Paul Klee's syntax

     Mountain Movement (in three and four time) 1928 / Bergblume 1933

I did not see this work [Mountain Movement] when I checked more than four hundred books about Klee in the National Library in Bern in the seventies.
Almost thirty years ago, I found this work on the auctioneer's catalogue which I bought at Strand in New York.
General public had chance to see the work first time when the catalogue was made after the death of the owner of the work; the reproduction is artwork: a sort of print.
I obtained the catalogue raisonne in early nineties
Coming back to Japan after long stay abroad for searching art books, I found the digital image of the painting online in 2011.
So far I have not used the books I collected for writing blogs.
When the photo of artwork becomes digital, it becomes digital artwork free to see anywhere any time.

It is not easy to guess how many digital image works there are online.
I did not find  [Mountain Movement] on image site pages.
Then I realised that I can search with English title since I have the catalogue.
I found the second work [Bergblume] not under the English title but the German title.

I always had desire to own the reproduction of the work in order to see it whenever I want.
The more I see his works the more I appreciate him.
I think the twenty century art criticism is totally outdated because there is no proper criticism possible without seeing and analysing the whole work.
I have convinced that all masterpieces have reason to survive: unique structure.
To understand such structure has been my passion for fifty years.

 Klee made many mosaic works with polygons.
this work [Mountain Movement] is a sort of cubist landscape; his interest in architecture made some compositional device.

The arch of a half circle can be seen as one evenly warped curved line and the centre of the half circle is easily spotted.
So I enclose the quarter circle with a rectangle.
The diagonal of the rectangle can be bent evenly in many ways and the quarter circle is one of them.
In this way any line segment can be expressed as a warped diagonal of a rectangle (diagonal is a curve with zero warp).
A horizontal line is a diagonal of a rectangle with zero vertical line and a vertical line is a diagonal of  a rectangle with zero horizontal line.
I call this unit the smallest unit of visible line: the element of artwork.
The diagonal model of Modern alphabet [B] and ancient Greek alphabet [B] is the same.
Both are made of six elements.

We can easily see that these two works are made of countable straight line segments.
There is a most noticeable point D(5) where five lines out from; the rest are four D(4) and fourteen D(3).
We call them polygonal mosaics, because each face is made of countable sides: a triangle is a tri-side.

Let me try to construct the design of [Mountain Movement] systematically.
As the subtitle suggests it seems that the triangle and the rectangle must be the two cores of the structure.

Firstly I extended one end of the four sides of the rectangle counter-clockwise to make a wind mill-like shape.

Extend each four side clockwise like a spiral to make a mosaic of five faces.

This is the first group.

Loosening the lines like string, stretch it on grid( string network model).


The second group with the triangle is more complicated.
Extend the three sides counter-clockwise.

Attach another triangle with one corner between the two corners of the first triangle.

Make a rectangle to fill the cavity.

With the same procedure make symmetrical pattern like this.

Move the horizontal side of the rectangle on the left.

String-grid model of this pattern is like this.

now we can see that the whole structure is made of three groups.

The spiral model can be extended; new spiral can be shooting out from each corner clockwise.

The extended spiral can bind the other two groups.

Though I picked two examples there are several more related works with polygons.


Goethe said, "(Painting ) lacks any established, accepted theory as exists in music".
Many syntactic devices in Klee's works suggest that Klee conceived such theory, though he did not have time to write down.
I expect that such total theory may be revealed when the analysis of his work progress.
The music theory largely relys on visible sign system, which means that we must have preceding innate ability to see the shapes of such signs, which I call visual grammar.

Klee's interest in syntax can be seen in his diary.

"When in Italy (in 1901), I learned to understand architectural monuments....Even the dullest will understand the commensurability of parts, to each other and to the whole, corresponds to to the hidden numerical proportions that exist in other artificial and natural organisms. It is clear that these figures are not cold and dead, but full of the breath of life; and the importance of measurements as an aid to study and creation becomes evident."

