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The Philosophy of Lines:

From Art Nouveau to Cyberspace

Palgrave​ 2021

This book offers a philosophical exploration of lines in art and culture and traces their history from Antiquity onwards. Lines can be physical phenomena, or cognitive responses to observed processes, or both at the same time. Based on this assumption, the book describes the “philosophy of lines” in art, architecture, and science. The book compares Western and Eastern traditions. It examines lines in the works of Paul Klee, Wassily Kandinsky, and Henri Michaux, as well as in Chinese and Japanese art and calligraphy. Lines are not merely a matter of aesthetics but also reflect the psychological states of entire cultures. In the nineteenth century, non-Euclidean geometry sparked the phenomenon of the “self-negating line,” which influenced modern art; it also prepared the ground for virtual reality. Straight lines, distorted lines, blurred lines, hot and cold lines, dynamic lines, lines of force, virtual lines… Lines narrate the development of human civilization.


Lines can represent realities not only through affirmation but also through negation. This idea has been developed by many Western philosophers and artists as well as by representatives of Buddhist and Taoist thought. Virtual reality is the last stage of this “adventure of lines.” Virtual Reality is not a random phenomenon enabled by advanced technology, but that it has been prepared over hundreds of years. But I also criticize the abstract space of Virtual Reality, which no longer offers a concrete place in which lines are embedded. There is no hand, no body, no gesture, no surface, and no environment. The instantaneously configured geometrical output of Computer Assisted Drawing leaves no traces. I show that some Western and Eastern philosophies offer alternatives and help us to think lines differently.


Wassily Kandinsky, Composition 8


Paul Klee, Highways and Byways

Much has been written about lines from an anthropological, philosophical, and scientific point of view. In this book I concentrate on the most peculiar characteristic of the line, which is its ambiguous ontological status. The twofold status of the line as both a physical fact and a concept grants the line a unique place in human civilization. The line is not simply empirically given but also an experience constantly developing future events out of its own past. 

Despite the big advancements in geometrical presentations of lines, nobody (not even Euclid) attempted to give a serious definition of the line. While in mathematical terms, a line is a consecution of points, in non-mathematical contexts, lines can appear in many different ways. A line can be material, for example when it is represented by a string, or it can be non-material as in geometry. Lines can be mere thought products or closely linked to bodily action as it happens in calligraphy or in dance. A line can be visually (or acoustically) assumed though at the same intellectually challenged. Furthermore, lines produce spatial experiences not only in the one who draws them but also in the observer. However, as long as lines are produced or seen by humans, all lines have one thing in common: they require some form of spatial reasoning.


The adventure of lines that I describe consists in a process of self-negation through which the line – paradoxically – attains its full potential. How do we experience a phenomenon shifting back and forth between the material and the non-material? As the word “line” is often used in a metaphorical fashion, these questions concern not only art but also politics, science, and many other human activities. The procedure culminates in the phenomenon of the virtual line that is decisive for today’s virtual reality in which we go “online.” 



1. Introduction




2. Strings, Traces, and Structures

What is a Line?

Ancient Lines

Renaissance Lines

Virtual Lines

The Graphic and the Geometrical

The Idealized Line

Non-Euclidean Geometry

Lines, Strings, and Structures

Western Lines and Eastern Dots

The Unbearable Straightness of Lines


3. Lines in Modern Society

McLuhan on Hot and Cool Lines

Distorting Lines

History is a Line

Linear and Non-Linear Science

Liquid Lines (Lines as Networks)

