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Integralsm symbol
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DIALOGUE DIALOGUE DIALOGUE DIALOGUE
ON INTEGRAON INTEGRAON INTEGRAON INTEGRALISM SYMBOLLISM SYMBOLLISM SYMBOLLISM SYMBOL
BYBYBYBY
ARMAHEDI MAHZARARMAHEDI MAHZARARMAHEDI MAHZARARMAHEDI MAHZAR
http://integralisme.wordpress.com
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Dialogue on integralism symbolDialogue on integralism symbolDialogue on integralism symbolDialogue on integralism symbol Part OnePart OnePart OnePart One
Ni Suiti and Ki Algo is my Anima and Animus which are
the feminine and the masculine unconscious subpersonalities
within myself. They are always in continuous conversation.
even if I am in the front of one of the million eyes of the
giant Tenretni and fingering her YTREWQ fingers.
In the following, is the dialogue of them concerning
the picture that I put as myself in the my blog
integralist.multiply.com. The Q(uestioner) is Ni Suiti and
the A(nswerer) is Ki Algo. Hopefully you can all enjoy it.
Here we go!.
Q: What is the picture that Arma used as his blog
identification?
A: That’s a part of a geometric pattern in the cover
of his first book : Integralism.
Q: What geometrical pattern?
A: Aperiodic tesselation.
PERIODIC TESSELATION:
CRYSTALS
Q: What is tesselation anyway?
A: Tesselation is the covering of infinite plane
with a finite set of tiles. If the tesselation periodic,
from infinitely many possible regular polygons
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only three can tile a plane periodically: triangle, square and
regular hexagon. Periodic tesselation means that you can shift
the tiling pattern translationally to get the same pattern
We have 3-fold rotationally symmetric tesselation
and the 4-fold rotationally symmetric tesselation
and the six-fold rotationally symmetric tesselation
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There are exactly 17 possible tilings of the plane
with 3, 4 and 6-fold rotation symmetry. 13 of them
can be found in Alhambra (I see it in here )
The combination of triangles, squares and hexagons
can make a periodic tesselation such as
Q: Beautiful!. All of them is periodic, meaning
you can shift the pattern and get the same pattern.
I see it in many mural decoration in islamic mosques and palaces
and it can also be found naturally in the structure of
all crystals. But your pattern is not periodic.
APERIODIC TESSELATION:
FROM 20.426 to 2
A: That’s why I call it aperiodic, following the
tradition of mathematics literature.
Q: Why do you interest in such tile?
A: Well, an article in Scientific American in the 60s
was caught in Arma’s eyes. It was written by the
logician Hao Wang. He asked if an infinite set
of finite kinds of 2-way square color domino can
fully cover all the plane periodically. It is called
tiling problem.
Q: But the tile in Arma’s integralist symbol are
not square, right?
A: Please do not interrupt me. The story to follow
is an amazing story of mathematical simplification.
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It began in 1966 when Robert Berger demonstrated
that the question Wang asked is unanswerable.
Mathematically the periodic tiling problem is in fact not
decidable. He proved that if a finite kinds of tile can be
used to cover all the plane, then it can not be filled in
periodically. Such tiling called aperiodic. To prove
it, he construct very large set of prototiles consisting
20,426 prototiles. He showed that that they can
tiles a plane aperiodically. Fortunately, he was able to
reduce the number of aperiodic tile
to relatively small set containing 104 tiles.
Following Berger discovery, there is a rush of simplifications
of the prototiles set. For example in 1968 Donald Knuth was
able to reduce the number to 92, then Robinson reduced it in 1971 to 35.
Roger Penrose improved it to 34. Robinson made more improvement
by reducing to 24. Karel Culik II finally in 1996 gave an example of the set of 13 tiles
which tile the plane aperiodically.
By allowing rotation, Robinson had been able to reduced the number
of prototile to just 6 tiles as it is shown in the following
Amazingly, three years later in 1974, the well known
UK physicist, Roger Penrose had been able to reduce
further the total number of required prototiles to just 2.
The trick is to change the shape of the tiles from
squares to rhombi. Well for me it is very impressive,
reduction from more than 20,000 to only 2 in less
than a decade.
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The two rhombi are the thick and the thin ones.
Let us call them Thicky and Thinny
By joining the edges of thickies and thinnies we can form
the following tesselation covering the infinite plane
Q: See, it is interesting but it is not the same as
your integralism symbol.
A: Well, the Penrose rhombi can be each cut in half
and rejoined edge by edge to form these Penrose
Kite and Dart tiles. Joining two halves of Thinny
makes the Dart and joining two halves of Thicky
makes the Kite:
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we can form the following aperiodic tesselation.
I hope you will see that my integralist symbol is nothing but a subconfiguration
of the above tesselation.
