Klee, Vasarely, Stella made the same type of polygonal mosaics (Geometrical classification of abstract art )

                                                   Klee [Village on the rocks], 1932
                                                   Vasarely [Quami], 1950
                                                   Stella [Irregular polygon], 1965
All the artworks can be classified into one system according to the geometrc structures of the work.
The upper three works are polygonal mosaics.

                                                 three works in the Irregular polygonseries
I introduce a new model called proto-shape: a polygon made of definite number of line segment with free angle and free size.

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Both look like overlapped polygons. ( This is a seeing of iconic level.)
But on perceptional level, the two shapes can be seen as two  polygons to share sides.
The left is made of a triangle and a six- sided polygon and
 the right is made of a triangle and a eight- sided polygon.
A concave polygon  can easily be identified by paying attention to the concave.

There is only one proto-shape for any polygon without concave but several proto-shapes for concave polygons .

  The left shape can be smaller.

These are the same as proto-type.


 Concave polygons can be classified.
A triangle does not have enough sides to make a concave.
A quadrilateral can make only one concave.
Polygons with more sides can be made by paying attention to the concaves and the convexes.

        polygonal table




I used to buy many art magazines and Auctioner's catalogues in Strand for this type of Vasarely's works.
Now I find more digital works than my collection .
Without extending the polygonal table I made the nine- sided polygon by cutting off two concaves on a five- sided polygon.

The other polygons can be found in the table.

                                               polygonal distribution diagram
                                               mechanical compositional method
Vasarely seems to have used some algorithmic method.
The below illustration shows such imginary process.
1)  Draw  five six- sided polygons surrounding a five- sided polygon.
2)  Add concaves on the polygons.
3)  Move some shape parallel.


Stella and Vazarely used polygons as motif.
The polygon distribution diagram below shows that Klee used the same method.

                        klee [Village on the rocks] polygon distributio diagram
There are several many-sided polygons: three eight-sided polygons(vermillion) and three seven-sided polygons(brown).
None of them have remarkable feature as main role.
They look like the back ground of the surrounding convex polygons .
So I draw auxiliary lines to show the likely hidden parts of the polygons.

                        klee [Village on the rock]  complimentary line model
In this way I can eliminate all the concave polygons.
The convex polygons which are made with cardboard can be overlapped to make the design.
There are one hexagon and the rest are pentagons, quadrilaterals and triangles.
This basic design was constructed firstly
 and the secondary surface structure was  made by extending and overlapping the surrounding polygons.
The structural model below confirms my asumption.

                         klee [Village on the rock]  structural model of intersecting polygons
The  overlapping shapes make three dimensional illusion.
For example the two polygonal corners  are lined on a vertical line near the center, which look like steeples
and  the parts on the lower right look like eaves structure.


Finally I make two models for harmony and balance: scale model and line extension model.

                        klee  [Village on the rock]  scale model

                              

As a line segment has not only the size but the angle, I extend all the line segments

                       Klee  [Village on the rocks] extension model
The extended lines  intersect at many points, which makes the structure balanced.


The diagram of modern art movement by Alfred Barr shows that all the movements until 1935 converged to two types: non-geometric abstract art and geometric abstract art.
Most followers took this diagram as a model to fablicate more movements for extending the network of the diagram.
The importance in this diagram is that the final stage is formal: artwork became geometrically analyzable.


Without any analysis we can recognize that these three works belong to the same genre.
It seems that they were made without any direct influence though the makers had the notion of making abstract art.
The world of internet overflows with art-images which themselves are new medium of art.
But geometrical patterns can easily be classified on the level of seeing shape.
The same type of works  can be collected to study the difference of syntax.
Creating new artwork means inventing new syntactic device.

Klee's musical serialism in linear composition

Many artists who claimed that they were influenced by Klee's work understood only symbolical aspect of his painting.
Some modern composers like Boulez and Stockhausen showed interest in Klee's formal technique.
Klee did not express musical feeing but composed structures like a composer.
Though there are many writings about "Klee and music", words cannot explain the syntax of art.
I am going to use visual grammar for showing Klee's syntactic devices.

