Proportion in art is integer ratio, not Golden Section / Vasalery [Pebble Series]

Since the early days of Greek philosophy men have tried to find in art a geometrical law, if art (which we identify with beauty) is harmony, and harmony is the due observance of proportions, it seems reasonable to assume that these proportions are fixed,
The geometrical proportion known as GoldenSection has for centuries been regarded as such a key to the mystery of art, and so universal  is its application, not only in art but also in nature, that it has at times been treated with religious venaration.
  Herbert Read   (The meaning of art)

The problem is that even if the golden section is used for a hidden structure, the effect is indistinguishable from integer ratio.

I have shown in my blog [Mondrian did not use golden ratio but integer ratio for proportion] that even Mondrian who is most associated with Golden Section used simple grid for composition.

The geometrical laws of art must be in every work.
All artworks have a simple natural number relationship.
Art can be explained scientifically if the smallest unit is a geometrical element that can be easily counted visually.

Flute is an instrument that can be found anywhere in the world, and we can listen to each sound separately.

The musical scale is naturally devised, unified by the equal temperament, and finally theorized by the pure temperament.

The curved lines in a painting can be seen as a series of several arcs.
The arcs can be extended to circles.
Since a limited number of circles are used for one work, it can be called a scale.

This time I pick up a work, in which the scale is consciously used.

“Pebble Series” by Vasalery is a series of masterpieces as pure abstract art.
Dozens of them now can be found if we bother to search online,

                       Vazalery [Pebble series]
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                       Vasalery [Pebble series]circle-cross model

 

    Vasalery [Pebble beries] concentric circle model

 

                                                   Vasalery [Pebble series 2]
Though the design is asymetric the sense of balance is extremly good.
The trick must be due to the hidden structure.

 

 

                                                   Vasalery [Pebble series 2] circle-cross model

 

                                         Vasalery [Pebble series 2] concentric circle model

 

 

 

Looking at these concentric circle models, we can see that the painter had decided the size of the circles in advance.

A loop shape is counted as one.
If two or more shapes overlap, it becomes a network, and strings and dots are counted.
The number of concave and convex in a loops is the same and countable.

By quantifying the quality of the shape in this way, all the works can be classified with distinct differences in shape.

The complexity of the structural devices seems to explain why some shapes are very interesting.

Though there are countless artworks,
a sensor called taste judgement distinguishes only a small number of masterpieces.
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The geometrical structure in Matisse's 'The Dance' in Barnes Foundation / Ornamental paintings and pictures are made using the same technique

We can easily find a masterpiece by looking at Matisse's art book,
The time taken for the aesthetic judgment of a picture is almost instantaneous.

Though the number of artworks becomes almost infinite when we recognize that digital images are also works,
the masterpieces can be easily taken out of these by the taste judgment.

The picture below is “The Dance” by Matisse from the Barns Collection, which is a decorative painting.

 

As we know that this was made as mural, I can make a circular cross model with more confidence.

 

     Matisse [Dance] Barnes Foundation / circle-cross model
The blue circle that forms the arch is used as the main harmonic circle.
Seurat used the same technique for [Cancan].
(The color has been changed from blue-purple to blue-green for explanation sake.)

When the circle  (blue purple) that makes the back of the left nude is shifted horizontally to the right, it overlaps the circle that makes the middle nude shin.
The left nude waist and right front lower leg units overlap on the vertical line.
The second largest red circle unit (bright pink) also contribute to balance.

 
The upper left thigh of the nude on the left, the back of the left foot of the middle nude and the lower part of the right thigh are lined up on the horizon.


Furthermore, the leftmost unit and the unit that creates the right calf are lined up on the vertical line.
Two third largest units (orange) are placed on the horizon

As I know better about Matisse's model making now, try to make a model for mural [Dance 2].

    Matisse [Dance 2]  / circle-cross model

It is no coincidence that I have taken up [Dance 2]and [Pink Nude] for analysis.

The masterpiece is something I want to have a copy at hand to enjoy seeing repeatedly.
Consequently I started thinking  why it looks so interesting. 
It is scientific practice to create and explain structural models.

