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."

Another Link

[feed_me_cereal]

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.

Computational Origami and protein folding

[megazoid81]

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.

Re:Everything can be related to math.

[b0r1s]

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

Re: Pi

[Dylan Zimmerman]

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:Another Link

[Wingie]

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.

Re: Pi

[Dylan Zimmerman]

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.

Re:Everything can be related to math.

[makapuf]

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)

the pattern of pi

[js7a]

This is a good page on the pattern of pi:
http://fabrice.bellard.free.fr/pi [bellard.free.fr]

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

Re: Pi

[Omkar]

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...

Polygons from circles

[pongo000]

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.

Re:Origami pick-up lines

[Zirnike]

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: Pi

[Ed Avis]

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?

Re:Origami + Math = Tom Hull

[spiffy_guy]

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]

Re:Modern origami artists familiar with math

[Our Man In Redmond]

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.

Re: Pi

[Smurf]

How is constructing an isoceles right triangle to compute sqrt(2) different from constructing a circle to compute pi?

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.

Feynman invented Flexagons...

[Invisible Now]

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...