                                                          Cezanne/Picasso/Klee
Obvious stylish progress can be seen in these three paintings.
In Cezanne's landscape with buildings the linear structure is dominant.
Picasso neglected perspective and natural color, and the space looks like a structure made of cardboard.
Klee made a mosaic of quadrilaterals; all the geometric elements are quantifiable.
I made a network model.
The model neglects the size and angles of the line segments.

                     Klee CastleHill  line network diagram
These illustrations are the equivalent at the network lebel.
Naming the intersection as "dot", each point is expressed as D (N) [N=3, 4, 5: N is the number of line segments].
There are six D (5). ten D (4) and twenty D (3).
Adding the elements of size and angle, the design is made.


Any artwork is classifiable at this level.


A composition with straight line segments needs a new model to explain the syntactic device.
Any line segment can be considered to be a radius with a pair of two circles which centers are the two ends of the line segment.

                                  Klee  Castle Hill  linear scale model
The linear scale model suggests  that the balance is due to three devices;
1) The extension of some line segments pass through the dots.
2) Some circles form pivot structure.
3) Some dots are alined on the horizontal lines and vertical lines of the frame work.

                                  Klee  Castle Hill  linear scale diagram
Placing one end of each line segment on the center of concentric diagram, we can have horizontal line segments to compare.
The diagram also shows that
1)  there are twenty different size.
2)  the number of the line segments of the same size.

                                  Klee  Castle Hill  Colored line model
In this painting the size of the line segments are important.
The colored line model confirmes that the effect is based on the schematic design.
There are several regular patterns like:
1) a chain of three same sized line segments: the shortest pink ones and vermilion ones.
2) a chain of two same sized line segments: lightest pink, vermilion, yellow, yellow ocher.
3) three light blue ones are attached at a point.
They are fragment of decorative pattern.
Decorative art is not different from painting; the difference is a degree of regularity.


This model can be constructed with colored straw: cutting straws in size and connecting them with strings.
In such a model the angles and the positions are changeable but harmony remains the same.




Now we know that Klee used serial technique in 1929.
This is almost the same time when Schoenberg invented twelve tone music.
It does not mean that there were any relation between Klee and Shoenberg.
Both music and art are based on innate universal grammar.
Plato said " I would teach children music, physics, and philosophy; but most importantly music, for the patterns in music and all the arts are the keys to learning."
My assumption is that the ability for sight precedes ability for hearing.
To understand any design we must have pattern-recognition ability beforehand.
The degree of sensing outside phenomenon is not analogue but digital.
Artist's expression  therefore becomes digital.
Digitaly expressed sounds and shapes form a scale.




After knowing that artwork is made according to the grammar, it is obvious that we cannot establish art theory without examining  the syntactic level of artwork.
The critics in the last century are like people arguing about classical music without reading the musical scores.

The same universal syntactic technique that was used for Ukiyo-e and Modern art

                                        Modigliani  [Caryatid]  pivot-line model 
Art making is not  sketching an object  but  designing a shape.
When he draws a curved line he is conscious that the line is made of several line segments each of which has different curvature.
Each curve becomes an arc.
The arc can be extended to a circle.
Using the circles he can adjust the balance.
Some artists were aware that the intersection of the circles can be fixed for balanced structure.
When I made the model of Modigliani's [Caryatid] I noticed a straight line going through more than three intersections(pivots).


This time I am going to check all the models I have made.

                               Modigliani  [Caryatid]  pivot-line model
Modigliani is generally known as a portrait painter.
But some people consider that [Caryatid] series is the most important.
The model supports such taste judgment.

                              Matisse  [Dance Ⅱ]  pivot-line model
We enjoy played music.
The effect is due to the structure which is expressed on the musical note.
Matisse painted [DanceⅠ] spontaenously.
Then he constructed [DanceⅡ] with unit arcs.
He chose a few circles to make a scale.
Matisse used the device to fasten the circles: to make the whole structure he intersected the circles on several spots, most of which are lined.

               Picasso  [Woman with a flower]  pivot-line model
Searching Picasso image online we find many imitation Picassos.
The reason why we can easily find out real Picasso is
the quality difference.
And the quality judgment can be supported by the model.

                             Klee  [Support for a shield]  pivot-line model
This masterpiece is not well known.
The reason may be because there is not so much to say in this picture of string on sign level.
The model reveals how the harmony is generated.
Now we know all the greatest modern artists used the same syntactic device.

