The entire evolutionary process of Piet Mondrian’s oeuvre and its substantial meanings can be explained by examining three key works:

1908-10

1915

1942-43
We shall examine these three works to demonstrate how from 1908 to 1943 the artist has been searching the most suitable forms and colors to express the same basic idea of a dynamic interaction between multiplicity and unity, that is to say, between the endless variety and mutability of the world around us and within ourselves on the one hand and the need to contemplate the intrinsic unity of all things on the other hand.
The Red Tree
In the figure of a bare tree we see the horizontal line of the ground presenting a sequence of small vertical strokes that consolidate to form the trunk from which blossoms the set of branches that gives birth to a manifold space that on the right then returns toward the ground line from which the trunk had been generated.
The same process can be observed in:
Apple Tree in Blue
Final Study for The Red Tree (Evening)

The horizontal line of the ground continues to the right and to the left suggesting the boundless extension of nature which was emphasized in the landscapes and seascapes of the same years with a dominant horizontal space:
The tree trunk is vertical like the architectural volumes of mills, lighthouses and church towers of the same period:
The artist ascribed to the horizontal the value of everything that can be defined as the “natural” not only in the sense of natural landscape but also as anything that before our eyes and within ourselves in the course of our lives constantly changes, transforms, evolves. In the vertical the artist instead saw a symbol of the “spiritual” that is, of the all-human propensity to seek stability, constancy, unity. Consciousness arises from this interaction between mutable and constant, multiplicity and unity, instincts and reason, that is to say, between opposite drives that the painter expresses plastically with a horizontal expansion and a vertical concentration of space. We talk about a dynamic interaction.
In the figure of the tree the vertical trunk tends to concentrate the horizontal extension of the ground and of the branches.
Through the unifying action of the trunk a manifold extended space is condensed into a set of branches which appear as a complex miniature universe. The tree can therefore be contemplated as a manifold space (the branches) symbolizing the multifarious, mutable space of the “natural”, while maintaining a synthetical wholeness at the same time.

Through the unifying action of the trunk, the horizontal extension of the ground, which would continue indefinitely as the all-horizontal landscapes of the same years, is concentrated into a complex and changing whole (the branches) but all within the canvas. The canvas is for the painter the space of consciousness, which therefore can make its own the virtually infinite and elusive space suggested by the unlimited ground line.
The multiplicity concentrated within the canvas then flows back to the right toward the ground line from which the trunk and the branches had formed. It is indeed a circular process. From an endless multiplicity (the line of the ground) to a measured one (the branches) which then flows back to the endless.
Circularity seems to be reasserted by a circle that can be seen in the upper right section of the canvas.
Pier and Ocean 5
About five years later we find the same idea of the manifold becoming one and then turning manifold again in a new work that although inspired by a landscape, is already in fact an abstract composition.
The work belongs to a series of drawings, and gouaches inspired by a pier jutting out from the beach into the sea. From the painter point of view the pier appears as a vertical which interpenetrate the horizontal flow of the sea.

The interaction between the upward vertical progression of a pier (recalling the tree trunk) and the horizontal expansion of the sea generates a whole variety of relationships between horizontals and verticals where something changes every instant (recalling the manifold set of branches of the tree):
The unstable signs find a more balanced and lasting situation in a square where the opposite directions assume the same value and the contrasting duality which animates the whole composition is transformed into an ideal unity. Same as the tree trunk, which unifies the set of capricious set of branches, the square unifies a variety of ever-changing orthogonal signs.
The sign of equivalence between opposites is born inside a square and thus suggests an inner space. The square is a plastic symbol of the unifying consciousness of man dealing with the multifarious aspect of the world symbolized here by a multitude of different relationships between opposites drives. In the square the changing external space is captured for an instant in synthesis.
This is another reason why modern painting turns abstract: how can you simultaneously represent the outer and inner world if you just look at the external appearance of things through the so called realistic or figurative painting?
It is, however, obvious that the consciousness can only produce partial syntheses; it clearly cannot exhaust all the possible relations with the external world. Every synthesis generated by thought is necessarily partial and temporary, and must therefore open up again to the multiform and ever-changing aspect of physical reality. This is what all sensible people do when they call their certainties into question in the light of experience. This is what philosophy has been doing for centuries, as have the arts and above all the experimental sciences.
A second square can be seen in Pier and Ocean 5 above the square that we have identified as a unitary synthesis of the composition as a whole:

Inside the second square we see a vertical segment divided by two horizontal segments that extend beyond the boundary of the square to the right and left. These two signs tell us that unity is opening up to duality. The unitary synthesis achieved for an instant in the lower square in the form of the equivalence of opposites is again broken up into a duality that then flows back toward the variety of different situations marked again by the alternating predominance of one direction or the other.
The unity generated with the first square opens up again to manifold space with the second. The composition evokes the manifold and controversial space of life which attains measure and a harmonious condition in the space of consciousness (the square proportion) before opening up again to the multifarious nature and unforeseeable events of life.
Here too we therefore see the ideal circular process that we have observed in the figure of the tree: Through the unifying action of the trunk, the horizontal extension of the ground (a virtually endless and manifold space) is concentrated into the branches (a complex but finite space). The finite multiplicity expressed by the branches then flows back to the right toward the ground line from which the trunk and the branches had formed.
From a virtually endless multiplicity (the line of the ground) to a finite one (the branches) which then flows back to the endless space of the ground. It is a circular process. In Pier and Ocean 5 the horizontal flow of the sea is concentrated by the vertical pier into a synthesis (the square) that then opens higher up to the horizontal before flowing back toward manifold space. In both works multiplicity becomes unity and then unity turns back to multiplicity.
There are of course also substantial differences between The Red Tree (1908-10) and Pier and Ocean 5 (1915) like for instance the use of an oval form which plays a significant in the relationship between multiplicity and unity in the latter composition. For a detailed examination of Pier and Ocean 5 see this.
Broadway Boogie Woogie
Let us now examine the third work which, in a totally new form, shows the same process form multiplicity to unity and from uniti back to multiplicity.
At first quick glance we see in Broadway Boogie Woogie a multitude of small gray, yellow, red, and blue fragments randomly moving along the lines..

