The Anonymous Architect of Euclid's Retreat
 

The work I am going to present today is part of a study of C.L. Dodgson's oeuvre, a study which I am pursuing in the context of a graduate program in the History and Theory of Architecture.
 
 
 

Introduction
 

Because my concerns are within the field of architecture, I am interested in the notions of space and time, the connection of these to human perception, and how space is conceived and created. Looking at the transformations of Geometry and focussing on the idea of the border as a place to dwell, this work is an attempt to grasp the « making » of space and the conditions in which it occurs. I will try to illustrate how these notions are rendered tangible in Dodgson's oeuvre.
 

An understanding of the crisis of modernity (or of modern science and art) cannot be pursued without looking closely at the transformations of Western metaphysics. And geometry occupies a central position in the development of western philosophy, both in the way geometry tends to be related to the expression of ideas--its inextricability from language-- and in the way we understand these ideas, somehow, in geometrical terms⁽¹⁾. I use the word geometry in a specific sense that has to do with our spacio-temporal existence as humans. I will try to clarify this notion with examples from Dodgson's work.
 

In the modern epoch, geometry was gradually transformed by the development of infinitesimal calculus in the 17th and 18th century and, ultimately, the new geometries that develop in the 19th century and begin to be applied in Dodgson's time. These are signs that render manifest the changes in the notion of perspectival space. The relation between the image and what it represents is gradually transformed. The situation of contemporary art-- and from my angle, architecture-- can be highlighted by looking closely at the problem of architectural representation.
 

In the scientific milieu of the 19th century, Euclidian geometry was not rejected in an outright fashion. Rather, it became a particular instrument, one of many, within a broader an more generalized geometry that claimed to be explaining every phenomenon. The very understanding of what the Euclidian principles were also tends to be misconceived. Today , they tend to be understood as stiff, instrumental and systematized explanations of reality, while non-Euclidian geometries are seen as opening new realms of form that supposedly have more to do with reality and a fair explanation of the universe.
 
 
 

Why was Charles L. Dodgson defending this old and dusty Euclid who had nothing to do with the world, its progress, its future ?
 

Some geometricians, Dodgson being one of them, were then expressing their deep belief in the somehow « original truths » of Euclidian goemetry, precisely because they preserved a link between the world of lived experience and the realm of ideas, and therefore could prevent the self-referentiality of the scientific quest⁽²⁾. In his book, Euclid and his Modern Rivals, Dodgson clearly expresses the importance of Euclidian geometry as a basis for our perception of the world. He speaks of a world in which parallel lines never meet. He criticizes the concepts that were then developing; these concepts acknowledged infinity as part of the world and considered numbers as mere instruments emptied of any symbolic value. Why would Dodgson be so resistant to the non-Euclidian geometries ? I do not think that he was simply rejecting them. I would claim that for him they were essential to an understanding of the world but at a poetic or fictional level.
 

Hence, Dodgson's work as a mathematician and a geometrician-- a defender of the principles of Euclid-- is very important in order to understand the other side of his work, his fiction books. Both sides of his work are intimately related. Putting them in parallel through a careful reading, they might eventually shed light on each other.
 
 
 

What is the space of Euclidian geometry?

A story about time and space.
 

For the pre-Cartesien thinker (as well as for some thinkers after Descartes⁽³⁾ who were reacting against his ideas), the geometric figures were not, in their ideal essence, part of the world of lived experience. They related the absolute and irrevocable truths of the world of Ideas. Like numbers in the realm of mathematics, the figures of geometry expresed, by analogy, the order of the cosmos, the order of things, and in that sense, revealed the harmony that governs nature. But there could not be such a thing as a perfect triangle either in nature or in the artifacts of men, for the human world remained mysterious, uncontrollable and overwhelming. In this chaotic world, space was not a preexisting and autonomous entity. It had to be ordered, created but mostly, it had to be kept alive and recreated now and then. Space and time were works of art.
 