Klee knew every shape around him was artistic more or less..
He seemingly convinced that art theory formally must be one for all things he sees and that the measurable elements of structure are responsible to give us sense of beauty.
He also wrote, "I cannot be grasped in the here and now, For my dwelling place is as much among the dead as the yet unborn"
He can be grasped now; his works' dwelling place is on the Internet among the other masterpieces.

smallest unit of artwork (Matisse's Pink Nude)

Using the Drawing Tools of Microsoft Word, with the left hand which is not my dominant arm, I copied Matisse's Pink Nude painted in 1935.
We can reproduce almost exact shape on the available grid of PC.
( We need to use finer grid to draw the face.)

Any shape can be reproduced as digital image on small grid.
As most  of the music in the world can be notated with twelve tones, it seems that most shapes drawn by human are made of limited curves, i. e. most artwork is made of limited units.

Cutting coloured papers with scissors, Matisse in his later years, made shapes without relying on manual dexterity.
If Matisse lived now, he would use PC to draw.

As we see in Bar's famous Diagram of modern art movement, the american formalism could not find proper position for Matisse, Klee and Picasso.
The way of criticism that invents isms and connects them to make a tree diagram is not structured on formal level but on semiotic level.
Formal art must not be founded on knowledge but on shape which is the common denominator of every visible thing, which is potentially artwork.

Matisse's abstraction was inductive: the simplification of natural shape.
Matisse noticed the interesting effect of Cezanne's deformed nude.
Cezanne taught Matisse that natural shape can be intentionally distorted.
" I am also correct if Cezanne is correct.", Matisse said.
He painted Dance experimentally.

I picked the dancer on the right side as the most deformed body.
I can imagine the notoriety when it was exhibited for the first time.
We need a formalism to include Matisse's abstraction.

To explain Visual Grammar, so far, I used very simple shapes taken from abstract paintings as example.
now the terms and the grammar are ready for figurative artwork too.
So I start from Matisse.

Pink Nude is well known because the painting process of twenty-two was recorded.
Black-and-white photos of some versions have been introduced in a number of books but now we can see all online.

Matisse painted, using the same model frequently.
According to Bar's book, Matisse made drawings for the painting beforehand.
But I found a pastel which is much similar to the first version of Pink Nude.
So I start comparing it with Pink Nude.

We know this pastel was painted by Matisse because the documents tells so( this is the level of reference).
We also know that this is not a portrait of special woman( level of symbol).
To see a naked woman is a level of icon.
To see the imperfect body is also iconic level and to see the shape as it is is a level of sense.
Pure vision sees the geometric characteristics of shape: anyone sees the same thing.
Picture becomes pure art on this level.

At the stage of the pastel, the image is already non realistic: illustrative.
There are neither actual colours nor tones.
The contour is drawn with black line.
The face is  like cartoon, much simpler than Ukiyoe painting.

In the oil paintings, the sense of reality is further removed.
At the first stage, the picture started with new motifs: flowers in a vase and a pot on the table behind the sofa.
Then the corner of the room disappears and the sofa becomes background.
The table disappears and the vase, the pot and the flowers becomes decorative element of the background.
The body is enlarged and some parts becomes outside of the frame.

Without the parts of the body,the figure becomes new shape.
And as a consequence the number of the intersection of the shape increases; the design became more complicated.
The head rises straight and is surrounded by the lines of the arm; the outline of the body is simplified.
Double lined buttocks become single in the final stage.
Being free from the reality, act of creating becomes playful with geometric elements.

Living in a country side, I have not been able to read all the books about Pink Nude.
I only hope that they will become digital and easily searchable in future.
Anyway in order to discuss the problem of art, words have serious limitation.
Not only that we define the meaning of art terms precisely but also we should devise grammatical models common to all shapes.

Now I try to analyse the network structure of the two pictures mathematically
.

① Comparing the two pictures in the same size, we notice the nude of Pink Nude has become unusually large.
We can hardly call them the same type of figurative art.

② Mark the intersecting points of the line.( The detail can be treated as a secondary design because they often disappear at the process.)

③Thinking that the lines are rubber strings, stretch them on grid.
Then separate the pattern in colour.

④Neglecting concavity, simplify the pattern.

⑤ In order to determine the minimum lattice pattern which represents the network, organize the nude part and the back ground separately.

