Topological painting

Topological painting is a painting style that began with the expressionism school and presents reality in a distorted fashion. The aim of this caricature-like reality distortion is to emotionally intensify reality. Israeli painter Igal Vardi enhanced and developed the style, and added his own unique characteristics.

About

Topology as a branch of mathematics represents reality through distortions - contracting, stretching, magnifying, and the tearing of form. In Igal Vardi’s topological painting reality is presented as fluid, chaotic, hybrid and collage-like. It is also shown as a circular transformation of the unraveling of form, in a parallel process of de-construction and construction of reality, as two complementing opposites – chaos as a condition for order. Vardi’s topological painting also presents distorted reality by a simultaneous convergence of three painting styles - impressionism, cubism and expressionism, which represent, respectively, the innocent eye, the thinking eye and the painting hand.

History of Art Influences on the Development of Style

Various artists have expressed topological distortion in their work: Pablo Picasso in his surrealist presented an exaggerated distortion of reality, which was in fact based on his simultaneous cubistic style that represented reality from different perspectives concurrently. Topological distortion can also be found in the futuristic works of Umberto Boccioni and the painting style of Francis Bacon. The biomorphic paintings of Joan Miró, Paul Klee and Jean Arp also echo the topological view of reality, and the paintings of Maurits Cornelis Escher reflect topological thinking by presenting reality in closed circularity based on the structure of the Möbius strip and a closed sequence of metamorphoses.

The style of Salvador Dalí’s last paintings was called “topological painting”, and presented a distinctly topological view of reality.

The Baroque and the Classical: Topological Painting as a Baroque Archetype

Artist historian Heinrich Wölfflin attempted to describe the history of artistic styles based on what he called “two modes of vision” which represented two archetypal styles: the baroque and the classical. The baroque style is characterized by movement, round lines and blurred shapes in relation to the background. In contrast, the classical style features staticity, straight lines and clear contours creating the boundaries of the object in reality. From a trans-historic perspective that presents the history of art based on a-historic archetypes, the baroque style is topological in essence as it presents reality with a distortion of movement in a process representing the beginning of form (morphogenesis). The archetypic, trans-historic style is realized in time and place in various ways, and takes on the unique character of each period.

While topological painting as represented in Igal Vardi’s style has shared common characteristics with classical baroque painting in which reality is not static but rather a process of formation, it differs from it in its unique attributes.

Leonardo da Vinci: Topological Caricatures

Leonardo da Vinci’s style was complementary, between reality depicted in painting as harmonious, symmetrical and refined, as reflected for example in the Mona Lisa, and the caricature paintings with their faces distorted to the extreme. Da Vinci studied the morphogenesis of nature, the secret of the genesis of forms, which he expressed in drawings showing the mutual relationships between convex and concave lines which create the movement toward formation. Da Vinci’s drawings clearly show topological thinking, which he also expressed in his written studies concomitant with his drawings of forms that represented a process of formation and gradual distortion from a moderate to a more polar state.

Salvador Dalí: Melted and Distorted Reality

Salvador Dalí was the only painter in the history of art who used the term “topological painting” to describe his artwork, naming his last four works in life “Topological Abduction of Europe - Homage to René e Thom”. These works in effect summarize his artistic journey, depicting reality as “melted” and distorted. Dali was influenced by Rene Thom’s theory, known as “catastrophe theory”, which shows the “tear” in the non-linear sequence of reality formation, and also by D'Arcy Thompson’s theory of transformation where he applied mathematical models and thinking to show how biological shapes are connected to each other in their formation.

Pablo Picasso: The Surrealist Period as Topological Painting

In his book “Viva Picasso – An Aesthetic Interpretation of his Art”, Igal Vardi maintains that Pablo Picasso represented reality from a topological perspective, particularly during his surrealist period. Picasso’s surrealist style depicts extreme distortions in reality, up to a state of extreme de-personalization. In fact Carl Gustav Jung wrote that Picasso’s work characterizes a schizophrenic mental state.