In some ancient mythologies, space and time were gods that could not be separated-- they were two expressions of the same order⁽⁴⁾. Today, the big revolving door of time is out of its joints⁽⁵⁾, off its hinges, fragmented, dislocated, no longer ordered into a circular movement in space according to the cardinal directions. Time, as we now conceive of it, is the linear time of modern history. The line of time is an infinite line in both directions.
 

Space and time have lost their qualitative aspects. The so-called « Cartesian space » is homogeneous, infinite, a mere quantity or combination of coordinates. It is a quantity that can be mesured and reduced to planes, horizontal and vertical, and their intersection. Cartesian time is a quantitative and autonomous notion that is no longer dependant on-- or predicate to-- space.
 

In the modern mindset, the notion of infinity is gradually appropriated by man. As the possibilities of knowledge become infinite, the world of man is also extended to infinity. Once infinity is part of the world, the desire to describe not only the simple figures of Euclidian geometry but all the possible figures that are in between these ideals (in the conic sections), gradually appear to the geometrician. To what results the desire to describe the whole creation in algebraic terms⁽⁶⁾. The universe, according to this conception, is there to be deciphered and understood through mathematics and scientific experiments. The language of science attemps to resolve the distance between the words and what they describe. Gradually, the language of poetry looses its value of truth.
 

Despite the changes implied by this modern conception, these notions are expressed in metaphysical terms up until the 19th century and they remain at the level of Ideas, of discourse and human knowledge until they begin to be tangibly applied⁽⁷⁾. In the context of the scientific milieu of the 19th century, the scientific utopia becomes reality. Scientists, each in their own specialized and autonomous disciplines, all participate with great enthusiasm in the scientific endeavor; for them, it is the only possibility of finding the true nature of the world.
 

The scientific endeavor becomes frightening once its abstract conceptions are equated or supplant lived experience. The real that we know through mathematical models is an approximation of reality that claims to be more real than experience itself. In Sylvie and Bruno, the image, the map of the country, becomes the country itself for the German professor and this, as we recall, has certain inconveniences.
 

The industrial revolution is the theater of an unlimited technological innovation that leads to the development and building of machines at a disorienting pace : utensils that render the life of men easier.
 

This new status of technology has had an important impact on architecture. Buildings also become utensils. They have nothing to do with either their location or any influence of the action of time, the elements, aging-- products of a utopia of universal and atemporal architecture. In the process of the building project, the drawing become a precise instrument of measure. The published picture, the image of the finished building tends to stand on its own. Outside of the realm of imagination, the image, as means of presentation, reaches its end.
 
 
 

History and time, linearity and cyclicality.
 

In a conception of space as something that preexists, the wall becomes a denuded limit that subdivides space. The notion of temporality and the evanescence of things are eclipsed.
 

In the linear conception of time, our epoch becomes historically bound. There was an origin and there will be an end. Events, facts, are isolated entities that occupy a certain point along the line. In this objectified conception the present is no longer vivid and meaningfull. But what about our experience of time? Is it only linear? What we experience everyday is time in its cyclicality as it is rendered tangible by light and shadows, by movement and rhythm.
 

In Lewis Carroll's work, this cyclicality is always present. I read it in terms of time-space fragments. The spaces that Alice experiences are always different expressions of the same space. These spaces oscillate between worlds that are different but equally real. The limit, the borderline between these worlds, contains-- or is , itself-- another world. The limit cannot be reduced to a plane, it expands itself. This limit is actually where things happen-- a trace of the passage of time.
 
 
 

Is this limit a wall ? What is the nature of doors ? What are the consequences of the homogeneity of time and space ?
 

In homogenized space, horizontality, verticality and depth are equivalent, they don't have any particular qualitative aspect. A tower is the same as a tunnel. In The Vision of the Three T's, Dodgson criticizes the modern conception of space as homogenous, removed from time and in which gravity doesn't matter. The architect comes to the construction site dressed in an outfit that he claims is atemporal-- completely outside the questions of fashion-- with ribbons that defy gravity. He can find inspiration in a piece of stilton-- the materiality of the building is irrelevant. Cheese or stone, its all the same, only form matters⁽⁸⁾.
 

