Cisim genişlemeleri ve origami çizimleri
Field extensions and origami constructions
Yazar: BURCU ŞANSAN
Danışman: DOÇ. DR. ERGÜN YARANERİ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Mühendisliği Ana Bilim Dalı / Matematik Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2019
Konu: Matematik = Mathematics

Özet:
Japonca bir kelime olan "origami"; oru - katlamak ve kami- kağıt birleşiminden
oluşur. Kağıt ile yapıldığı için kağıdın icadıyla doğal olarak ortaya çıktığı düşünülüyor.
Göçlerle birlikte kağıt ve kağıt katlama Çin' den Japonya' ya, Kore' ye; ipek yoluyla
Avrupa' ya yayılmış. Origaminin canlanması 20. yy da ya¸sayan Akire Yoshizawa (
1911-2005) tarafından katlama işaretlerini geliştirmesiyle hız kazandı. Bilinen klasik
origami haricinde daha karmaşık modeller üretilmeye başlandı. Dekorasyondan güneş
panellerine, robotikten hava yastıklarına kadar birçok katlama günlük hayatta yerini
alıyor.
1990 yıllarından sonra modern origami matematikçilerin de ilgisini çekmeye başladı.
Bu dönemde geliştirilen origami aksiyomları ile geometrik, cebirsel, denklem
çözümleri, hata payı fonksiyonları gibi detaylarda akademik çalışmalar yapılmaya
başlanmıştır.
Robert Lang tarafından geliştirilen treemaker yazılımı ile origami modellerindeki
katlamaların izlerinin algoritması çözümlenmiştir. Bu sayede daha karmaşık
modellerin yapılması artmış; diğer yandan da teknolojik ve endüstriyel tasarımların
üretilmesine yol açmıştır. Japon bilgisayar mühendisi Tomohiro Tachi, OSME
6 sempozyumundaki konuşmasında herşeyin origamisi yapılır mı sorusuna uzun
araştırmalardan sonra evet cevabını vermiş, peki herşeyi origami ile yapan makine
olur mu sorusuna da düşünmeden evet cevabını vermiştir. Origaminin geldiği noktayı
kısaca bu cümle özetliyor.
Origami, günümüzde çeşitli branşlarda ele alınıyor. Matematik, ürün tasarımları, uzay
araştırmaları, mimari, tekstil, biyoloji gibi ana başlıklar verebiliriz.
Bu tez, bir inceleme çalışması olup, pergel cetvel çizimleri ve cisim genişlemeleri
arasındaki ilişkilerin benzerlerinin, kağıt katlamaları ve cisim genişlemeleri arasında
da olduğunu göstermeyi amaçlıyor. Tez çalışmasında geometrik çizimler sorularına
origami ile çözüm aradık.
İlk bölümde Yunanlı matematikçilerin geometrik sorularını ele aldık. Pergel-cetvel
ile hangi şekil ve çizimlerin yapılabildiği, hangisinin imkansız olduğunu nedenleriyle
inceledik.
İkinci bölümde benzer soruların origami ile çözümlerini araştırdık. Huzita
aksiyomlarını kullanarak pergel-cetvel ile yapılan geometrik şekilleri oluşturduk.
Origami çizilebilir sayıları tanımladık ve bunları cisim genişlemeleri ile ilişkilendirdik.
Üçüncü bölümde katlanabilir origami hakkında geliştirilen origami teoremlerine yer
verdik. Katlanabilir origami kavramı genellikle hareketli tasarımların alt yapısını
oluşturuyor.
Tez okuyucusuna cebirsel olarak bilgi verecek olan bu çalışma ile origami ile diğer
açık sorulara çözüm bulma fırsatı verebilir.

Abstract:
Origami is a Japanese compound word, from oru meaning "folding" and kami meaning
"paper," and refers to the art of paper folding. Japanese origami crane is most famous
of all origami models, international symbol of peace. It is believed that it originated
naturally following the invention of paper. Paper and paper-folding spread from China
to Japan and Korea and, via the Silk Road, to Europe through migrations. The
revitalization of origami accelerated when a system of notation for origami ( dotted
lines and arrows) was developed by Akira Yoshizawa (1911 - 2005) in the 20th century.
The standard for origami instructions in which valley folds are indicated by dashes line
and mountain folds are indicated by chain line. These origami instructions are known
by world widely and become origami language. This led to the creation of complicated
models that go beyond conventional origami, and many such folds enter daily life in a
range of areas from decoration to solar panels to robotics to airbags today.
In the 1990s, modern origami began to strike the attention of mathematicians.
