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.

Art-making is logical(Mondrian puzzle)

Anything we see becomes the image on retina, which is two-dimensional shape.
On this level we can build up art theory of everything we see.
Though the retinal image is the whole view which the two eyes see, shape-seeing is selective: we pick up what is interesting, neglecting the surrounding image.
We see images and read articles disregarding the surrounding ads on PC screen.
And when we see a sculpture we do not see the background.

The eyes follow the outline of a mountain range.
The contour of a nude can be the motive of a picture.
The outline of a nose has various expressions: Roman nose; aquiline.
Sports car's sleek silhouette is commonly appreciated.
These examples show that we daily see lines selectively from the whole view.
The fovea can pay attention on a line, moving along it, whilst the retina sees the whole simultaneously.

The smallest element of shapes including art work is a string.
We can trace any shape with string.
When a string is attached to the side of another string, we see three strings coming out from the touching point, dot.
In general a shape is a network of strings; the atom of design is a string and the molecule of design is a branch.
String shape appears as essential element of the images of art and nature.
Shape seeing is seeing strings.

Digital image has given us new habit .
I used to print new found digital images as reproduction collection but soon realised that seeing them on PC screen is more convenient..
Internet became part of visible world to find interesting images.
When all the pictures we want to see are stocked online we do not need picture books; we can pick only images we like.
Then we realise that we do not need the authorities' selection any more, as it has happened in the musical world.
We have entered the digital art age.
Redefining that art is very pleasing shape, the new art theory can be built up based on huge amount of digital images available without historical reference.
The digital images are increasing constantly.
And comparing the images became easier; making the images smaller on the same plane.
Being able to enjoy seeing digital images is the necessary condition to be artist.
The task of artist is to invent the method of making new pleasing shapes.
I have been writing these logs using digital images online solely though I developed my idea on reproduction.

The artists' interest is a line within the frame.
So the model for these images can be like this:

I searched and found easily online the illustration in the book"Tristram Shandy" which I saw more than forty years ago.

The models of each lines can be expressed like this:

The model is stretchable like rubber string and bendable like lead wire.
A string alone can make art: one stroke drawing.
Picasso made some such drawings.

I remember seeing young Klee's drawing of a pig drawn blindfolded though I failed to find it online.
He started from a muzzle, passing the back to the tail then coming back to the starting point.
This method works because the hand moves tracing the imagined pig drawing.

Easy way to remember this image above is to imagine tracing the line from the top to the bottom end.
Instead of tracing it with the index finger, moving the eyes along the line works.
Then closing eyes and imagine to move eyes works as well.
It seems the eyeballs are actually moving under eyelid.
It is not easy to imagine the whole image.
Imagining tracing the line is easy way to conceive the image, which suggests that seeing a line is moving the eyes along it.
Seeing a line is primitive: tactile.

We Japanese can read hundreds of Chinese characters but be able to write less than fifty percent of them.
We remember those as sign but to know how to write, the structures must be remembered.
Obviously seeing art work is not sign seeing but shape seeing.
The reason why so many people fail to enjoy abstract art is that people lose the childhood habit of sense-seeing.
Sign-seeing is seeing the whole image at once and art-seeing is moving eye tracing the lines.

A string can be expressed on grid, gaining proportion and angle.
The simplest  network has a touching point of three strings, which I call D(3), D meaning dot.
When a string is on the grid, only three or four strings can be attached on grid point, dot.
Now it is ready to explain a Mondrian puzzle.

① I chose a Mondrian with simple grid line from Google image.
② Disregarding the proportion, express the design on grid.
③ Make the grid except the frame line with match sticks( 7 by 7 horizontal sticks and 8 by 6    vertical sticks)
④ Make a new pattern by removing a stick. The number of the new patterns are the same as the number of the sticks.
⑤ Remove one again from each of the new patterns.
Repeat the same procedures.
The Mondrian pattern has 28 horizontal sticks and 36 vertical sticks.
So the pattern can be found in the group of ( 49+48) -(28+36) removal.

Let me take smaller structure as example to save the space.

The match structure of these is a branch of D(4)+D(4).

There is a way to make proportional variations automatically.

The smaller the grid is, the closer the design becomes to the original,


String model D(3) below can represent the letters T, Y, E, F.

And a dot can hold infinite strings although only eight on grid.

These string grid model except 7 are used for Chinese characters: 丁；十；大；木；米.

More complicated patterns can be made by attaching more than two string model patterns.


Let me introduce another match stick puzzle.
Make two branch model D(3), then connect them.

The shapes can be transformed each other, turning a branch 90 degrees around the dot.

Chinese character 工、上、下、and alphabet H are made, which suggests they have the same basis, the grammar.
The turned images and the mirrored images are considered to be the same as the rule.

Think of adding a new string on the model of H.
There are three possible points to add it.

Turning a branch around the joint 90 degrees, the shapes can be transformed.

All turned out to be the same.

To make 4 D(3) there are 6 possible points to attach a string.

These three become the same branch pattern.

There are many possible patterns  made like these:

" For Baumgarten aesthetics is the science of sense experience, a younger sister of logic, and most perfect kind of knowledge that sense experience can have."(Aesthetics from Wikipedia)

To make variation is intuitively logical process, which can be further progressed to logic.
Creativity is the ability to use a kind of logical process.

art and math have the same basis: visual grammar

Art and math have the same basis.
I do not mean that art has mathematical basis as many authors have written.
My claim is that there is the same visual-grammatical basis for all the mental activities, including math and art.
The proto visual grammar is formed by experiences which enter through the visual sense organ.
To see a picture we customarily see it on perceptual level.
When we enjoy it formally however, we forget time passing and the work being a thing in space: shape seeing is not practical.
Art seeing (shape seeing) is not perceptual seeing but sensual seeing.
Any association from seeing is the secondary activity of perception.