⑥Colour the nude in pink and the back ground in blue.
In the pastel, the nude network fits into a rectangular frame.　
The body in Pink Nude touches to the frame and forms eight dots.
The pattern becomes complex mosaic.

We can clearly see the difference on grid level.
The lattice indicates how many squares are necessary.
This is the minimum lattice structure of the network  minimum lattice model of network.
The minimum network of Pink Nude is a grid of twenty times twenty and of the Pastel ten times eleven.

Instead of Alfred Bar's tree diagram,the new diagram of shape, in which each artwork has its own position, should be made.
The outline of a body is a loop, so it is possible to classify this type of artworks based on the number of concavity.
Let me try to classify Matisse's human body.
(The figures on the right are referential; archaic idols and modern artworks.)
The number of concavity of the nude in the painting of Renaissance is approximately two times of Pink Nude.

Online I tried to find photos of Mt. Choukai which I enjoy seeing whenever weather is fine.

 It was easy for me to find photos of the similar look though there are dozens of photos taken from different sides.
I traced the outline of the mount.
The top has a cavity which is the crater, the left slope descends without so much dents but the right side has a large lump breaking symmetry.

"Nature does not create art.It is we, and the faculty of interpretation peculiar to the human mind, that see art."
   Man Ray

I see Man Ray's photo the same way.
Features of the curve is more simplified in the outline of nude,
A number of smooth curves are connected like a melody.
Each curve is recognisable by difference in curvature, which makes the outline expressive.
The body curve can be seen easily without the head and legs, which makes us realize that a chain of curves is the cause of enjoyment of seeing.

When we see a shape, the eyes tend to follow along the contour.
Seeing a shape is tactile and seeing colour is optical.
The ends of a line attract our eyes, so think about the eye movement starting from the left end.
The line rises to the right then falls straight bending a little to the left.
We recognize the line's turns: right, left, right, left; the turn is countable.
We also see some parts of the line specifically: not only the two ends, but also each bend.
When a line is closed, we can count the number of concave(the number of convex is the same.)

Henri Matisse in 1908 wrote in 「Notes of a painter」

"The entire arrangement of my picture is expressive: the place occupied by the figures, the empty spaces around them, the proportions, every thing has its share. Composition is the art of arranging in a decorative manner the diverse elements at the painter's command to express his feelings. In a picture every part will be visible and will play its appointed role,whether it be principal or secondary."

It is a great surprise for me to know he had a such insight in 1908.
My plan is to explain his great ideas grammatically little by little.
Now we are ready to see the body shape in the Pastel.

This shape is a network of 18 intersections where 3 lines meet.
Each line can be seen separately from intersection to intersection or intersection to the end..

①　A curve in the same curvature can be recognised as a unit.
the eyes tend to go to the top of a line.

② Looking at the top of the curve , we notice that the right curve is different from the left curve.

③ Each distinguishable curve can be enclosed in a rectangle.

④ A curve keeping the same curvature can be seen as one curved segment.The top and the bottom points and the two sides points attract our view due to the eye muscle structure; our view is rectangular; all writings in the world is framed in a rectangle .

The natural shape  like the outline of a mount is usually very complicated and mathematical curve like parabola has very gradual change of curvature
We do not see these lines as succession of a few divisible segments of curve.
But an artwork is made of simple curves: the shape of plates in Cezanne's  still life is not oval shaped but of divisible curve.

At a glance, the outline of  the pastel has more unit rectangles than Pink Nude.
The shape of picture became simpler after Cezanne;  the outline is divided into smaller number of rectangular units.
This often happened in so-called art history; art history is the history of artistic artefacts.
When simplification of shape happens, artefact have more chance to have higher quality because it is easy to see the structure.

We cannot see the shape of a flying bird.
When we see Moor's sculpture we have to stop walking around.
Art is a static image, not an object.
Digital image online becomes potential artwork for everyone.
We are surrounded by potential artworks ever time every where.
We see an image in every visible thing: the outline of mountain; nude; calligraphy; decorative pattern; figurative and non figurative painting.

Every shape seen as image has lines made of curved line segments as unit.
A series of curved lines makes melodious line, which affects us enjoyment.
We learn at early stage innate visual grammar  by looking at shapes around us.

