Australia "maths" is used without exception to represent mathematics (or even mathamatics, if that's how you're comfortable spelling it), not "math". I guess by your definition, our country is full of anal instructors in a misguided attempt at regularization. Or is that regularisation? So maths looks funny to you; well math looks strange to me. The English language is a work in progress - just look at the annual additions to the OED as an example. Just because a huge chunk of the English-speaking world (North America) says math, doesn't make it an absolute.
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.
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?)
-- Ed Avis ed@membled.com
"Pi can be found everywhere".
;->
Hence, my theory (er um, law) that, "Math can be found everywhere... Except in a women's head."
I didn't say anything about other parts of her body
Even though, leave it to women... Instead of sticking with the Greek letter Pi to represent, they switched the the letter "V" to represent thier pi.
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.
Insightful? You mean that the only patters are periodic? Of course, pi is irrational and hence has a non-periodic expansion in any base. But there are certainly other patterns in the world than periodic ones, since pi can be described using a finite amount of information, of course there are patterns to pi as to any other computable number.
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
Well, then the question is merely one of semantics. A pattern has to show up more than once, I'd say.
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.
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.
Why?
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
After I found this site on making business card cubes, I started doing more experiments and figured out how to make tetrahedrons, octahedrons, and icosahedrons using a really simple module.
Instructions are here
Now I have a nice set on my monitor.
-- stream of did I lock the front door consciousness
this place looks like they are making a profit from origami: check out the second thumbnail.
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?
Here is the business, Pi is not just some "dull assed" irrational number like (2)^(1/2). Pi is transcendental.
Therefore, Pi is irrational BUT!ALSO! Pi is not constructible. Like, say, sqrt(2) is the hypotenuse of right triangle with legs equal to 1. (To the guy who says it can be computed) Pi cannot be computed (see sentence about triangles and stuff)!
We can think about it, sort of.. A computer can approximate it.. I don't know, maybe that counts as "computing" to some. (like astronomers or engineers or compscientists)
Pi is merely the limit of an infinite, nonrepeating sequence of real numbers.
It should be called 'paper folding', because that's exactly what it is.
"It take 9 months to bear a child, no matter how many women you assign to the job."
Isn't it a theorem that every rational number has a decimal expansion that either repeats or terminates?
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.
((lambda (x) (x x)) (lambda (x) (x x))) http://www.endpointcomputing.com a scientific approach to custom computing.
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 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!
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!
Probably no more than black people saying the "N" word is racist.
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).
Spoken like a LISP guy.
As someone that tutored college students who were lost in calc, I sympathize with your last statement. But my experience appears to tell me that its not strictly the mathematicians fault but that students are already "ruined" by the time they get to college. The aparent reason would be that math is taught to be all about numbers and getting the right answer. Ugh, to cut a page long rant into the short form, let's just say there need to be more word problems and at the high school level there should be more proofs done in a more prose like fashion.
Let's refine your conjecture to say that every rational number repeats eventually (terminates means repeating zeros). Every decimal place produced is the previous remainder modulo the divisor. Since a rational number has a finite divisor there can only be a finite number of remainders before one will be repeated. For bonus points the reader can explain how many before it repeats and what it has to do with relative primes.
If I've gotten something wrong above, give me a minute to rewrite my axioms.
t
Origami is one of my favorite hobbies, but I never thought about it being related to science.
The Good Book clearly states in II Meshugginahs 3:16 that paper cannot be creased, heathen. As with Galileo we must threaten these idolatrous reprobates with excommunication unless they recant their immoral paper-fondling! Glory!
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
Like the above, there is a pattern, and like the above, it would seem reasonable that a formula could be derived to return the n digit. .011235813215475129...
Food not Bombs is a nice platitude but it breaks down when you notice that the Bombees are usually well fed
"Sanity is not statistical", George Orwell, "1984"
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.
*sigh*
I'll bite.
A rational number is any number that can be expressed as an integer divided by an integer. You're right that Pi is not rational, which is to say that it cannot be represented as an integer divided by another integer.
You are *completely* wrong in your assumption that an irrational number cannot have a pattern to it. Consider, for example, the number 0.10011000111000011110000011111......
The pattern in this number is very obvious. It cannot, however be represented as an integer divided by another integer and is thus irrational.
"My religion is to live --and die-- without regret." -- Milarepa
Reminds me of something Richard Feynman said about the term 'computer science' being a misnomer. Science is the study of nature through observation, whereas computers and math are human creations and therefore not natural. They were created by us to do useful things and are quite helpful for placing the natural world in terms that are explict and understandable. However, the laws that these disciplines are built from are, in the end, created by us for their utility. Mathematics is changed and altered by our scientific research, but it is morphed to fit by our doing. The new rules were not 'discovered', but created by necessity. It is like a bridge we design to span a particular space. We cannot 'discover' the bridge because it did not exist before we saw a need for it. Math has no carte blanche. When it does not help us illustrate the natural world it is useless, though pretty, mumbo-jumbo- like numerology.
The difference seems subtle, but it is profound. We cannot blindly take math to be the 'language of god'. It is our language and nothing more or less.
Anyone with me?
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 friends for their amusement. I'd also do Game of Life by hand on graph paper (home computers were not around yet). I sure had a lot of free time back then.
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 +.
OS Software is like love: The best way to make it grow is to give it away.
I enjoyed this article. Not so much math, but some great paper-folding puzzles.
One of these days we are going to strap 4 scientists into a machine whose design the SETI telescopes picked up ... and you are going to be so wrong.
Note to the humourless: http://www.iblist.com/book.php?id=741
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.
No repeating pattern. From there, I'd say it depends on your definition of pattern. Your pattern only occurs in base 10.
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..."
We offer a math & origami class at the university where I teach: Origami of Math It's writing-intensive too, if you can believe that! :)
-megan
but what do i know, i'm just a model.
My comment was actually geared towards higher math than the college level calc classes.
I did quite well as a math major through college, but when it comes to trying to read professional math books and learn anything much from them, it's virtually hopeless.
Even with the standard mathematician's trick of skimming first, and repeating several times in deeper and deeper depth, there comes a time when you start to say "Why are they doing this, and what were they trying to show, and why does it matter?"
Those are the questions that mathematicians almost never answer. How many times have you read a chapter in a math book and had it say something like:
"By the end of this chapter you will have a good feeling for the well known theorems in basic knot theory. we'll proceed by showing blablaba which will require 3 theorems from basic algebra. Then we will show xyz which is a consequence of well known theorems from topology which we will remind you of as we go along. Then in the end we'll show that all knots are pdq, this requires that you know some basics about coloring theorems but we will state those theorems approximately rather than proving them in depth...."
And then have them complete the chapter with the same level of explanation of why and what they are doing throughout? Almost NEVER.
Otherwise even to a dedicated and competent mathematician, it starts to look arbitrary if you haven't seen something before. The fact is that the bulk of math papers place NO emphasis on cognitive understanding, and ALL emphasis on short concise (dense) proofs of apparently arbitrary facts followed by a final proof of the main issues, and then move on to an apparently separate part of the authors favorite research topic.
I agree that our math education needs a major overhaul. I think it should concentrate on two things: mathematical modelling, and computational mathematics. These are the things that people will need the most.
Specifically students should be able to take a problem about the real world, and create formulas that describe it. They should have a sense for what statistics mean about the real world, and they should be able to get approximate answers quickly and correct answers efficiently.
When it comes to explaining math though, the mathematicians take all the blame for simply ignoring explanation in favor of correct minimal proof.
((lambda (x) (x x)) (lambda (x) (x x))) http://www.endpointcomputing.com a scientific approach to custom computing.