About Axiom 7 in Origami Math

[samueltam]

Recently I read about there are 7 basic operations of origami constructions.

I am able to figure out, to certain extend, what axioms 1-6 are about.

When I look at axiom 7, I think in most cases it can be replace by using the operations of axiom 4 twice and then the operation of axiom 2.

If an axiom can be replaced by something else, it cannot be qualified as an axiom.

What I need is an example of fold that can only be achieved by axiom 7?

Can anyone help?

[Sushy]

I dont understand how you want to apply axiom 4 twice. since this only produces a straight line through one point that is perpendicular to a straight line and there is only one possibility?!

I found something, although it is in German, it might help you with the pictures
http://www.origami-imagiro.de/downloads ... axiome.pdf

[samueltam]

I have made an example to show what I mean.

http://dl.dropbox.com/u/12788610/axiom7.pdf

If an axiom can be replaced by something more fundamental, it can't be qualified to be called an "axiom".

My problem is I cannot find an example of origami folds that can only be achieved by the operation specified by axiom 7 but not everything else.

[Sushy]

Now i see what you meant.

With applying axiom 4 twice, you just produce a parallel to line 2 which goes through that point, and then folding that point to the intersection with line 1 (axiom 2).
Hm you are probably right with saying that this can be applied for every case, but I'm not sure if this dismisses the axiom 7.

[Razzmatazz]

...If an axiom can be replaced by something else, it cannot be qualified as an axiom...

You are wrong about that.
An axiom could exist as theoretical proof of another axiom. You'll find this occurs a lot more in Euclidian Geometry.

[Bugfolder]

There's something of a problem with calling any of the seven Huzita-Justin operations "axioms". Normally, in mathematics, an "axiom" is something that is assumed to be true at the outset. A "theorem" is something that is proved from axioms and other theorems.

If you have a collection of axioms, assuming that they are *consistent*, you can derive a geometry from them and can construct various geometric figures that represent numbers (e.g., so-called "origami numbers.")

The issue with the HJAs is that they're not truly independent. In fact, Martens, in his book, shows that they all can be derived as special cases of the "two-points-to-two-lines" axiom (O6, depending on who's counting). So, most of the axioms are not, in fact, independent.

Another problem is that if you're talking about Euclidean paper, some of the axioms don't admit solutions. You can't fold two points onto two parallel lines if the distance between the points is less than the distance between the lines. So it's kind of problematic to call O6 an "axiom" if it isn't always true!

That's why I've preferred to refer to them as mere "operations". As operations, it doesn't matter if they're not truly independent or don't have solutions; we can still use them and talk about them in mathematically interesting ways, and there's not the baggage associated with the special meaning of the word "axiom."

[origami_8]

Damn, now we'll need to buy it even though there are already two copies of ODS in our bookshelf.
There's no possibility to trade an old copy for a new one?

[Pop pop]

Mr. Lang sorry but i dont understand what had you said could you please cut that down for me

[The Average Folder]

Yeah, I could pretend that I know what that means, but I don't. Huh?

[gachepapier]

Yeah, I could pretend that I know what that means, but I don't. Huh?

"Axioms" is a word with a loaded meaning - it normally is meant for rules that you chose to 'believe in' without demonstration, from which you can derive other rules.

A set of axioms is ideally chosen in such a manner that there is
- independence (meaning that you cannot derive one axiom from another) ; and
- consistancy (meaning that you cannot disprove an axiom on the basis of the chosen axioms).

If you really want to have a fun read, try looking up Gödel

[sinubuj]

Hi,

is there not also a problematic if we talking about Euclidean paper?
You can't fold a point onto an other point if the distance between the point is greater than the distance between the corner from the Euclidean paper. So it's kind of problematic at 01, 02, 05 and 06 "axiom"!
Also ther is a problematic at axiom 04, e.g. if the point is at the left lower corner and the line is at right lower corner (as a small dor ear).

Gruß Dominik

... Another problem is that if you're talking about Euclidean paper, some of the axioms don't admit solutions. You can't fold two points onto two parallel lines if the distance between the points is less than the distance between the lines. So it's kind of problematic to call O6 an "axiom" if it isn't always true!...