There is a frequently quoted topic in Hermann Weyle's book "Symmetry".
When the mathematician proved that there are only seventeen ways to cover a plane with tiles artists already had used all the possible way.
Since the time of Ancient Egypt until modern time, the patterns had been used.
This suggests that the ability to make patterns is universal.
Making the pattern solved the mathematical problem, which means that applied art can solve applied mathematical problem.
Everyone can see shapes, which  teaches him  apriori self organized system of shape.

Seeing a shape is to see the quantifiable geometrical characteristics of the shape, which have number concept.
Most noticeable characteristic is countability of shape.
String, loop, dot, cavity and coil are countable elements:

Then can the proportion represent number?

Effect of the proportion in art work has been popular; especially of golden section.
I put it aside because the length of a line segment cannot be exactly measured by sight.
It is as hard as weighing a thing by holding it or hearing absolute pitch.
What the example shows is the orderly size difference of the shapes.

The proportion can be sensed exactly in this way by the help of countable squares.
The proportional element appears at the last stage of art style; there are many art works ignoring the exact proportion.

Visual sense is the most important sense organ because it can count the characteristics of shape and  grouped shapes.

Babylonia, Maya, Ancient China, and  Ancient Egypt had similar numerical system: grouped shapes represent natural number.
The notations of the small number like 1, 2, 3 are similar in all of these ancient civilizations.
The inconvenience to write large numbers made invention of sign; each civilization has different sign.
It supports the idea that the concept of number did not come first but sensing shape made the concept of number.
The representation of number concept inevitably became same everywhere.
The same shapes of the ancient numerical system is the archaeological remnant of what ancient people had before language, which suggests that the universal grammar is not language but shape.

Before learning any sign system every child self-teaches the structural relation of shape by visual experience.
Though every child masters the basic visual grammar it is usually replaced for sign system by education.
A few grown-ups do not forget the great pleasure of seeing shape to become artist or mathematician.
Clive Bell writes,
"I wonder sometimes, whether the appreciators of art and mathematical solutions are not even more closely allied. Before we feel aesthetic emotion for a combination of forms, do we not perceive intellectually the rightness and necessity of the combination?"

There is a story that Socrates could instruct an ignorant slave boy to solve a geometrical problem.
The boy had innate ability; how old could he be?
Around ten year old since he could serve as a house servant?
There is an IQ test with shapes for the lower grader of elementary school, which means that  children of younger than ten must have the ability to see the shapes in the test.
There is some evidence to lower the age further: dozens of You Tube movie about child pianists.
It seems that three year old kids are ready to learn playing the piano, which means they can understand the musical structure.

                                                         (three year old Hiyori plays)
A child player must be curious about hearing musical structure.
Then where the curiosity and ability come from?
There must be a preceding ability: the visual sense.
All the children are curious about shapes.
My guess is visual sense has innate ability of recognizing structural system of shapes.
This structural understanding becomes the model of all the mental activity: all the other abilities are moulded on this formal system.

Shape seeing is the innate ability every child can do.
But analysing three year old child's drawing cannot show us how the child's visual sense ability is; to play the piano a child does not have to create sounds since the piano prepares all the sounds;  it is like a child who can sense a circle can draw a circle using  PC.

This visual ability remind me of Chomsky's universal grammar.
"....the theory suggests that some rules of grammar are hard-wired into the brain, and manifest without being taught.
Chomsky has speculated that UG might be extremely simple and abstract, for example only a mechanism for  for combining symbols in particular way, he calls Merge. ....the human brain contains a limited set of rules for organizing languages....all languages has a common structural basis. This set of rules is known as universal grammar."(Wikipedia)

My assumption is that the set of rules can be self-taught by the experiences of seeing shapes.

I am more familiar with Piaget, so let me quote from Wikipedia about Piaget's "The Child Conception of Space".
"Children at the earliest stage, when asked to reproduce certain geometric figures, preserved only the topological features; that is, they were accurate in terms of continuity/discontinuity, enclosure/ exclusion, inside/ outside, nearness/ farness."

I use this idea as the principle to explain visual grammar.
The topological features of shapes can construct number concept and logical system: proto visual grammar appears at this stage.

there is shape relation to show us number concept: the nearness of shapes makes a group and isolate from other groups.

The grouped shapes show the number concept and such group can be arranged in order: the lower grader at elementary school can learn counting with the picture of apples.


There is another obvious relation of shapes which can teach us basic logic.
When I say we have innate logical intelligence, I mean there is logical structure of shape and our sense experience can notice it.
Kids can draw a face with two eyes, which means they understand the number 1,2 and inside- outside relation.
I call such faculty of counting and of logic proto visual grammar, which can be developed to visual grammar.

Let me show a simple logical example: the possible position of two loops( loop shown as square).
When there are two loops one is either inside or outside of the other.

To add one to the left pattern there are three possible positions: outside of the large square; between the two squares; inside the small square.

To add one to the right pattern there are two ways: outside two squares; inside the either one of the squares.

(the same pattern is shown in blue)

There are good examples in Arp's works which show his grammatical awareness.
I listed up some examples.

Arp writes,


"....I wanted to create new appearances, to extract new forms from man.
We do not wish to copy nature. 
We do not want to reproduce, we want produce.
We want produce as a plant produce a fruit and does not itself reproduce.
We want to produce directly and without meditation.
As there is not the least trace of abstraction in this art, we will call it concrete art."

His genius picked patterns from pre-existing list of shapes.