The structure that protrudes from the picture creates a balance effect in Matisse's [Pink Nude]

The document tells about [Pink Nude],
“After he finished a painting, the assistant took a picture then washed clean the canvas for the next session .
the same process was repeated 22 times.”

I had thought that this is why Matisse’s [ pink nude] has finished look with a simple manner, as if the lines were traced on a sketch drawn in advance.

But the good balance can not be acquired on blank canvas with no traces even after 22 trials because the balance is largely depending on position

I use the same illust from the second blog about [pink nude].

                                         Matisse [pink nude] and the first version
The black and white photo is the first version.
What is common to the two pictures is that the body parts protrude outside the canvas.
Besides many curves are the same arcs.
Here is  the circle-arc pivot model I made for the same blog.

 

                                            Matisse [pimk nude ] circle-pivot model
The model shows that
A) the main body can be made of five different sized circles.
B) the green circles are fastened at several points (pivots).

When I wrote about [Dance] ２
I found Matisse reached more advanced stage of syntax which I named circle-cross model level.
Forgetting about the document I guess  how to achieve this balancing effect.
To use the unit (a set of circle and the vertical ,horizontal diameter) device, the picture screen must be extended.

    Matisse [Pink Nude] circle-cross model extended outside the frame

It is obvious that thegreen circles are most important structualy.

    Matisse [Pink nude]  same circle-arc framework model

The model shows that the seventeen parts of the smooth body are made of the arcs of the same circle.

The same circle-arc framework technique can be seen in Hokusai’s [Tomegane] and Seurat’s [Cancan].

Masterpieces are no drawn but designed.

The universal technique rediscovered by Modern Artist was not understood grammatically,
which is the reason for the stagnation of the current art world.

The beauty of nude is geometrical 2 / Raphael

 Leonardo, Michelangelo and Raphael are generally considered as the three greatest Renaissance painters.
The former two are famous for their interesting episodes,
but Raphael is most important as a formal artist.
German art historian Walter Friedländer says,
"With the death of Raphael, the classical renaissance began to decline".(wikipedia)


  I make a model of the central figure of Raphael's completed drawing, [Three Graces]: more important work than [Mona Lisa]

                                     Raphael [Three Graces]

   Raphael [Tree Graces] circle-cross model

 The outline of the body is made up of five different circles.

The biggest purple unit that forms the lower part of the left thigh is the most important structurally.
On the horizontal diameter,
  A) another purple unit is attached ontop.
  B) two taching yellow-green units are placed .
There is another yellow-green unit on the vertical line of the left unit.

Four pink units are connected vertically and horizontally.

More than three intersecting points (pivots) are linned up.

I overlay the four circles of the scale on the arc model so that they pass through the top of the head.

  Raphel  [Three Graces]  body-proportioned circle model

I have made two models for the previous blog.

Granach [Venus]  body-proportioned circle model

   Seurat [The Model]  body-proportioned circle model

I made the three models together to compare.

   three body-proportioned circle models

Placing three models together, we can see the difference clearly.

Theｙ all use small Scale (type of circle required to make all arcs).

Raphael 5
Cranach 4
Seurat 3

The proportion of the circle to the body creates the character of the body.
  
① The size of the largest circle relative to the height of the body can be compared.

Seurat＞Cranach＞Raphael

 A large circle makes a long curved part of the body.
in Seurat the circles make upper arms and thighs  to make the body longer.
in Cranach this is used only for exaggeratedly long,thick thighs.
in Raphael this circle, being the diameter is close to two-thirds of the height, is very harmonious with the other circles.


② Second largest circle

C＞S＞ R

in Cranach this circle make the long part together with the largest circle.
 it is useful for creating proportions where the contrast of  parts is conspicuous.
in Seurat this is used for most of the slender body .
in Raphael this  makes most undulations with the third largest red circle.

③Third largest circle

R＞C＞S

in Raphael this circle is used most frequently at regular interval throughout the body.
in Cranach this is most expressive; undulating lines of left arm, nallow waist and fleshless knees.
in Seurat this is used only for the shoulder.

Limited arcs effectively expresses body features.

The contour is a looped melody line.
Just as music is made of musical notes, the masterpiece seems to be made of arcs

 

The beauty of nude is geometrical 1 /Cranach and Seurat

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The most widely used motif throughout history may be a nude figure.
Why can we recognise by sight who painted the masterpieces of a nude?
Nudes by Seurat and Cranach can be easily distinguished.
Looking at a shape is looking it geometrically.
The circle-cross model should give us some clue.