                   Sharaku(Hokusai)  [Yamashita Kinsaku]  pivot-line model
Many people believe that Sharaku was Noh-actor and only early half body portraits are significant.
It seems that they see only facial expression and fail to see the whole body.
I consider that Sharaku style is the result of Hokusai's interest in early ukiyo-e.
The quality is equal to the best early ukiyo-e painters like Totorii Kiyonobu.

            Kaigetsudo Ando  [Tatibizinzu]  pivot-line model
I have chosen Kaigetsudo because his style is most distinct among early ukiyo-e painters.
Wikipedia says about Kaigetsudo, "it is probable that many of the works attributed to him were painted by his disciples".
The taste judgment can tell us which are by Ando himself and the pivot-line model can confirm the judgment.
The study of Ukiyo-e painting can be started with structural analysis of  works.
The pivot-line model shows us how artwoks are balanced and harmonised.

Cezanne's intuitive syntax and Modigliani's intentional syntax

                                  Hokusai [Ryakuga haya-osie]
                                  Cezanne [Six Bathers]/ Modigliani [Caryatid]
For these works, the same type of harmonizing syntax was used.
Painting, ornament, sculpture and archtecture are based on the same syntax.
As we can see in [Hokusai Manga], Hokusai knew that painting and ornament are designed in the same way.
Renaissance artists were experts of all the types of art.
There is no such thing as pure art but artwork is made for some practical purpose.
Art survived even after the practical purpose outdated or forggoten because the shape remained like a shell.
We get artistic sensation by seeing visibly perceivable geometric characteristics.
We perceive them quantitatively.
Artist is an inventor of new syntax.
Most people are accostomed to see shapes on iconic level.
In figurative painting the outline is not noticed as geometric because the line is a sequence of arcs.
Unlike painting , ornament has easily recognizable symmetric structure of circles.
A good listener can only perceive the effect of the musical note but the composer  can　use the musical syntax.
A good viewer can only perceive harmonious effect  of shape but the artist can use the syntax of shape.
Klee wrote in his diary,
"I learned to understand archtectual monuments....the obvious commensurability of parts, to each other and to the whole, corresponds to the hidden numerical proportions that exist in other artificial and natural organizms.....the importance of measurement....becomes evident."

                Cezanne [Six Bathers] lithograph
This drawing above seems to have been drawn after the lithograph below.
Althogh the legs are missing the foot is in proper position.
The curvature of the lines is very clear.
Arcs are the smallest elements.
The sequence of arcs becomes a melodious line.
The closed or almost closed line is a motif.
Ornament has pivot structure for balance and harmony.
At least Cezanne knew that deliberately drawn line becomes a curve with single curvature though he was not aware that the composition was forming  pivot structure.


The litograph can be reconstructed with five circles as he had excellent sense of harmony.
The modern artists could easily recognize Cezanne's scale method.

                Cezanne [Six  bathers] arc model
The circle-arc model I made before was to show that the largest and the second largest circles make the basic composition.
This time I add the smaller circles to show that each motif is well balanced.

                Cezanne [Six Bathers] pivot model
This model shows that same colored arcs are grouped at specific parts, which suggests that he paid very much  attention on the arcs of the same circle.
To make the point clear ,I enlarged two parts.

               Cezanne left Bather
The three pink circles overlapping the body are the main structures with pivots(intersection of more than three circles).
The highest circle which is the extension of the bottom thigh has six pivots.
The middle circle which is the extension of backbone has four pivots.
The lowest circle which is the extension of the front part of lower limb has also four pivots.
When a designer trys to balance the closely grouped arcs, the extended circles of the arcs tend to form pivot structure like ornamental design.

                            Cezanne right bather in [Six bathers]
The largest circles (green) which make the arm parts is the basic.
The second largest circles (pink) are not used as a whole circle but covering partially from both sides.
Many of the smaller circles intersect at the pivots.


The holes of ancient flute are made to sound scaled tones.
A musician of flute makes scaled music.
The document says that Cezanne was very fond of visiting The Loubre and enjoyed to draw marble statues.
Cezanne's sense of harmony further developed as he painted  the same motif  for sevral months.
Cezanne discovered the harmonizing device unintentionally.
Modigliani learned Cezanne's technique of harmony.
It is almost unbelievable that Cezanne used pivot structure consciously but surely Modigliani used it
deliberately.