Oil on Canvas, cm. 127 x 127
..which join up with others to generate some symmetrical configurations (Diagram A)

symmetrical configurations that then give birth to planes (Diagram B) where the relationship between horizontal and vertical appears more stable and durable than in the initial fragments moving along the lines.

New planes are born, as shown by diagram C, that differ from those observed in diagram B by presenting an inner space marked with a different color:

Diagram D shows how the self-internalization of space continues and the three primary colors concentrate in the area of just two planes while to the right we see a single large plane in which the three primary colors interpenetrate to form a compact unity:

The manifold space made of yellow, red and blue fragments expanding on lines toward opposite directions which disrupted our visual field at the beginning of the process by keeping the eye in constant motion, attain now a unitary synthesis (Diagram E):

Recapitulating what we have seen so far: A multitude of small gray, yellow, red, and blue fragments become monochromatic planes which transform into two-colored planes that then become a single plane constituting a synthesis of the three primary colors. Space undergoes uninterrupted transformation from a condition of multiplicity to one of unity; from the natural to the spiritual.
Same as the tree trunk of The Red Tree and the central square of Pier and Ocean 5 the unitary plane in Broadway Boogie Woogie is a plastic symbol of the unifying consciousness of man dealing with the multifarious aspect of the world symbolized here by virtually endless lines and an ever-changing variety of shapes and colors.

The controversial and virtually infinite space of the lines is transformed into a finite and lasting space with the unitary plane. It would, however, be a mistake to see this as calm in the sense of a total absence of inner tension. The unitary plane should rather be seen as a temporary synthesis which, like the square of Pier and Ocean 5 reopens to the manifold space around as we shall see in diagram F:

Plane 11 is the same size as plane 10 but consists solely of red and gray rather than the three primary colors. Moreover the horizontal line running suddenly through the vertical plane tends visually to disrupt the previously attained balanced interpenetration of horizontal and vertical (10).
After the equivalence of the opposite directions and the synthesis of three primary colors attained in plane 10, the colors are again reduced in plane 11 and the external dynamism of the lines reappears to generate new opposition. After the degree of comparative calm, constancy, and unity achieved in plane 10, spatial movement thus seems to reappear in plane 11.

The indication provided by plane 11 finds further confirmation in plane 12, where blue, yellow, and red are juxtaposed but no longer interpenetrate as they did in plane 10. The juxtaposition produces the impression of three separate planes, whereas the interpenetration combines the three colors in a single structure of greater stability. Note how the yellow on the right of 12 already seeks to cross the perimeter of the plane and flow into the yellow of the surrounding lines. Plane 12 can therefore be seen as plane 10 in the process of dissolution.

Configuration 13 possibly represents the conclusion of the process of reopening the unitary synthesis in that it can be seen as a continuation of the disintegration of 12.
Space proceeds from a comparatively static and wholly internal condition (10) toward one of growing instability (11) that is gradually transformed into the more dynamic and variable external space of the lines (12, 13).
From expansion toward increasing concentration (Diagram E):

and then from concentration back to expansion (Diagram F):

This is the way Broadway Boogie Woogie breathes. Through a dynamic process the one and the many (the spiritual and the natural) merge and transform into each other.
(For a detailed examination of Broadway Boogie Woogie see this)
A circular process
The Red Tree (1908-10), Pier and Ocean 5 (1915) and Broadway Boogie Woogie (1942-43) show a similar process from multiplicity to unity and from unity to multiplicity, that is a circular process. In all the three paintings the horizontal opens up the concentration previously exercised by the vertical:
The horizontal (symbol of the natural) re-opens the synthesis generated by the unifying action of the vertical (the spiritual).
A dynamic unity
The unity that Mondrian strove to express is a temporary synthesis generated momentarily by the subject’s inner world in its changing relationship with the outer world. It is not something to be attained once and for all. Establishing equilibrium between the manifold, ever-changing appearance of nature and the synthesis invoked by the consciousness does not mean attaining fixed points and immutable truths. The square of Pier and Ocean 5 and the unitary plane of Broadway Boogie Woogie are not potentially static and all-inclusive unities but dynamic unities intrinsically linked to the manifold space in which they are born and toward which they return a moment later.
The tree of 1908-10 thus already reveals in wholly embryonic form the process from multiplicity to unity and from unity to multiplicity that took shape in 1915 (Pier and Ocean 5) and was expressed with the brightest colors in 1943 (Broadway Boogie Woogie).
“Life is a continued examination of the same thing in ever-greater depth” (Mondrian)
“The evolution of his work is certainly the most eloquent of this century. I know of no other example of such acute finalism.” (Seuphor)