Arp and Mondrian have common structure

Any visible thing becomes two dimensional design as retinal image.
The lines of every design can be traced with string.
Any network of string has geometrical features.

①There are certain Arp drawings which look like broken networks.

②Imagining that the network is stretchable, fix it on grid.
A string is the smallest element of shape and a  loop is the closed string.
And the smallest network has a point D (3):
D (3) means a dot from which three lines coming out.
On the grid there are only two different types of dot: D (3) and D (4).
Mondrian composition is the typical structure of this type and some Arp belong to this group.

③The model being free proportion, any number of squares and lines can be located anywhere.
So the squares and lines can be put aside in order to check the network.

④Link structure appears,expressing joint parts as different unit.
Every network has its own link structure.
Usually Arp is associated with Surrealism but formally he was the follower of Cubism.
Almost all the modern art originated from Cubism.
These Arp works have the same type of link structure as Mondrian: link structure with D (3) and D(4).
Grammatically both belong to the same type on the basic level.


" Others have seen what is and asked why. I have seen what could be and asked why not."         (Picasso)
Abstract art indeed started from Cubism.
As Picasso was a draft man by nature; everything had to be drawn,
Cubism started as linear composition with clear outline.

                                         (Picasso, Mondrian, Halley, Stella and Tuttle)

Picasso was so much interested in every type of shapes that he could almost foresee the pre-existent grammar.

" I like all painting. I always look at the paintings- good or bad- in barbershops, furniture stores, provincial hotels. I am like a drinker who needs wine. As long as it is wine, it does not matter which wine." ( Picasso)

Ancient Chinese characters passed similar abstract process as Picasso's abstraction.
But Picasso had to rediscover abstract art because the European art world was mainly concerned with painting only.

 In digital art world the viewer can take initiative.
Most of the interesting images appear as digital image nowadays,
so we should call them digital art work.
Definition of formalism is, according to Wikipedia,
"The context for the work, including the reason for its creation, the historical background, and the life of the artist, not considered to be significant."
Without knowledge of the history, the medium, the size, and whereabouts of the original of digital images online, I enjoy seeing them.
Art is not art because the authorities said so but because the the viewers enjoy the image.
Herbert Read defined 'visual art as anything which being seen giving pleasure'.
So I define pleasing digital images online are art.

I found online a group of images which shows the abstract process of the character 'horse'.
I took the liberty of adding the string structure models underneath.

The original of the four on the left are very small incised images on bronze vessels and the right  two are of calligraphy.

We reached the smallest structure which I call 'link structure'.
The grammar is not only for literature but for all writings.
Likewise the visual grammar is for all shapes.
Inventive artists always used intuitively the pre-existing grammar to make variation.

After sufficient visual experiences, we notice that a loop is a connected string and that a string is a broken loop.
Then we acquire ability to connect a string and disconnect a loop, which can be called syntactical ability.
There are Arp works with real string, one of which I picked below.

Real string may be easier to make variation because most of the geometrical characteristics are not fixed.
There are two ways to make a loop from the simplest network.

 To connect 2 D(3), there are several ways.

Now we have four 2 D(3).

Adding one string to the possible locations of each 2 D(3), 3 D(3) can be made.

Likewise 4 D(3) can be made.

Ancient Greek failed to establish aesthetics because they thought that painting and sculpture were the major, pure art and neglected much simpler, purer art like Ancient Greek icon.

Sciences always  started after finding  the smallest elements.
In biology the notion that every living thing was made of cells was important starting point.
With sufficient digital art works, we can easily arrange them from the simple shapes to the more complicated ones according to the countable characteristics, which is the beginning of art science.
Universal and timeless formal grammar is the basis of aesthetics.
Formally there is no difference between representative art and abstract art: every image is two dimensionally geometrical.
There are certain quantifiable geometrical characteristics common to all shapes.
The difference is the quantity of the characteristics.
Representational shape which is due to utilitarian purpose may have larger number in the countable characteristics.
The characteristic which distinguish Arp from Mondrian is concavity: Mondrian shape has no concave.
I will write about concavity next time.