                              Modigliani [Caryatid] pivot model
There are  three distinctive syntactic devices in the model.
1) It is made of six circle scale.
2) The second largest circle(green) is the base of the pivot structure.
3) There are eleven lines with more than three pivots and six of them intersect at a point(pivots on radial lines).


As nobody studied  artwok syntactically, the development of Modern art stood still.
Universal Art is a masterpiece based on unique syntax.

               


          

Picasso and Matisse belong to the same school called Cezannism

It is commonly said that  Abstract Art is the final stage of  Modern Art.
But some artists like Kandinsky and Kupka are not directly connected to Cezanne.
Picasso and Matisse are direct heir of Cezanne though they are not abstract artists.
Picasso said,
 " There is no abstract art.
You must always start with something.
Afterward you can remove all traces of reality."
The classification by isms is superficial ; it is based on sign level and fomal art is based on sense level.
The re-classification by the structure of artwork is necessary.
I show the common element of Picasso and Matisse,
.  
        Cezanne  Five Bathers / Matisse  Pink Nude / Picasso  Woman with Flower


Until the last time I showed that the pictures on the left and the center are made of arcs and that the circles of the arcs are structured with pivots .
    Cezanne  Bathers(litho)  pink and green circle-arc model
This lithograph is composed with lines only.
And Cezanne must have thought the composition satisfactory, as it was signed.
Therefore we can expect the work represents his syntactic idea .


Matiss spent several months to finish this painting and he left 22 photos of each session.


In the thirties Picasso made many  harmonious paintings.
I chose [Woman with Flower] to make the circle-arc model and the scale diagram.

Matisse  Pink Nude  circle-arc model / concentric scale diagram



    Picasso  [Woman with Flower] circle-arc model   scale diagram


The largest yellow green circle is dominant in [Pink Nude].
The third largest light blue circle is dominant in [Woman with Flower].
I used lighter blue for the two circles with many pivots; the right circle with five pivots and left circle with four pivots.
The concentric circle scale diagrams show that [Pink Nude] is more harmonious than [Woman with Flower] because of two reasons.
1) the number of circles in the scale in [Pink Nude] is five and the number of the circles in the scale in [Woman with Flower] is eleven.
2) the proportion of the circles in the scale of [pink Nude] is very even and the proportion in [Woman with Flower] is uneven.


Unlike a book a blog should be readable without refering the previous blogs so much.
So I show some pivot structure models from the past writings.

        Sharaku(Hokusai)  [Otani Hirotugu]

    Sori(Hokusai)  [Namiura first version]

    Kaigetudo 

The same pivot structure was used by some Ukiyo-e painters.


The present problems in art criticism are due to the careless use of the term 'art' for many different meanings.
'Art' should be used for one meaing.
In my writing it means formal art which is universal (regardless of when and where).
Some time before I started conceiving three problems  to think over.
1) What is modern art? (hundred year problem)
2) Who is Sharaku? (two hundred year problem)
3) What is fine art? (ten thousand year problem)


For long time I had the notion that abstract art is made of lines and curves,
When I wrote about Matisse's [Pink Nude], I realised that I see the contour of any shape as a chain of curves.
Abstract art and figurative art became the same type.
Any consciously drawn curve has single curvature which is an arc.
After defining that the smallest element of art is arc, I could solve the second problem.
Establishing the grammar  and finding syntactic devices like pivot structure, 
the other two problems may be solved .
The examples of syntactically similar works form a group of jigsaw pieces like an island and many islands may be connected to become one.
I further expect to find the pivot structure in other styles of art like ancient Chinese Jade design, early mediaval miniature, early renaissance painting and so on, which makes unification of art.


Universal artwork is the shape of the image projected on retina.
Digital image online is ideal artwork.
As any shape is geometrically analysable,
All artworks should be classified into one system.

Harmonious scale in Matisse's Pink Nude

Any painting has a structure on formal level.
Everyone knows that painting is not always artwork.
Only masterpieces deserve to be called pure art which can tell us what art is.
As the taste judgement can find masterpieces immediately,
there is no difficulty to find interesting works among ever increasing digital images.
I expect that making models can help to think why great artworks are so distinctively attractive.
The principle of visual grammar is that redrawing the same shape makes the lines more smooth, almost  chains of arcs.
Here is a good example of such case.
Matisse spent almost six months to paint Pink Nude and twenty-two photos were taken for each sesession.