Blurred Lines


Part II: Lines: An Aesthetics of Disappearance


4. Differential Lines

Charles Baudelaire

Heinrich Wölfflin                                            

Broder Christiansen


Adolf Loos


5. Dynamic Lines

Maurice Merleau-Ponty

Paul Klee

Wassily Kandinsky

Henry van de Velde

Piet Mondrian

Henri Michaux

Paul Valéry


6. Drawing as Thinking

The Lines we Draw

Drawing as Thinking

Film-Thinking, Drawing-Thinking

Lines versus Images

The Calligraphic Sign as Thought

Stylistic Lines

7. Non-Euclidean Geometry

Euclid vs. Non-Euclid

The Line as Cognition: Fichte

The Non-Euclidean Revolution

Non-Euclidean Geometry and Art

God and the Map of the Universe

The Crisis of Geometry and the Crisis of Science


From Lines to String Theory


8. “The Movement that the Eye Cannot See:” Flexuous Lines

Félix Ravaisson

Bergson and Merleau-Ponty

Futurist Lines of Force

Latent Lines

Further Developments: Rhizomes, Twistings, and Virtualizations

Traces and Deterritorializations

Twisting the Lines: Heidegger’s Verwindung

Virtual Lines in Virtual Reality


Part III: Living Lines of the East

9. Calligraphic Lines

Calligraphy and Chinese Painting

Lines, Traits, and Strokes

Vision and Bodily Presence


10. Organic Lines of the East



Lines and Surfaces

Alan Watts on Lines

Calligraphy as Thinking


11. Dream Lines

Ma and Kire

The Lines of Virtual Architecture


12. Two Kinds of Virtual Realities

Technical and Existential Virtual Reality

Existential Lines


13. Conclusion



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“When I see the tiles at the bottom of the swimming pool through the depth of the water I see them not in spite of the water and the reflections, but rather through them, by them.” (Maurice Merleau-Ponty)

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Virtual landscape


Most thinkers interested in this intriguing aspect of the line are working in the realm of aesthetics. Some approaches towards lines have become famous. Heinrich Wölfflin perceives the “depreciation of the line as a boundary,” and Maurice Merleau-Ponty sees lines as dynamic phenomena. Henri Bergson, whilst creating his philosophy of time, speaks of the line as of a succession of events. Ludwig Wittgenstein suggests that lines challenge the human intellect not through their blurred character but through a more sophisticated procedure of self-negation. More radically, Piet Mondrian believes that in his paintings lines “destroy” each other through an effect of mutual opposition. Finally, the French painter Henri Michaux relies on the “divestment” (désaisissement) of the line, that is, on the line’s “negative values.” For all these thinkers and artists the line is not simply present (abstractly or concretely), but its existence is linked to a complex ontology employing both affirmation and negation. This process culminates in the phenomenon of the virtual line. Virtual lines, which have no physical existence, are neither actual nor potential but still they are real because they can be intellectually assumed. They are not just appearances created by our imagination but acquire a new status generally called “virtual.”

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Blurred  line


A part of my analysis focuses on the use of lines in Far Eastern art and culture where a similar idea of the “self-negating line” is central. The line drawn in calligraphy has an ontological status similar to that of the “divested” or “negative” line that has been on the agenda of the above-mentioned Western artists. In East Asian aesthetics, the “self-negating line” is determined by Taoism but also by Buddhism, especially by the Zen-Buddhist branch.


Geography is the science of lines measuring the three-dimensional space of objects. “Cyberspace,” which is the space in which different pieces of Virtual Reality are imbedded, has no map because it knows no physical limits. In virtual reality, Euclidean geometry, which remains essential for any kind of geography, becomes non-Euclidean. Four-dimensional theories match perfectly well with the idea of Virtual Reality: time and space are configured through mutable lines that are constantly negated as “real” lines. I show that throughout human history, lines have always been – at least to some extent – virtual in precisely this sense. The latest stage of this development, which is virtual reality, must be seen as the accomplishment of a long process.

Paul Klee, Chambre habitée

Japanese architect Tadao Ando designs houses with labyrinthine structures in which “walls become abstract and negated.”  Darell Wayne Fields invites us “to discover the lines that are the essence of Ando’s work – [which are] the lines that are not seen.” Though Fields finds that Ando’s manner of sketching is “more attuned to carving than to drawing,” he also states that Ando’s lines constantly fluctuate between “ideas of materiality and voidedness.” 

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Air Architecture (1961). Yves Klein was looking for a “cartography” able to make visible the invisible, and, paradoxically, he found it in the architecture of those elements that have neither lines nor clear limits: air, fire and water. 

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Tadao Ando, Church of Light

SANAA Glass Pavilion Toledo Museum of Ar

SANAA Glass Pavilion Toledo Museum of Art. Continuation of Yves Klein?


Rembrandt, Seated female nude, 1658


Albrecht Dürer, Female Nude, 1493

“The difference between Dürer and Rembrandt [is that] in the one case the masses appear with stressed, in the other with unstressed edges.” (H. Wölfflin)

Klee, Fliehende Linie

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The rhizome has best been visually realized by a famous painter of errant lines: Jean Dubuffet. In his paintings called Recits (1974) and Non-Lieux (1984), Dubuffet explores the interweaving of auto-generating lines. These lines are also erratic traces because they are not submitted to the constraints of a preexistent shape but simply materialize the rhizome.

Dubuffet, Recits (1974)

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Dubuffet, Fluence (1984) 

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Mondrian, "Composition with Large Red Plane, Yellow, Black, Gray and Blue" 1921,


Musical improvisation on Piet Mondrian. Click here.


Peter Schwenger: Asemic Writing.