Márta Turcsányi-Szabó

Eötvös Loránd University, Faculty of Natural Sciences

Department of General Computer Science

HUNGARY

E-mail:TURCSANYINE@LUDENS.ELTE.HU

Recently I have spent some time in Japan so I had the opportunity of getting acquainted with japanese culture, the language, and the people. These three interact strongly on each other, so to understand one urges the learning of the other. Getting acquainted with people and their thoughts meant learning their language, the beauty of the language is enhanced by the admiration of calligraphy, calligraphy is the companion of sumi-e, to learn sumi-e one has to learn more about paper, touching paper leads to folding paper, and folding paper is origami. I have always admired arts and I especially loved to do handicrafts myself. Origami gave me the delight of expression which led my research on this ancient art. Being an old LOGO-ist I often saw the world through the Turtle. One learns a lot about nature through playing around with fractals and vice versa. [1]. The patterns produced by some recursive procedures [2] give inspiration to origami structures which I would like to introduce.

Let us start from the simple procedure that every beginner in LOGO tackles sooner or later.

The result represents a 200 unit square piece of paper which is the starting point of every origami object. To produce a three dimensional piece of origami work is a child's job, of which japanese children are very talented. A lot of objects are well known basics, while the rest depends on imagination and talent. The most famous piece of origami is probably that of folding a crane or tsuru, which is a lucky bird in Japan. Here are the basic folding lines of the crane:

...but the completion really depends on artistic view. As Kunihiko Kasahara, my favourite origamist wrote in his book [3]:

Or see Eiji Nakamura's further variations [7].

Let us go back to our piece of paper or in other words our SQUARE. Let us divide the sides of the paper into half and cut with a scissors till nearly the centre. We can do the actual cutting of one side with the Turtle itself by replacing FORWARD :SIZE with the following generalised Koch curve procedure [8] where N=4 (No. of segments) & r=1/2 (scale ratio).

Let us construct the NEW_SQUARE procedure with this ONE_SIDE.

... results in 4 squares actually, but the road the Turtle takes represents the future edges of the paper.

Do not cut all the way till the centre, leave out a few millimetres to keep the paper together.

Creativity solves the problem of choosing the corner for the head of the crane. Do not hold back on experimenting with other cutting lines or trying out combinations. In the following page, some examples of slit designs from the oldest manuscript Senbazuru (A Thousand Cranes) of traditional origami in Japan can be seen [9]. It was written in 1797 by Rokouan Gido Ichien, the chief Priest of a temple at Kuwana in Ise-no Kuni. It contains 49 kinds of crane-folding methods with the use of one square or rectangle sheet of paper with slits to produce a fairly good number of mutually connected cranes. Each amulet is accompanied by Akizato, the famous poet's Haiku to emphasise lyrical expression, and illustrations are drawn by Shunsensai. (A Haiku is a type of short poem, having lines of 5-7-5 syllables, in which the poet attempts to express his deepest emotions by describing nature in simple but beautiful language.)

A lot of these origami emblems appear in family crests, which is possessed by every Japanese family. The origin of Japanese family crests (mon) goes back to the eleventh century. Each of the high-ranking countries in those days began using a specific textile design for his most formal costume worn at the Imperial court [10]>.

To increase the number of cranes you should go back to your LOGO procedure and experiment. First of all if you would like to see the actual cutting surfaces produced, it is better to substitute the ONE_SIDE procedure with a slightly different one:

Using this as an element for your NEW_SQUARE procedure helps you visualise where and how to cut or vice versa guiding through the lines of cut gives you the LOGO procedure.

Understanding recursion is not so easy but paper cutting techniques can help a lot. Start cutting the resulting bits of squares in the very same way for every square to produce the basics of the next and next levels. At the same time try to put down the algorithm in LOGO too:

TO REC_PATTERN2 :SIZE

REC_PATTERN1 :SIZE / 2

RIGHT 88

REC_PATTERN1 :SIZE / 2

LEFT 176

REC_PATTERN1 :SIZE / 2

RIGHT 88

REC_PATTERN1 :SIZE / 2

END

TO NEWER_SQUARE2 :SIZE

REPEAT 4 [REC_PATTERN2 :SIZE

RIGHT 90]

END

This produces a new level of cutting. Continue the next level of cutting and algorithm writing.

Since each level of cutting is done after the previous cutting level, the procedures can be nested and the algorithm continues till we reach the desired level:

Try this new procedure of recursive squares and decide on the level of cutting. As you can see the :

SQUARE1 300 3 commands do the job perfectly.

You should try to do each origami masterpiece from a single piece of paper, so do take care while cutting to keep the paper together. When you are finished comes the more difficult part of folding the birds separately without tearing the paper.

Patience and gentleness should be your guide.

Choosing the corner for heads and tails enhances the product as well as choosing cutting slits.

This page describes another beautiful example from Rokouan. It is called Hyakkaku

(A Hundred Cranes). Actually you end up with 97 cranes, but the base itself is a 100 unit-square paper.

The Haiku accompanying this marvellous piece of origami is difficult to translate but it certainly gives the impression of force. Actually Asahina has a complex internal meaning and it can be a name of a place, a family or crest too.

If you try a number of easier variations, and manage a few of the complicated ones too you will definitely get some expertise. But I must tell you that, although Rokouan's book of origami is indeed very popular, most admirers remain at this stage and only a few persistent fellows complete the book. If keeping paper together is what hinders the project, you might as well complete the design from separate pieces of paper. I guess Rokouan's spirit would not be insulted by doing so.