Mathematics of the origami

Foldings of Origami S can present mathematical minor problems at the very least sympathetic nerves.

Formalization of the origami

The formalism to which it is generally refers is that of Huzita . It contains 6 axioms which are in fact 6 basic foldings making it possible to break up any origami. Here is the list:

Axiom 1. a single fold passes by two point P and Q specified.
Axiom 2. a single fold brings a point P on a point Q.
Axiome 3. a single fold superimposes two lines m and N .
Axiom 4. a single fold passes by a point P and is orthogonal on a line m .
Axiom 5. Is a line m and two points P and Q; a single fold passes by Q and brings P on m .
Axiom 6. Is two lines m and N and two points P and Q; a single fold brings P on m and Q on N .

A4 sheets and origami

By noting $a$ the height and $b$ the width of a rectangle.

That is to say $B$ the point of carryforward of the width on a height ( $O$ and $A$ two corners of the rectangle such that contains $B$ , $A$ is closer to $B$ than $O$ is it). By noting $B'$ the opposite of $B$ (respectively $O'$ that of $O$ and $A'$ that of $A$ ).

$(O, B, B', O')$ is a square. One defers $B'$ in $C$ on (one notes $C'$ its opposite on ) to form the square $there (A', C, It, B')$ .

There remains the rectangle $(C, has, B, It)$ ; which are its properties?

Here lengths of some segments of this figure:

is length $a$ ,
and length $b$ ,
$=-$ is thus length $a-b$ ,
is also length $a-b$ ,
and thus $=-$ is length $b- (a-b)$ .

Let us note $r$ the report/ratio length of by : $r: = {B (a-b) \ over a-b}$

Let us express $r$ according to the report/ratio of on (that one notes $\ alpha$ ):

$\ alpha: = {has \ over B}$

Then:

$(1) \; \; r= {2 \ alpha \ over \ alpha -1}$

By construction ( is its length and its width), $\ alpha$ is larger than 1; the equation (1) says to us that it is smaller than 2 (if not it is impossible to carry out this figure; $C$ cannot be traced).

For information, one can trace the variations of the behavior of $r$ according to $\ alpha$ . A value of $r$ is particularly interesting: $r= \ alpha$ , that means that the proportions of the rectangle remaining after having withdrawn the two successive squares (initially $(B, O, O', B')$ then $(B', It, C, A')$ ), are the same ones as those of the original rectangle .

In this case, $\ alpha$ checks the equation:

$\ alpha = {2 \ alpha \ over \ alpha -1}$

Who has only one single solution: $\ alpha = \ sqrt {2}$ .

Coincidentally, they are precisely the proportions of the sheets $An$ (for example $A4$ , standard rectangular sheets):

Thus the sheets $An$ make it possible to carry out origamis fractales , because in the rectangle remaining, with the same proportions that the first, it is still possible to withdraw two squares, then to start again, theoretically until the infinite one.

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