The analytical cubism revolution, characterized by presenting reality from different perspectives simultaneously using “fractal fragments” of pictorial facets, signified the beginning of reality presented as distorted and topological. Historically this is evident in the conversations Picasso (the father of the cubist painting revolution, together with Braque), conducted with intellectuals, among them the mathematician Maurice Princet, to discuss Poincaré’s topological mathematics. Picasso, who listened and absorbed information about mathematics and physics from these conversations, translated them into a new language of painting paralleling the mathematical theory. This influence appeared later in the topological painting in his surrealist style, and was also reflected in his “linear sculpture”, for example his sculpture as a tribute to Apollinaire. In fact, Picasso thought topologically without declaring it, as reflected in the painting revolution he initiated, from analytical cubism through his surrealist style.

Umberto Boccioni: Adding the Fourth Dimension (Movement) to Cubism

The introduction of movement into the static art of analytical cubism was reflected in the futurist style, particularly that of the painter and sculptor Umberto Boccioni. Boccioni visited Picasso’s studio and absorbed the influences of the attributes of static analytical cubism. Upon his return to Italy he added the “fourth dimension, movement, to Picasso’s static analytical cubism. Boccioni’s work, in both painting and sculptor, is in essence topological, based on power fields found in movement and in distorting the spatial coordinates of reality. In the beginning, It was Picasso who influenced Boccioni, and later Picasso adopted the “fourth dimension” of movement conceived by Boccioni and the futurists, which served as the catalyst for the birth of his surrealist style in which he depicted reality in extreme, distinctly topological distortions.

Francis Bacon: Distorted Portraits

Topological painting found in Picasso’s surrealist style directly influenced the English painter Francis Bacon. He was consciously and declaratively affected by Picasso, and in his footsteps developed a painting style that clearly topologically distorts the portraits and figures in his paintings, although most scholarly literature about Bacon does not offer a clear explanation for these distortions. The existing influence of the cultural climate with respect to Henri Poincaré’s non-Euclidean mathematics during the cubist revolution also influenced Bacon’s art in England as he embedded Picasso’s aesthetic code into his style.

Maurits Cornelis Escher: Reality as a Möbius strip

Topological theory influenced the works of the painter Maurits Cornelis Escher. His work used the principles of the non-Euclidean hyperbolic surface, informed by the French mathematician Henri Poincaré who clearly studied the intricacies of topology. In his works Escher depicted metamorphoses in which shapes gradually metamorphose to something entirely different. He also explored the concept of infinity through a series that becomes smaller up to infinity, examined issues in the study of symmetry and depicted reality as a Möbius band taken from the topology field. Escher also based his work on the physicist Roger Penrose in showing reality based on the Penrose triangle, a triangular impossible object also taken from topology.

Joan Miró: Biomorphy

The painter Joan Miró drew his works in the biomorphic style as influenced by Art Nouveau and the architectural work of Antoni Gaudi. His approach was characterized by an attempt to trace the beginning of forms in nature (morphogenesis), rather than a static depiction of shapes as they are revealed in the natural world with regularity. The biomorphic approach distinctly shows the formative distortion of reality according to the topological method, and Miró’s paintings allude to topological painting.

Paul Klee: From the Amorphic to Form

The painter Paul Klee, in some of his black and white drawings, depicted what he called the attempt to “take the line for a walk”. Klee made his drawings without lifting his pen from the paper, and in doing so created the amorphous, shapeless line that in the course of its (topological) movement creates the more formed shapes. This movement of the amorphous line represents Klee’s way of showing the “hidden reality” found behind visible reality.

Jean Arp: Entropy and Art

The works of the painter and sculptor Jean Arp were influenced by entropy laws from the physics field, representing reality as falling apart according to thermodynamic laws. In his work Arp showed the “tear” created in reality when extreme distortion occurs which is beyond the possible limits of matter. The extreme distortion that causes the tear was presented theoretically by the mathematician Rene Thom in his “Catastrophe Theory”, which preceded chaos theory in putting forth the concept of the fractal which represents the same sudden morphological fracture. As noted above, Thom’s theory directly influenced Dali’s topological work.