Q: OK. I see it now.
It is wonderful. It is Mathematical.
Mathematics is beautiful.
But it is really unnatural.
There is no such pattern in the crystal.
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Well, I could hardly hear what Ki Algo said in
answering Ni Suiti. I think he was mumbling,
and there was long silence after that.
I will try harder later to eavesdrop their conservation
and report it to you to be enjoyed. See you next time.
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Dialogue onDialogue onDialogue onDialogue on integralism symbolintegralism symbolintegralism symbolintegralism symbol Part TwoPart TwoPart TwoPart Two
Ni Suiti is my Anima who loves visual arts, chemistry and geometry and Ki Algo is my Animus who loves music, physics and algebra. They are still discussing my integralism symbol in the cover of my first book which is factually a Penrose tiling. In the following dialogue S is for Suiti and A is for Algo. Let us see, what are the facts known by Ki Algo on the realization of Penrose tiles in nature.
PHYSICAL REALIZATION:
QUASICRYSTALS
S: Arma’s symbol for integralism is a Penrose tiling and it is
beautiful alright, but it is just a mathematical game. I think
it will not appear in nature such as the 17 patterns
of periodic tiles appear in crystals. It is forbidden. Am I wrong?
A: Well, you are wrong. It is true that there are only 17 patterns of crystal symmetry,
none of them have 5-fold rotational symmetry, but exactly one year after
the publication of Arma’s first book Integralism, the physicist Dan Schechtman
announced the discovery of a phase of an Aluminium Manganese alloy which
produced tenfold symmetric X-ray diffraction photograph. It may be similar to
this photograph.
This symmetric pattern can only be explained if the atoms are
arranged aperiodically in the form of three dimensional
generalization of Penrose tiling as it is discovered by
the amateur mathematician Robert Ammann.
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S: So, I am sure that they are not crystals. What are they?
A: Steinhardt and Levine, shortly after the announcement of Shechtman’s discovery,
used the term ‘quasicrystal’ By the end of the 1980s the idea of quasicrystal became
acceptable and in 1991 the International Union of Crystallography amended its
definition of crystal, reducing it to the ability to produce a clear-cut diffraction pattern
and acknowledging the possibility of the ordering to be either periodic or aperiodic.
With the new definition, quasicrystal is just a kinds of crystal: the aperiodic crystal.
S: How many types of quasicrystal are there?
A: In the same year of Schechtman publication, Ishimasa and coauthors
published their discovery of twelvefold symmetry diffraction pattern of Ni-Cr particles.
Soon another equally challenging case presented a sample which gave a sharp
eightfold diffraction picture. Lately, a team led by An Pang Tsai from Japan’s
National Research Institute for Metals in Tsukuba has discovered quasicrystals of
cadmium- ytterbium that are stable and exhibit three-dimensional icosahedral symmetry.
So there are four types of quasicrystals:
• 8-fold or octagonal symmetric quasicrystals
such as
lV-Ni-Si
Cr-Ni-Si
Mn-Si
Mn-Si-Al
Mn-Fe-Si
• 10-fold or decagonal symmetric quasicrystals
such as
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Al-Ni-Co
Al-Cu-Mn
Al-Cu-Fe
Al-Cu-Ni
Al-Cu-Co
Al-Cu-Co-Si
Al-Mn-Pd
V-Ni-Si
Cr-Ni
• 12-fold or dodecagonal symmetric quasicrystals
such as
Cr-Ni
V-Ni
V-Ni-Si
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• Dodecahedral symmetric quasicrystals
such as
Al-Mn
Al-Mn-Si
Al-Li-Cu
Al-Pd-Mn
Al-Cu-Fe
Al-Mg-Zn
Zn-Mg-RE
Nb-Fe
V-Ni-Si
Pd-U-Si
S: Beautiful, but what’s the use of quasicrystals technologically?
A: Until now there are several applications, for example
• thin film quasicrystal coating are used as a non-stick surface for saucepans.
• razor blades and surgical instruments are made from quasicrystalline material
• Quasicrystals have also been associated with hydrogen storage
• Metallic alloys are reinforced by deposition of icosahedral particles
to improve the materials ability to withstand strain
S: I wonder if quasicrystal patterns also occur in cultural artifacts in history?
A: I think you can asked Arma’s friend Tenretni immediately
I think we have to wait what Ni Suiti find as her answer. Bye, for now.
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Dialogue onDialogue onDialogue onDialogue on integralism symbolintegralism symbolintegralism symbolintegralism symbol Part ThreePart ThreePart ThreePart Three
In the previous dialogue, Ki Algo was explaining mathematical and physical aspects of my integralism symbol. In the following dialogue, Ni Suiti report her findings of the cultural realizations of the aperiodic tiling in history. Listen!