When time is removed from space-- like in basement spaces lit by artificial light (the first space into which Alice evolves after falling down slowly in the tunel)-- space becomes frightening as if it was created by something overwhelming and horrifying-- something entierly « other ». Progressively, Alice returns to the surface, reconquers space and time and creates spaces that allow for things to happen-- in between spaces.
 
 
 

How does this creation of space occur ?
 

In the same way reality cannot be objectified, man cannot be reduced to a subjective seeing machine. It is amazing how in Lewis Carroll's work one can find clues about the essence of space and the way its existence lies in embodied perception⁽⁹⁾. The perception of space has to do with someone implicated in his surroundings, his state of distraction or concentration. But mostly, it has to do with the postures that the body adopts in movement, the mood, the bodily humours, humour. The perceiver is not in space. Space does not preexist. The perceiver, like Alice, is actually and literally creating spaces-- a succession of time-space fragments which cannot be isolated but one continuously becoming.
 

In Lewis Carroll's work, the characters of the books are continuously transforming themselves-- successively becoming other. The same phenomenon happens to the reader. As Octavio Paz writes it in The Bow and the Lyre, Lewis Carroll's prose, in the way it is a rhythmic creation of images, becomes poetry. The reader is constantly recreating these images, following the rhythm of the poem that not only affects his imagination, but puts his whole body into a different posture. According to Paz, this allows for the space of the book to come to the world. Even though in Lewis Carroll's work almost everything is expressed through nonsense, the spatial-rhythmic dimension of his writing conveyes a deeper meaning.
 

For poetic language to remain outside of any methodological application, it has to speak about something common. Poetry does not follow a linear path, but one that is discontinuous and fragmented. Poetry is not limited to literature and art. Even the modern scientific mindset, even technology can be imagined as this potential for poetic expression rather than as a burden. This is, I believe, what Lewis Carroll was trying to say.
 

Lewis Carroll wrote for children. When I say for, I don't mean it in the sense to the attention of, but I mean, as Gilles Deleuze puts it, on the behalf of children. What they have to say can reveal the essence of things. In the same way, one could say that Lewis Carroll wrote for Euclid, on his behalf, trying to express the essence of geometry, its unquestionable truths. In the Alices, geometric figures become characters. Space and time also become characters, they speak to us--revealing to the reader their original union. Euclid finds, in Wonderland, a retreat where he can escape from his modern rivals and possibly enter into a dialogue with the new geometries.
 

Can there be such a dialogue induced within architecture ?
 

If time is conceived in terms of movement and rhythm as it is rendered tangible by light and shadows, then light cannot be conceived simply as another material for the use of the architect. It can easily be displayed in an ostentatious way that reduces it. And even though space is bound to time and to human perception, the materiality of the building does not disapear-- it is what one perceives. Things come together is a kind of non-fixity, a flow, a tide... The rhythm is created by a succession of aggregates-- concentrations of matter-- and silences. The rhythmic-matter is continuously transforming itself.
 
 
 

Inhabiting the limit
 

The notion of the limit between things is important in order to conceive of notions that are no longer caught in a dialectical thinking. Between fiction and the real, between day and night, in the space of ambiguity, opposites come together and our perception becomes what it always was : a hallucinatory experience. The limit, the border, is a world of possibles.
 
 
 

Working within the limits of language, Lewis Carroll creates a new language in order to express the lingering questions of humanity. They sound, they appear completely new, newborn. Language ceases to be a fixed system but can be conceived as continuously groing into something else-- a language that is alive. Words enter in a dance, they play and sometimes they eat each other (Snark). A story lives when ever it is retold and a building comes to life with its inhabitants.
 

