Fujita Rules

Rules Fujita is a set of seven rules that formally describe geometric constructions using flat origami , similar to constructions using a compass and a ruler .

In fact, they describe all possible ways of obtaining one new fold on a sheet of paper, by combining already existing various elements of the sheet - points and lines . Under the lines are meant the edges of the sheet or folds of paper, under the points - the intersection of lines. The essential point is that the fold is formed by a single fold, and as a result of folding the figure remains flat.

Often these rules are called "axioms", although from a formal point of view they are not axioms .

Content

Rules

Folds in these rules do not always exist, the rule states only that if such a fold exists, then it can “be found”.

Rule 1

Let two points be given $p_{1}$ ${\ displaystyle p_ {1}}$ and $p_{2}$ ${\ displaystyle p_ {2}}$ , then the sheet can be folded so that these two points will lie on the crease.

Rule 2

Let two points be given $p_{1}$ ${\ displaystyle p_ {1}}$ and $p_{2}$ ${\ displaystyle p_ {2}}$ , then the sheet can be folded so that one point will move to another.

Rule 3

Let two straight lines be given $l_{1}$ ${\ displaystyle l_ {1}}$ and $l_{2}$ ${\ displaystyle l_ {2}}$ , then the sheet can be folded so that one straight line goes into another.

Rule 4

Let be given a straight line. $l_{1}$ ${\ displaystyle l_ {1}}$ and point $p_{1}$ ${\ displaystyle p_ {1}}$ , then the sheet can be folded so that the point falls on the crease, and the straight line goes into itself (that is, the fold line is perpendicular to it).

Rule 5

Let be given a straight line. $l_{1}$ ${\ displaystyle l_ {1}}$ and two points $p_{1}$ ${\ displaystyle p_ {1}}$ and $p_{2}$ ${\ displaystyle p_ {2}}$ , then the sheet can be folded so that the point $p_{2}$ ${\ displaystyle p_ {2}}$ gets on the crease, and $p_{1}$ ${\ displaystyle p_ {1}}$ - on the line $l_{1}$ ${\ displaystyle l_ {1}}$ .

Rule 6 (fold Protein)

Let two straight lines be given $l_{1}$ ${\ displaystyle l_ {1}}$ and $l_{2}$ ${\ displaystyle l_ {2}}$ and two points $p_{1}$ ${\ displaystyle p_ {1}}$ and $p_{2}$ ${\ displaystyle p_ {2}}$ , then the sheet can be folded so that the point $p_{1}$ ${\ displaystyle p_ {1}}$ will get straight $l_{1}$ ${\ displaystyle l_ {1}}$ and point $p_{2}$ ${\ displaystyle p_ {2}}$ will get straight $l_{2}$ ${\ displaystyle l_ {2}}$ .

Rule 7

Let two straight lines be given $l_{1}$ ${\ displaystyle l_ {1}}$ and $l_{2}$ ${\ displaystyle l_ {2}}$ and point $p$ ${\ displaystyle p}$ , then the sheet can be folded so that the point $p$ ${\ displaystyle p}$ will get straight $l_{1}$ ${\ displaystyle l_ {1}}$ and straight $l_{2}$ ${\ displaystyle l_ {2}}$ will go over into itself (that is, the fold line will be perpendicular to it).

Remarks

All the folds in this list can be obtained as a result of the sequential application of rule number 6. That is, they do not add anything to the mathematician, but they allow to reduce the number of folds. The system of seven rules is complete, that is, they describe all possible ways of obtaining one new fold on a sheet of paper, by combining already existing various elements of the sheet. This last statement was proved by Lang ^[1] .

Possible and Impossible Constructs

Possible

All constructions are nothing more than solutions of any equation , and the coefficients of this equation are related to the lengths of the given segments. Therefore, it is convenient to talk about the construction of a number - a graphical solution of an equation of a certain type. Within the framework of the above requirements, the following constructions are possible:

Construction of solutions of linear equations .
Construction of solutions of quadratic equations .
Construction of solutions of cubic equations (rule 6).

In other words, it is possible to construct only numbers equal to arithmetic expressions using the square and cubic roots of the original numbers (lengths of segments). In particular, with the help of such constructions it is possible to carry out a doubling of the cube , the trisection of an angle , the construction of a regular heptagon .

Impossible to

The solution of the quadrature problem of a circle, however, remains impossible, since π is a transcendental number .

History

The main rule (number 6) was considered by Margherita Piazzolla Belok ^[2] , she also owned the first constructions of the trisection of the angle and quadrature of a circle using origami constructions. Folds Protein is enough to get folds in all other rules.

A complete list of the rules appears in the work of Jacques Justine ^[3] , who later also referred to Peter Messer as a co-author. Almost simultaneously, rules 1-6 were formulated by Fumiaki Fujita ^[4] . The last seventh rule was added even later by Kosiro Hatori ^[5] .

Variations and generalizations

The list of possible constructions can be significantly expanded if you allow the creation of several folds at a time. Although a person who decides to carry out several folds in a single action, in practice will face physical difficulties, it is nevertheless possible to derive rules similar to those of Fujita for this case ^[6] .

Under the admission of such additional rules, it is possible to prove the following theorem:

Any algebraic degree equation

n

${\ displaystyle n}$ can be decided by simultaneous

(n-2)

{\ displaystyle (n-2)}

-fold folds.

It is of interest whether it is possible to solve the same equation by folding, involving a smaller number of simultaneous folds. This is undoubtedly true for $n=4$ ${\ displaystyle n = 4}$ and unknown for $n=5$ ${\ displaystyle n = 5}$ ^[6] .

Notes

↑ Lang R. Origami and Geometric Constructions .
↑ Beloch, MP Sul metodo del ripiegamento della carta peri risoluzione dei problemi geometrici / Periodico di Mathematiche. - Ser. 4. - Vol. 16. - 1936. - pp. 104-108.
↑ Justin, J. Resolution of the lee de l'equation de troisieme et al. - H. Huzita ed. - 1989. - pp. 251-261.
Z ita ita Ax z - Humiaki Huzita, ed. - 1989. - pp. 143-158.
↑ Koshiro Hatori Origami Construction .
↑ ¹ ² Alperin RC, Lang RJ One-, Two- and Multi-Fold Origami Axioms .

Links

Petrunin A. Plane Origami and Constructions (Rus.) // Kvant . - № 1 . - p . 38-40 .
Lang R. Huzita Axiomas . (eng.)
Hull T. Origami Geometric Constructions . (eng.)

[1] Lang R. Origami and Geometric Constructions .

[2] Beloch, MP Sul metodo del ripiegamento della carta peri risoluzione dei problemi geometrici / Periodico di Mathematiche. - Ser. 4. - Vol. 16. - 1936. - pp. 104-108.

[3] Justin, J. Resolution of the lee de l'equation de troisieme et al. - H. Huzita ed. - 1989. - pp. 251-261.

[4] Z ita ita Ax z - Humiaki Huzita, ed. - 1989. - pp. 143-158.

[5] Koshiro Hatori Origami Construction .

[s-6] ¹ ² Alperin RC, Lang RJ One-, Two- and Multi-Fold Origami Axioms .