CULTURAL REALIZATION:
ISLAMIC ARCHITECTURE
A: Hello Suiti. Do you have some answers from my giant fried Tenretni.
S: Yes, she told me that the physicist Peter Lu from Harvard University did
some field research in Iran, Turkey, Azerbaijan and India and found a surprising fact
that Islamic maths was 500 years ahead. See ABC News in Science webpage
http://www.abc.net.au/science/news/stories/2007/1855313.htm?ancient .
A: That’s big news, but the data is too little to be significant.
S: There are so many discoveries To convince you I will list some of the strange
ancient artifacts chronologically ordered. The decoration of the artifact
is in the leftside and the aperiodic pattern is in the rightside.
• The Gunbad-i Kabud tomb tower in Maragha, Iran (1197
CE.),
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• Abbasid Al-Mustansiriyya Madrasa in Baghdad, Iraq (1227-34 AD),
• The Ilkhanid Uljaytu Mausoleum in Sultaniya, Iran (1304 AD),
• The Mamluk Quran of Sandal (1306-15 AD)
• The Mamluk Quran of Aydughdi ibn Abdallah al-Badri (1313 AD),
• The Timurid Tuman Aqa Mausoleum in the Shah-i Zinda complex in Samarkand,
Uzbekistan (1405 AD).
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• • the Ottoman Green Mosque in Bursa, Turkey (1424 AD),
• The shrine of Khwaja Abdullah Ansari at Gazargah in Herat,
Afghanistan (1425 to 1429 C.E.) (3, 9),
• The Darb-i Imam shrine in Isfahan, Iran (1453 CE)
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• The Friday Mosque, Isfahan, Iran (late 15th century AD).
A: Wow! So many forms in ancient artifacts structurally similar to the postmodern
physics discoveries. Where do you find such data?
S: The above information are from supporting online material for the article in
the magazine Science 315 (2007), 1106-1110 by
Peter J. Lu and Paul J. Steinhardt,
Decagonal and quasi-crystalline tilings in medieval Islamic architecture,
The article is also available online in one of the author webpage
http://www.physics.harvard.edu/~plu/publications/
A: I think, only scientists will read the article.
S: Oh no, the findings are so surprising, so it was reported all around the world in newspaper,
magazines, radio and television broadcasting
• Firstly in Lu’s campus
http://www.news.harvard.edu/gazette/2007/03.01/99-tiles.html
• But there is also an article in New YorkTimes
http://www.nytimes.com/2007/02/27/science/27math.html?hp
• Another in Newsweek International
http://www.msnbc.msn.com/id/17553752/site/newsweek/
• The BBC news in bahasa Indonesia is in
http://www.bbc.co.uk/indonesian/news/story/2007/02/070223_geometricislamicart.shtml
The list for other worldwide news on the discovery can be found in Peter Lu webpage
http://www.physics.harvard.edu/~plu/research/islamic_quasicrystal/
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A: Oh my goodness. I have never expected such explosion of news.
It’s very exciting. I’ve never known that art, mathematics and physics
has such a common underlying form.
I hope you enjoy my report of eavesdropping Ki Algo and Ni Suiti dialogues. I will report more on their dialogues, if I find something interesting to be shared in cyberspace. See you later
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Dialogue onDialogue onDialogue onDialogue on integralism symbol:integralism symbol:integralism symbol:integralism symbol: Part FDourPart FDourPart FDourPart FDour
In the last dialogue, Ni Suiti was explaining her discoveries in the answers of the she-giant Tenretni about the aperiodic pattern by the ancient muslim architects. It surprised Ki Algo. The following dialogue is their philosophical reflection ignited by such surprising discovery.
A: Why did aperiodic tiling has so many realizations? The concept, as mathematical entity, is
realized in the mind of mathematicians. Afterward, it was discovered that it is realized physically
in quasicrystals.
S:
Astonishingly, it has also been realized in the mind of muslim architects for about five
centuries. Because it is discovered in nature, then it should be there before it exists in the
mind of mathematician or even the architects. How can it be?
A:
Looking at these facts, I think the aperiodic tiling pattern is outside human mind and it is
also outside natural world. I suppose that it must be in some kind of Plato’s World of Ideas
of Mathematical Forms.
S:
Well, if it is really there in the outside, it seems to me that it can be intelectually ‘seen’ with
our mind’s ‘eyes’ mathematician sees them because they use their intuition. With the same
intuition, they accept undefined concepts and underived axioms of any mathematical
formal system such as geometry and arithmetics. The traditional muslim architects saw it
there first because their remembrance practice, so their intuitive eyes are clear so the
“remember” the form first before the mathematician. Plato called it ”anamnesis”
A:
Well, reflecting more deeply, I see that your analogy begging a question. If our mind can
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‘see’ the form behind the thinks we see, can the mind also “see” the “Forms” behind the
forms behind the things with our innermost eye?
