Duchamp  and  The  Allure  of  the  Fourth  Dimension  

Dr Tom Loveday

Art Association of Australia and New Zealand Annual Conference

“Alpha Alpha Alpha November Zulu,” November, 2008

Griffith University, Brisbane, Australia

Introduction  

There have been many explanations of what Duchamp may or may not have

read, studied or known.1 I will be using only two texts from a complex web

that have been connected to Duchamp. They are Claude Bragdon’s (1866-

1946) A primer of Higher Space and Jules Henri Poincaré’s (1854-1912)

Science and Hypothesis.2 But I am not arguing, as researchers have done,

that these are definitely and directly connected to Duchamp’s thinking. In

that sense this paper is speculative or theoretical rather than historical.3 I am

using the ideas in these texts to suggest a line of investigation and experiment

rather than to draw historical conclusions.

This paper argues that to understand Marcel Duchamp’s claim that the Large

Glass is a four-dimensional work, the variable use of motifs, such as shadows,

must also be understood.4 The result should be a better grasp of the

attractiveness of the fourth dimension and why it was no longer so attractive

by the 1950s and why Duchamp’s explanations of that time seemed so off-

hand.

1 This paper is not seeking to attribute motives to Duchamp, nor to claim that Duchamp was intimately aware of any of the work mentioned but rather to try to come to terms with some of the more sophisticated thinking through which his work has been viewed at the time it was made. 2 Claude Bragdon, A Primer of Higher Space, Alfred A Knopf, 1938 (first published 1913) and Henri Poincaré (WJ Greenstreet trans), Science and Hypothesis, Cosimo Classics, 2007(first published 1905). 3 I am investigating the concepts in the context of Duchamp’s work in the same as is found in Gavin Parkinson, The Duchamp Book, Tate Publishing, 2008. 4 Marcel Duchamp, The Bride Stripped Bare of Her Bachelors, Even, also known as The Large Glass, 1915-23.

2

What  is  The  Fourth  Dimension?    

The fourth dimension is fundamentally a geometric concept. However, it is

also part of the general cultural movement of the European Enlightenment

towards a correlation between reason and experience and thus to find

“truth.”5 The search surrounding the fourth dimension is a search for evidence

of a geometry concept “in the world.” However, the fourth dimension

emerges from a consideration of the third dimension, a geometric concept

already well accepted as correlating with experience.6

The logic of three dimensions apparently correlates well with experience fo

the world. This correlation is called shape and is generally equated with form.7

One dimension correlates with a line in space, two correlates with planes in

space and adding a third provides a geometric explanation of the volume

and shape of objects in space.8 Placing the three dimensions as far apart as

possible, at 90 degrees, appears to isolate them so that they don’t interfere

with each other’s quantities.

5 The fourth dimension is one of many ideas that reveal a discrepancy between experience and representation. Others include: perpetual motion, electricity, magnetism, hypnotism, séance, the afterlife and auras. 6 Dimensional numbers, three-, four- and n-dimensional geometry should not be confused with non-Euclidean geometry, curved space and so on. 7 The equation of form and shape is one of the essential ideas of modernist thinking. Non-modern thinking tends to have a looser understanding of this equation. 8 This idea underpins several Bauhausian theories. See: Wassily Kandinsky, Point and Line to Plane, Dover Press, UK, 1979.

3

The notion that that are three dimensions leads to a feeling for linear

numerical progression that might extend the geometric description of space

and objects. There are three dimensions therefore there could be four

dimensions. This then leads to further extensions, using a form of “inductive”

reasoning, in which there could be 5, 6, 7 or indeed, “n” dimensions.

The third dimension of correlated with the experience of “depth.” The fourth

dimension is likewise correlated with the experience of “movement.” The

spatial implications of adding another dimension must be correlated with

measuring a movement, as they are with the other dimensions. The fourth

dimension completes its natural extension from the other geometrically

determined dimensions.