Topology and Psychology

The Psychological Aspect: Topological Symbolic Thought

Sigmund Freud identified two types of thought processes in an individual’s cognitive scheme: primary and secondary. The primary process is characterized by symbolic and illogical thought, while the secondary process is rational and logical. Secondary rational thought examines reality according to Euclidean mathematics, while the primary thinking process is irrational and examines reality based on non-Euclidean mathematics, in other words - topologically. Freud claimed that there is a circular relationship between primary and secondary thinking, and vice versa.

Jean Piaget: Three Geometries Constructing the Cognitive Scheme

Proof of Freud’s argument is found in Jean Piaget’s study of children’s drawings. Piaget framed the three stages of children’s drawing during child development as representing a transition between three geometries. According to Piaget, the first stage is the scribble, after which the child moves on to the second stage, a schematic drawing of reality in which objects are drawn as the child remembers them, and not as he or she sees them. At this stage, the child draws a table as a board with four legs slanted sideways. Finally, the child reaches the mature stage in which reality is presented in a proportional and realistic drawing. For Piaget the scribble stage represents topological geometry in which there is no inside or outside, and reality is depicted as a typological network. The schematic drawing stage represents projective geometry (the table with the legs slanting sideways), and the realistic drawing stage represents Euclidean geometry. Piaget maintained that an individual’s mental scheme contains these three geometries together: the topological, the projective and the Euclidean. He claimed that in order to observe reality realistically and sanely (i.e. according to Euclidean geometry) we must rely on deeper layers of consciousness on the geometric scale – the projective and the topological. Paralleling Piaget, Freud also maintained that rational and logical secondary thought processes take place as long as primary, irrational and illogical thought pulsates below and constructs this thought. The circular paradoxical relationship between cause and effect is presented in Shlomo Giora Shoham’s theory of myths, which shows the dyadic relationship between history on the diachronic timeline which shapes synchronic metatemporal myths; and that the minute myths consolidate into a structure, they once again influence the origins that constructed them.

Kurt Lewin: The Principles of Topology in Psychology

The psychologist Kurt Lewin, who wrote the book “Principles of Topological Psychology, described the individual psyche as based on psychodynamic theory, which is in turn influenced by topological mathematical theory. According to Lewin, the individual psyche functions based on the concept of the “topological field”. Thus he attempted to move topographic Freudian thought toward new psychological thinking based on “field theory”, in which the influence of the isolated psychological vector changes to show a mental picture consisting of a network of vectors functioning simultaneously in the “topological field”.

Jacque Lacan: The Mind is Topology

The psychologist Jacque Lacan significantly advanced psychology’s understanding of the human mind from the topological perspective. According to Lacan, the human mind is built topologically, and functions as such. In contrast to his teacher, Freud, who described the mind as built topographically, one layer above the other (according to the Euclidean model), Lacan maintained that the human mind is built as a network, in a non-Euclidean structure. He described the human mind as composed of three “registers”: the real, the imaginary and the symbolic, and maintained that these three registers function together based on the topological model known as the Borromean ring. The topological Borromean ring is characterized by three interlocked rings, each of which cannot exist without the other two. According to Lacan, the human mind functions psycho-dynamically according to the non-linear dynamic between the three abovementioned registers, at times in a Möbius strip structure. This approach breaks with Freud’s topographic-hierarchical conception, in an attempt to understand the causal relationship between the different parts of the human mind.

Igal Vardi’s Topological Paintings

Development stages in Igal Vardi’s topological paintings

The Eye, the Brain and the Hand

“Topological painting”, as developed by the Israeli artist Igal Vardi in his artistic work, is based on the model of the three topological Borromean rings. According to Vardi’s artistic theory, reality can only be perceived based on a triple dialectic simultaneously, between the innocent eye that looks, the brain that interprets and the hand that draws. The eye represents empirical epistemology, the brain rationalistic epistemology and the hand pragmatic epistemology.

