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art is very pleasing shape, the new art theory can be built up based on huge amount of digital images available without historical reference.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.

every shape is made of string

"In a black and white drawing the spaces are all white and all are bounded by black lines; in most oil paintings the spaces are multi-coloured and so are the boundaries; you cannot imagine a boundary line without any content, or a content without a boundary lines....therefore,when I speak of significant form, I mean a combination of lines and colours( counting white and black as colours)"
   (Bell "Art")
Seeing a line as a boundary line of content is perceptive seeing.
Seeing a line itself without content is sensitive seeing.
We can see an object and its two dimensional reproduction in the same way on sense level: the retinal images of both are two dimensional, which is why we can establish unified art theory for every thing we see.

 When we say that these shapes are same, it means we move the eye on both shapes in the same way.
Imagine the first one is made of string and the second of  a piece of wood.
We still can say that we see these things in the same way: we move the eye along the outline.
Seeing a shape is tactile: the finger can sense a shape along the contour.

Let me explain the eye movement using the model below: the line becoming wider and finally we see only one edge.

We can see a line because it has thickness, which means the line has two edges.
When the line is narrow both edges are seen at once which is the same as we see one edge.
When the line becomes wider we see the two edges, which is why we see the area.

Everything we see becomes a design network on the retina.
Every network is made of string.
The simplest network is a string and a closed string is a loop.
It can be said that a string is an element.
As atoms make a molecule and may become compound, string may compose more complicated patterns.

We can also start with a network as an element.
In this case a string or a loop is a broken network.
A network has countable strings and dots( meeting points of strings).
When two strings meet at a point, the result is a string.
Only when more than three strings meet at a point a dot appears
A dot is expressed as D(N), N=3,4,5....: dot with N strings.
As every design is made of string and dot, the eye counts the number of the strings and the dots when we see it.The model can be drawn on grid having the characteristics of proportion and angle.
For example D(0) type has only two models: 

Let me show some examples; there are many D(0) with two loops inside in Arp's paintings.

All the shapes belong to the same D(0) type with two loops inside:

Some of them look like creatures with two eyes, which is perception level of seeing.
Piaget wrote that young children acquire topological level of seeing at very early stage.
.I consider this is the beginning of sensitive seeing.
Associating two eyes which  consists of sensing and referring comes later.

The model works to classify alphabet:

There are ten types of models in this group.

D(0)

D(3)

2 D(3)

 D(4)

Not only the alphabet but any design has a string model.
For example using the two models above this new model can be made.

Imagine this is stretchable string then it is easy to make Arp's work bellow.

Many modern paintings originated from Cubism (including Constructivist painting, Dada ism painting) have distinctive types of string model, which suggests they belong to the same group.

"I wonder.... whether the appreciators of art and of mathematical solutions are not even more closely allied.... I have been inquiring why certain combinations of forms move us; I should not have travelled by other road had I enquired, instead, why certain combinations are perceived to be right and necessary, and why our perception of their rightness and necessity is moving."
     ( Bell)
Bell is showing which  direction I should go.