Both sides of Dodgson's work come together in the way he criticizes the Victorian society, the scientific mindset and its fallacies. Created through nonsense and humour, his work is resistant to a complete analysis or demystification and cannot be exhausted by the characteristics of any literary movement. It oscillates within the thick limit that seperates Romanticism⁽¹⁰⁾ from Surrealism. Is it outside of history ? I think it is speaking precisely about our historicity. Humour could be a way to understand the essence of things.

Illustrations:
 

fig. 1 The platonic bodies

fig. 2 The platonic bodies imbricated in the dodecahedron

fig. 3 The geometry of the mind, from Luca Paciolli's tretease (name and date)

fig. 4 Illustration of the conic sections according to the principles of Desargues.

fig. 5 Vertical City, Ludwic Hilberseimer, 1924.

fig. 6 Rococco doors, 15th century.

fig. 7 The Brooklyn Giant, photo (name and date)

fig. 8 Silent witnesses, John Hejduk, date

fig. 9 Silent witnesses, John Hejduk, date

fig. 10 Retreat masque, John Hejduk, date

fig. 11 Installations on Berlin, model, date

Fig.12 Daniel Liebeskind, (project and date)

1. ¹In his book Les origines de la géométrie, Michel Serres expresses how geometry remains outside of cultural differences, of dogmas and in the same way, outside of singular scientific moments. In that sense, geometry is common to humanity. But the logos it measures remains mysterious and, somehow, original to all origins.

2. ² In Euclid and his Modern Rivals, the principal argument of Dodgson is that, in their inductive or even abductive reasoning, the scientist are not working with the reality of lived experience anymore, but, in fact, creating this new idealized real through their hypothesis. Their argumentation is caught in a circular process where the means become the ends and the premises orientate the whole resolution of the problems. The singularities of life tend to be flattened in this will to come to a general formula of the world.

3. ³ Even for Descartes himself, figures remained at the level of Ideas, the only certain, clear and distinct reality. When I speak of Cartesian space, I refer to the generalized conception that appeared after Descartes.

4. ⁴ The images of the antique labyrinth are quite telling in regards to this union of time and space. The labyrinth is circular and is bound to the space created by the dance and the rhythm of this dance. There is an entry and an exit, a beginning and an end, still, it is speaking of this constant « being lost » of life itself. The modern labyrinth can be imagined as an infinite line as it is admirably described in Borges' work.

5. ⁵ Shakespeare's writes in (...): "Time is out of joint". The poet expresses the transformation of time in the modern world.

6. ⁶ Algebra becomes an absolute language and its signs are not referring to reality anymore. It is an abstract language in which numbers do not have any symbolic values. Even infinity (8) becomes a number for the mere end of solving mathematical problems.

7. ⁷ This might have to do with the fact that most people could not, before the 19th century, conceive of infinity as part of the sub lunar world. For two reasons: first, infinity was associated with God and only God was infinite in his essence and perfection. If his creation could be conceived as such, the human understanding of it could never claim to be infinite. The other reason could be that a conception of the world as infinite, of infinity in matter and substance, is antagonistic to lived experience and to an embodied perception of the world, as the world we can experience and perceive is actually finite.

8. ⁸ It is interesting to read here the influence of John Ruskin, for whom the importance of the human sensibility meeting the work of art was primordial. Ruskin was defending such an attitude agaist pragmatism. For him the materiality of architecture was primordial-- one should ask the stone what it has to say. A stone could tell the story of how it was crafted, reveal the passage of time upon its face.

9. ⁹ As M. Merleau-Ponty explains it in Phenomenology of Perception, our experience of height is very different from that of horizontal distances and depth has to do with movement not being perceived visually but (to some extend) with all the senses (touch, smell, earing...taste). The link between perception and reason (body and mind) and between man and the world, has to do with our temporal existence.

10. ¹⁰ In The bow and the Lyre, Paz expresses this idea of a romanticism that is not merely nostalgic of the past or a reactionary attitude against the industrial revolution and the scientific mind set, but a romanticism that is trying to reconcile the mythos and the logos. Such movements as Romanticism and Surrealism are visions of the world that can travel underground, through history, and reappear whenthey are least expected.