S:
The eyes that is behind the mind eyes behind the physical eyes arethe spiritual eyes. But
what is the FORM behind the Mathematical “Forms” behind the physical forms?
A:
Let me guess. All geometric forms are generated by using simple rules. I think there is the
“Forms” behind the forms. I can discover it with logic. You can say that I “see” it with my
mind’s “eyes.” But I can see it deeper. I see a FORM exists behind all the logical generating
rules namely all your ‘Forms.’ It is structured set of relations between all the rules. It is the
principle behind the rules. For such formal rules, the principle is symmetry. Periodicity is
just one kind of symmetry. In fact, the aperiodicty of quasicrystals can be described as the
projection of periodicity in higher dimensional space.
S:
Periodicity in space is like musical rythm in time. In music, the rythm is the framework for
the melody. The counterpart of melody in space is the mutual transformations of material
forms. That’s it the mathematical forms is forming a the great symphony within what the
ancient muslim philosopger called the knowledge of God.
A:
That’s a good metaphor. But what is the real reality?
S:
The symmetry is the mathematical version of Beauty. The Beauty is the ultimate value
beyond our world but penetrates to our world. Philosophers like to call such beyondness as
transcendence and such penetrateness as immanence. Religious person call the Ultimate
Transcendence as God. Beauty is just one of the characteristic of the Ultimate
Transcendence.
A:
What are the other ones?
S:
The other characteristics of the ultimate transcendence is Truth and Goodness. Without
Truth we can not get science. Without Goodness we can not have dynamically ordered
society. The Goodness in society is Justice. If the Beauty manifest in the symmetry, The
Truth manifests in the consistency and the Good manifests itself in the optimality.
A:
What an enlightening vision!
S:
Beauty, Truth and Goodness as the Atributes of the Ultimate Being are universal entities
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which are realized differently in different cultures. But they are also realized universally
and naturally in different forms of matter as natural Symmetry, Consistency and
Optimality.
A:
How can we see such Universal Trio?
S:
The important thing is to know how do we “see” Beauty, Truth and Goodness. Symmetric
Beauty can be realized by emotion or feeling, Consistent Truth by reason or Logic and
Optimal Goodness by intuition.
A:
Yes, I think we see such immaterial things through feeling, reason and intuition. But all
those immaterial things is revealed in material things which we can see with sensation.
S:
The late psychologist Carl Gustaf Jung said that those are the fundamental psychological
function: (1) sensation, (2) feeling, (3) thinking and (4) intuition. Arma think that they are
correlated to (1) matter, (2) energy, (3) information and (4) values respectively. We sense
matter, feel energy, think information and intuit values. So, the fundamental psychological
functions are correlated to the four categories of relative substances in integralism.
A:
What Arma did not k
now is that you are the archetype of Intuition and I am the archetype of Logic. He also did
not know that your grand-daughter Si Nessa and my grand-son Si Emo is the archetypes
the archetype of Sensation and Feeling respectively.
It is surprising to me that beside the two eldery archetypes, Ni Suiti and Ki Algo, there are two child archetypes, Si Nessa and Si Emo, who are accompanying them. Pondering on the names of those archetypes, I finally found out that their names are the anagrams of indonesian terms related to the four psychological function that was listed by Carl Gustaf Jung. Wow! Can you find it? See you later.
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AfternoteAfternoteAfternoteAfternote onononon integralism symbol:integralism symbol:integralism symbol:integralism symbol:
Dan Shechtman had obtained his Ph.D. from Technion – Israel
Institute of Technology, and in 1983, he managed to get Ilan Blech, a colleague at his alma mater,
interested in his findings of the "forbidden" 10-fold symmetric diffraction pattern.
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Together they attempted to interpret the diffraction pattern and translate it to the atomic pattern
of a crystal. They submitted an article to the Journal of Applied Physics in the summer of 1984.
But the article came back seemingly by return of post – the editor had refused it immediately.
Shechtman then asked John Cahn, a renowned physicist who had lured him over to NIST in the
first place, to take a look at his data. The otherwise busy researcher eventually did, and in turn,
Cahn consulted with a French crystallographer, Denis Gratias.
In November 1984, together with Cahn, Blech and Gratias, Shechtman finally got to publish his
data in Physical Review Letters. The article went off like a bomb among crystallographers. It
questioned the most
fundamental truth of their science: that all crystals consist of repeating, periodic patterns.
After a long waiting time, the Nobel Prize in Chemistry 2011 was finally awarded to Dan
Shechtman "for the discovery of quasicrystals".