                                                        Cranach [Venus]

                               Cranach [venus]  circle-cross model
Black diagonals indicate the center of the picture.
The red circle surrounding the lower belly and two largest blue circles pass through the center.
The largest blue circles and the second largest green circles form the basic structure.
The red circles are concentrated on the upper body and lower legs and
many of them are bundled together.
Three straight lines are made of lined units.

I deviced a new model made of arc model and scaled circles.

    Cranach [Venus]  body- proportion- circles model
I have written that "Arc is atom of art."
The technique is allways deviced using circles that is the extension of the arcs.

This picture uses five circles that look different in size.
All arcs are part of a circle, so the curvatuor is the same.
The long loose arc means that the circle of the arc is large comparing to the nude.
The nude has a plump part and a thin part.
This is because each part is made of different sized circle.
The largest blue circles form the long thighs.
The second largest blue circles form the arms, upper right thigh and the hind parts of lower legs.

The red circle is most useful to make unique shape of the nude.
The size is one fifth of the body height.
Chained arcs of the same size is very effective; the left arm and right knee are made of four red arcs.

I made body-proportion-circles model of the standing nude in the center of Seurat's [Model].

 

Seurat [Model]  body-proprtioned circles model

The standing figure in the [Model] can be made of only three circles.
As the smallest circles are only used for the shoulders,
it can be said that it is almost made up of two kinds of circles.
The largest purple circles have larger ratio to the body than the largest circles in [Venus].
The red circle is half the body, creating a smart figure.

It is clear that the ratio of the circles determins the character of nude.

 

 

Identification of Sharaku and Hokusai on deep structural level

                            Shunro(Hokusai) [Sumo-e (Kimenzan and Dewanoumi)]　arc model

This sumo-e made me convince that Sharaku is Hokusai.
When I finished [Who is Sharaku?] I had not discovered more advanced  model.
As I found another digital image for the sumo-e, I made circlecross model.

                  Shunro(Hokusai) [Sumo-e] circle-cross model

The model reveals why the paiting looks as solid as netsuke sculpture.

Extending each arc, I made circles.
Each circle with the horizontal and vertical diameters forms a unit (circle-cross unit).
Of the largest blue circles, I made the basic three circles in  sky blue.
The three units are on the same vertical line.
The top light blue circle that makes the left shoulder of the back wrestler passes the center of the picture.
The red unit and green unit are placed on the vertical line.

Moving the middle blue-unit to the right,it overlaps the unit that makes the left thigh of the back wrestler.

It is clear that this picture is made systematically.

For comparison, I chose early Sharaku: [iwai Hansiro].

               Sharaku [Iwai Hansiro] circle-cross model

There are five distinctive vertical lines.
Looking from the left,
the first vertical line is made by
the unit of the third sheet of four layer collar on the left and
the unit of the second sheet of of the four-layer collar on the right.
On this line smaller red unit is placed.

The second vertical line passes through the left eye.
The top blue unit makes the top part of the comb and
the bottom unit makes the first layer of the right collar.


On the third vertical line,
there are one dark red unit that makes the foehead and
blue unit that makes right　shoulder.
Another red unit makes right cuff.

The fourth vertical line passes through the center of the picture.
The top pinked unit makes the right cheek and
the second pinked unit makes the first layer of the left collar.

The fifth vertical line is made of three red units.
The top unit makes a line of hair.
The second unit makes the line between the collar and the sleeve.
The third unit makes the right sleeve.

The top part of the picture reminds me of the circle-pivot model of [osiokurihatoutuusen-no-zu].
There are two similar techniques .
Firstly the comb is made of four different circles,
and the ship is also made of four different circles,
Secondly the hair waves are made of bundled circles of the same size.

　　Souri(Hokusai) "osiokurihatoutuusen-no-zu [Fast Cargo Boat Battling The Waves] "

There is no art theory without studying individual artworks structurally.

Seurat's [the Model] 1

Unlike Cezanne's [Bathers], each nude is independently structured.
Besides the nude in the middle is centered: the composition is very classical.
The cotour of the body seems to be uniquely smooth with a succesion of the same curve: the circle-arc model confirms it.