4

In the search for a truth based on the correlation of rational geometry with

experience, this leads to the idea that movement actually is the fourth

dimension. This is further refined to movement as a difference in position of

objects, which, in its most abstracted form becomes time.9

Theoretical physics as a well as geometry then pursues the inclusion of time in

linear geometric models of the cosmos, eventually arriving at Einstein’s two

theories of relativity.10 In science there is a tendency to attach a moral value

to scientific truth, which then extends to the structures, geometric of otherwise

from which scientific truth is made. For example, objectivity is seen as an

ethical stance in science.11 However in popular culture, natural, scientific truth

is merged with spiritual truth that reappears under a new-age style of

mysticism, such as that of Madam Blavatsky. Mankind’s destiny emerges as a

natural movement above the actual world to a new level, a higher, more

enlightening plane or space by grasping geometric truths.12

But mankind is also bound to “reality.” In his definition of space, architect and

writer Claude Bragdon (1866-1946) assumes that three-dimensionality is

“reality”, at least as far as it can be imagined. In other words, he merges

analogy with reality using a mysterious occult style of argument, to find

“truth.”

What we think of as space is more probably only as much of it as is imaginable. It

may be that our space bears a relation to infinite space analogous to that which

the images of a magic lantern bear to the wall on which these images are made

to appear—a wall with solidity, thickness, extension in other and more directions

9 This can be compared with Kant’s idea that the form of the world is space and of the self, is time. See Immanuel Kant, Critique of Pure Reason. Using the Kantian intuition, one could say that the fourth dimension places the Kantian form of the self in the world and as such is a reconciliatory concept. 10 Albert Einstein’s (1879-1955), “Special Relativity,” (1905) form which comes the well-known, “E = mc2” and “General Relativity,” which sets out a general theory of the cosmos in terms of gravity and space-time (1916). See, Albert Einstein, ON THE ELECTRODYNAMICS OF MOVING BODIES, in, Frances A Davies, The Principle of Relativity, Dover Books, 1952. 11 My paper, The War on Bullshit: Art, Philosophy and Power, a Scatological Approach, Association of Art Historians conference, London 2008, deals with this in more detail. 12 All of the models of the fourth dimension rely upon geometric explanations that assume the acceptance of the notion of a universe, or single cosmos. The reduction of the cosmos to a single concept, or uni-verse, provides that dazzling thought that all of it, at a vast scale, is already out there, waiting to be discovered, consistent, impartial and geometric.

5

within the wavering circle of light which would correspond to our sense of the

cosmos.13

Shadows play more than a single metaphorical role in Bragdon’s spatial

consciousness. Through shadows intuition of the higher dimensions is

apparently possible, as Bragdon writes,

Man, a higher-dimensional entity is pursued by the necessity of representing

himself through his shadow or personality in a lower dimensional world.14

Bragdon’s fourth dimension retains the scientific passivity of the objective

observer, despite its mysticism and as such is a cosmology imbued with the

ethics of science, even though Bragdon was an architect.

In a completely different direction (from an actual scientist) is the idea that

dimensions are a resolution of complex sensory experience into a simpler,

more manageable world. Within such a reduction, the fourth dimensions

constitutes a failure to reduce spatial cognition to its most simple form and in

which to find a way to see a fourth dimension leads to a breakdown of the

capacity to adhere to reality and thereby to lose one’s “being.”15

Jules Henri Poincaré’s (1854-1912) fourth dimension is an active bodily

function, rather than merely the passive effect of experience in the objective

observer. Poincaré’s active bodily function produces a spatial framework that

Poincaré calls “motor space.”16 Thus the balance, movement and muscular

force all play a part in tactility.