The origami axioms developed during this period enabled the conduct of academic
study relating to details such as geometry, solutions of algebraic equations and
approximation method of error functions. Origami has an real geometry that s natural
subject of study. The oldest origami math book was 1840 by Rev. Dionysius Lardner,
which demonstrated geometric rules by folding paper. Other book was 1893 by Sundra
Row, which was more impacted by compass, ruler and paper folding. In 1936, origami
was analyzed in terms of geometric constructions, using compass and ruler according
to set of axioms by Piazzolla. This study might be first contribution to "origami
mathematics". In 1985 Humiaki Huzita and Benedetto Scimemi presented origami
6 axioms at the First International Meeting of Origami Science and Technology. Those
six operations became known as the Huzita axioms. The Huzita axioms provided the
first formal description of what types of geometric constructions were possible with
origami.
The algorithms of folding patterns in origami models were analyzed using TreeMaker,
a software developed by Robert Lang. This not only allowed for more complicated
models to come into abundance, but at the same time it brought about the production
of technological and industrial designs. In a speech he delivered during the 6th OSME
symposium, Japanese computer scientist Tomohiro Tachi answered yes in reference
to his long research on whether it is possible to make origami out of everything and
a definite yes to the question of whether a machine is possible that does everything
through origami. This briefly summarizes the current state of origami. Origami is
covered in a variety of areas such as mathematics, product design, space research,
architecture, textiles, and biology today.
This is a survey thesis and it aims to show that relationships that resemble those
between compass and ruler constructions and field extensions hold between paper folds
and field extensions. The present thesis seeks for answers to questions about geometric
constructions through origami.
The first chapter covers the geometrical problems of Greek mathematicians. It
examines which shapes and drawings are possible using the compass and the ruler and
which are not and why. Three geometrical problems in particular, often referred to as
the Four Classical Problems, and all to be solved by purely geometric means using only
a straight edge and a compass. They were "the squaring of the circle", "the doubling of
the cube", "the trisection of an angle" and "regular polygon". These problem' s actual
solutions had to wait until the 19th Century. In 1837 Wantzel showed that trisecting
an angle and doubling a cube by straightedge and compass were not always possible.
In 1882 Lindemann showed that squaring the circle was impossible with straightedge
and compass when he showed that p transcendental over Q .
The second chapter searches for solutions to similar problems through origami. The
results are surprisingly different from ruler and compass geometry. Firstly we give
basic definitions and theorems of field theory. It uses the Huzita axioms to form
geometrical shapes otherwise drawn using the compass and the ruler. Six main folding
method were developed by Huzita- Justin which allow constructions with circles and
lines. We showed that Three Classical Problems are solvable by Origami axioms. The
ability to trisect an angle shows that origami axioms are stronger than compass and
ruler. 5th axiom folds a point to a line over a crease. Because the point reflects across
the crease, its distance to the crease is the same as the distance from the crease to
reflected point. This shows that the crease is tangent to the parabola whose focus is
point and whose directrix is the line. We can use a parabola to construct square roots
with origami. It is clear that the creases in 6th axiom is tangent to two parabolas.
Finding a common tangent between two parabolas can be equivalent as solving cubic
equations. By the way we showed that 5th axiom and 6th axiom are not available
at special conditions. Additionally dividing n equal part method is explained. We
show that cubic equations can be solved using origami folding. It defines origami
construction numbers and associates them with field extensions.
The third chapter deals with the theorems of origami that have been proposed regarding
flat foldable origami. The concept of flat foldable origami lays the groundwork
for moving designs. What is the difference between with origami and flat foldable
origami? Origami models could be 1D, 2D or 3D. But flat foladable origami has some
special differences which are its paper is zero thickness and the folded form is flat.
They called this model of origami flat foldable origami. Flat foldable origami has
also been applied to other fields such as science and technology. Flat foldable origami
methods are used at solar panels in space, airbag at cars, self moving cells, robotic
engineering, furniture designs, enlarging fabric by body, architectural surfaces. It has
also been the inspiration of mathematical research into the properties of folded shapes.
Toshikazu Kawasaki, Jacques Justin and Jun Maekawa established several fundamental
theorems for single flat folded vertex. These theorems are introduced in this chapter.
The conclusions part, we give some future researches by add new opportunities.
Origami doesn't allow to cutting, using glue. We can give further studies by cutting
paper or using tracing paper. These ways make origami solutions more powerful than
traditional origami solutions.
The present study offers an algebraic account of origami and may provide an
opportunity for finding solutions to other open questions. In addition, origami is worth
studying and exploring in other math related fields. For example, there is a connection
between origami and topology, differential geometry, even to graph theory, something
that we don't usually assume origami would associate with.