     Seurat [the Model]  circle-arc model
The dark pink arcs are dominant with a few purple arcs.

I found the study for this painting online recently.
The appearance of the body is very similar though the proportion is different .

     Seurat  study for [the Model]

The circle-arc model can be made without the purple circles.

     Seurat study for [the Model]

The model has not only pivots (intersection of circles) but also lined up units
I find the same techniqus in the classical nudes.
[The Model] reached the same high quality as the classical art.

Cezanne's painting became the forerunner of modern art because he did not care about content.
His [Bathers ] are immediately understood by those who could sense the formal structure.

The meaning of arched border painted around Seurat's "Cancan"

The painting has painted border which top is arched.
Recently I noticed that this arch is important key to understand Seurat's method of composition.
This curve looks the same as the curve of raised hem of the skirt and the curves of the legs side by side.
The arc model can check how the curves are.

Only five types of circles are enough to make the model.
This creates harmonious effect.

    Seurat [Cancan] arc model

Circle-arc model can be made extending  arc-model.

     Seurat "Cancan" circle-arc model

Enlarged model show how exactly the circles are placed.

     Seurat "Cancan" enlarged circle-arc model

 Blue circles make the basic structure.

     Seurat "Cancan" blue circle-arc model

The two circles which are the extension of the arch pass near the bottom corners.
The circles which make the hem of the skirt intersect higher than the painting.

     Seurat [Cancan]  circle-cross model

The second largest red circles are dominant.
A circle with the horizontal and vertical diameter make a unit called circle-cross unit.
When more than three circles intersect at a point I call the point"pivot".
Circles fastened at a point can still be moved along the point for adjusting balance.
Lined units make good balance.

Seurat"s late composition is highly syntactic.
Syntax is ornamental.
I have classified the level of ornamental development into seven.
The highest level of work can be expressed as circle-cross model.
The most syntactically finished work by Cezanne is lithograph "Six Bathers" which is linear work.
Picasso started monochromatic Cubism.
Even Fauvist Matisse painted "Dance" with a few colors.
Early Modern artists all recognized that atom of art is arc.

           

Mondrian did not use golden ratio but integer ratio for proportion

I picked up this illustration from the previous blog "How to make Mondrian."

This suggests that I could express all the Mondrian in the big eyed grid.
I have read about Mondrian's use of Golden Ratio very often.
I searched in (Golden Ratio. Mondrian).
It seems that his use of Golden Ratio is accepted with no doubt.
I found one suitable blog to point the problem: [Katari-e no Sekai].

....[Your painting is made of squares only.]
Mondrian answered that I see no squares.
There are only Golden Squares.

There are two accompanied illustrations

The left illustration is recognized as Golden Rectangle as it is divided to squares.
I do not see Golden Rectangles in the right picture.
I see all the rectangles are different as Mondrian said.

I used soft "paint" to extract the rectangle.
Then I changed the size to find suitable one for grid.

The rectangle 8/5 is near perfect match:
  8/5=1.8 ( Golden Ratio is 1.618)

The  Mondrian on the right is expressed on grid.

Mondrian started Chinese Lattice-like work with four different rectangles.

The next type does not have regular units but looks like installed shelf;
 rectangles with same width piling up
and rectangles with same height lining on horizontal line.

 A  large almost-square appears near center.

Cabinet likeness disappears: all the rectangles are attached to the frame.

I picked works of high resolution for checking.

10/8

9/9

15/12

32/16

31/20

23/21

25/21

31/31

37/36

75/76

This suggests that all his works can be designed on regular grid when the good digital images are available.

Relatively small number of grid eyes is enough to design Mondrian composition, which is why the proportion clearly recognizable.

There is great consequence in the use of simple grid.

A) Historical art is applied art.
Pure art does not depend on the current art history.
It can be created on the innate sense ability.

B) golden ratio was used as temporal fashion.

On the other hand the visual grammar is innate ability of all human.
Recognition of natural number is not mathematical but pre-mathematical perception.
The grammar is the base of all intellectual activity including mathematics.
Mondrian re-discovered the visual grammar after following Cubism.