                                         Matisse [Pink Nude] and the photo of the first version


Firstly I make the circle-arc model of the final painting.

                   Matisse  Pink Nude  circle-arc model
There are seven pivots (intersection of more than three arcs), and
five circles are enough to make the whole structure.


Now I arrange the circles concentrically.

                                   concentric diagram of scale for Matisse's Pink Nude
The diagram shows that the circles  have very regular intervals.
The regular scale creates  sense of harmony.


I use these cicles to make circle-arc model of the first version of the painting.

                   Matisse Pink Nude  the photo of the first version
Surprisingly all the clearly drawn lines of the nude can be replaced with the circles in the scale.
It means that the five circles already had become the main motifs at the first stage.


I made the circle-arc models of all the versions to see the change of scale.


    Matisse Pink Nude arc models


I used only thirteen circles to make all the models.
The largest yellow green circle became dominant at the last stage.
This way of using the largest circles for unity is the same technique Hokusai used for [To-Megane].
It can be said that the twenty-one versions were painted to search the right circles for the unclear parts of the nude in the first painting.




I have made more than a dozen of circle-arc model for personally interesting artworks.
So far thirteen circles were enough to make all the models.
This number suggests that our ability to distinguish the difference in curvature of arcs.
It is not coincidence that musical scale has twelve tones.
Our sense is not analogue but digital.
Our sense organ is so primitive that twelve or thirteen scale is enough to express our perception.
Visual perception develops firstly and then we start thinking logically on this level.
The other abilities are constructed based on this structure.
This is the reason why every cultured world valued art.
It seems that art is not considered important anymore as education.
Such educational system fails to produce potential geniuses.

Matisse's DanceⅡhas ingenious syntactic structure for harmony [Rethinking modern art] Ⅱ

The second drawing  after the first tends to have very clear curvature.
We can pay attention on one curvature of one part at one time.
Therefore the curve becomes a chain of arcs when we draw it.
When we try to valance many arcs we are conscious of the extention of the both sides of the curve: circle.
As a result the circles almost start forming a pivot structure(with more than three circles  intersecting at a point).
Many genius noticed that this structure can be constructed to make balanced composition.
The good example of this process can be seen in Matisse's Dance Ⅱafter DanceⅠ.
The left  illustration is the first version.
Only the nude in the left is drawn with clear line.
It is obvious that Matisse intended to draw all the nudes in the same style in danceⅡ.
The right picture is made of round contours and looks more rhythmical and harmonious.
The outlines can be easily replaced with arcs of a few circles
.I used only six circles which function like a musical scale.              Matisse     DanceⅡarc model
Extending the arcs, the circle-arc model can be made.


         Matisse  DanceⅡ  circle-arc model
To see better the pink circle network stretching the entire surface, I make pink circle-arc model.



    Matisse  DanceⅡ  pjnk circle-arc model
The crosses show the intersecting points of more than three circles which are named pivot.
The pivots are systematically arranged.
There are two pivots on the green vertical line.
There are four pivots on the upper horizontal line and three pivots on the lower horizontal line.
There are two perfect circles; left one with four pivots and right one with three pivots.
There are six pivots on the light pink circle on the lower right.



The most dominant circles are shown in light blue.

    Matisse  DanceⅡ  blue circle-arc model
The largest pink circles make the base network, then  the network of the second largest blue circles overlap.
As the most of the arcs are the parts of the networks, the whole effect becomes  harmonious like the musical tones on the musical note.


Formal art history can be written as the history of inventions of such devices.
I wonder how many syntactic devices have been invented until now.
The present art world does not have the good atmosphere to produce artists.
I believe that the digital images are the medium of art in our time.
The painters for the gallery system are already outdated like chess masters.

Soon  the grammatical knowledge about art will be established online.
Then AI artist can be created.
As David Cope's Emmy composed almost-Bach, it will be possible for AI artist to paint almost-Picasso.
Besides brand new art styles may be invented as the AI chess master found many unexpected new moves.