Vardi’s topological paintings represent reality from three stylistic perspectives: the impressionist – in other words the empirical according to the innocent eye, the cubist –the rational according to the organizing brain, and the expressionist – the pragmatic according to the hand that draws spontaneously and freely.

Through these three artistic schools Vardi’s topological paintings represent the world of phenomena that appears to the eye (through the impressionist style), the real world that is concealed from the eye (by means of the cubist style) and what mediates between them (through the expressionist style). These three styles, each time anew, represent topological reality that is continuously changing.

As a young man Vardi studied with the painter-sculptor Cecilia Markowitz, a student of the sculptor Antoine Bourdelle (who was a student of Auguste Rodin) and the painter André Lhote. He began his artistic journey influenced by André Lhote’s school of realistic cubism, and gradually transitioned to the expressionistic stage which led him to focus on a distorted representation of reality based on topological principles. His “topological paintings” consolidated gradually, both theoretically and practically, in the course of is exhibitions: “Formative Realism” in 2003, “Mythical Paintings” in 2006, and “Variations on the History of Art” in 2008. The Formative Realism exhibition represents Paul Klee’s approach in an attempt to “take the line for a walk”, creating a transition from a shapeless amorphic painting to one that gradually constructs reality. The amorphic movement constructs reality in a distinct representation of a resumptive “topological network”, in the language of the philosophers Gilles Deleuze and Félix Guattari. The topological network is a covert depth structure constructing the overt surface.

“Myth Paintings” represents an additional stage in Vardi’s consolidation of the topological style in his work. In his last exhibition, “Variations on the History of Art”, which opened in 2008, Vardi attempts to present the history of painting through a topological stylistic prism which he developed, and includes the works of known painters and artists who received new stylistic expression through topological glasses. Thus, in his current works Vardi has reached the stage in which he represents reality with maximum distortion, resonating Picasso and Bacon.

Further reading

• J. Gleick, Chaos – Making a New Science, from English: Emanuel Lotem, Maariv Library, 1991
•  Z. Giora, The Unconscious and the Theory of Psychoneuroses, Translation: Dina Gil, Papyrus, 1988, pp. 23–33
• Jean Piaget - How Children Create Mathematical Concepts”, in U. Lust. & M. Nisan (Eds.), Psychology in Teaching, Otzar Hamoreh, 1978, pp. 179–185
• Mikhail Bakhtin, Later Essays and Notes, translated from Russian and added comments and Introduction Sergei Sandler, scientific editing: Helena Rimon, Resling 2008.
• T. Katz-Freiman, Exaggeration is a Double Truth, Wild Exaggeration, Haifa Museums, Haifa Museum of Art, 2009, pp: 7-11
• Y. Portugali, Contained Relations – Society and Space in the Israeli-Palestinian Conflict, Hakibbutz Hameuchad, 1996, p: 60-62
• Igal Vardi, Mimesis – The Psychology of Modern Painting, Yediot Sfarim Publishing, 1996
• M. Zuckermann, Chapters in the Sociology of Art, The Broadcast University, Ministry of Defense, 1996, p: 115
• Igal Vardi, The Personality Collage – A Synergetic Theory of Personality, Yediot Sfarim, 2013
• Graves D, The New Institutional Theory of Art, Common Ground, 2010
• Igal Vardi, Sketch, Teaching Painting and Mentoring the Artist, Yediot Sfarim, 2004
• C. Lévi-Strauss, The Savage Mind, Hebrew: Ilia Gildin, Sifriat Poalim, 1973
• Igal Vardi, Skitsa (Sketch) – Instructions for Drawing and Training the Artist, Yediot Sfarim, 2004

External links

• Carmit Saphhire-Witz, Imitation of the Mona Lisa, NRG, 21 July 2008.
• Igal Vardi, Variations on the History of Art – The Topological Painting, see Igal Vardi’s website

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