Each muscle gives rise to a special sensation which may be increased or

diminished so that the aggregate of our sensations will depend upon as many

13 Claude Bragdon, A Primer of Higher Space, Alfred A Knopf, 1938, p 15, “The Fourth Dimension Defined.” The wavering circle of light is a reference to Plato’s cave simile from Republic. Most of Bragdon’s geometry, if not mysticism is drawn from P D Ouspensky, Tertium Organum: The Third Canon of Thought, a Key to the Enigmas of the World. 14 Claude Bragdon, A Primer of Higher Space, Alfred A Knopf, 1938, plate 12, the second part of the text is presented as a series of hand drawing plates, including this text with a drawing of a figure pursued by its shadow. 15 There is a connection here to the postmodern notion of de-subjectivisation or dislocation of the subject that cannot be pursued but which might make a connection between the Large Glass and postmodern art through Duchamp’s reading of the fourth dimension. 16 Henri Poincaré (WJ Greenstreet trans), Science and Hypothesis, Cosimo Classics, 2007, p. 55.

6

variables as we have muscles. From this point of view motor space would have as

many dimensions as we have muscles.17

Motor space is reduced to representational space in the imagination, as if

motor space were becoming visual space in one’s imagination. For Poincaré,

muscular sensations are reduced and merged with conceptual

understanding of self and space. For Poincaré, vision does this in order to

minimise the criteria for perceptual judgments, decreasing complexity and

increasing perceptual speed and accuracy.18

These two versions of the fourth dimension are not necessarily an opposition

but certainly they come from distinct and incompatible and yet popular

points of view about the correlation of geometry and experience in

Duchamp’s world. But there were ways, using geometric concepts, to find

common ground between the two. One of these is “affine” transformative

geometry.

Affine  Transformation  and  Shadows  

The mathematician Christian Felix Klein (1849-1925) invented a system for

categorizing geometry according to transformations using the “affinity” of

similar shapes.19 Affine transformations describe changes to bodies and their

space according to certain mathematically defined methods, such as

reflection, rotation and translation. The similarities and differences, as

described by transformations, can be categorized according to the particular

transformations used.

Transformations that involve translations do not change line, length or angles

between lines and so retain shape and size. These transformations are

17 Henri Poincaré (WJ Greenstreet trans), Science and Hypothesis, Cosimo Classics, 2007, p. 55. 18 Henri Poincaré (WJ Greenstreet trans), Science and Hypothesis, Cosimo Classics, 2007, pp. 51-71. 19 David A. Branna, Matthew F Esplen and Jeremy J Grey, Geometry, Cambridge University Press, 1999, pp 45-93. It should be noted here that dimensional models of space and non-Euclidean geometry are not necessarily of the same kind. N-dimensional space is often thought of as linear and Euclidean. Non-Euclidean space does not necessarily concern itself with dimensions and their representation. In that sense, the cross over from affine geometry to four-dimensional space is original, at least as far as I am aware.

7

grouped together as Euclidean, which is to say conforming to Euclid’s laws of

geometry based on length and linear effects. These transformations appear

only as movement.20

Transformations that retain relative position but not angle and length can be

grouped as “affine.” Curiously Duchampian, the term affine refers to affinity

between shapes rather than any unity of similarity or difference, much like the

relationship between an object and its shadow, still the same but different

and definitely belonging together.

The point about affine translations is that they can be accurately represented

using projective geometry, including perspective, a topic that interested

Duchamp.21

In Affine translations which use planar rotations, linear transpositions and

reflections, objects change shape and size but retain some geometric

properties that have the effect of retaining similarity or affinity with their

original. For example, an object and its shadow is an affine transformation

(actually a complex one) from one set of spatial conditions to another. In a

shadow the number, angles and lengths of lines are changed while the

conformation of the shape remains.

As such, it might then be possible to represent a particular four-dimensional

object, such as a hypercube, using shadows as if this were an affine

transformation. (I am not proposing to do this mathematically here, but

instead use visual explanation of a hypercube.)

A hypercube is a familiar geometric shape for those interested in the fourth

dimension. However, in perspective, the hypercube resembles three cubes

joined face to face, in a line:

20 The representation of four-dimensional objects appears as a problem of representing movement, in these kinds of transformations. 21 Duchamp attitude to perspective and representation is covered thoroughly in Linda Dalrymple Henderson, Duchamp in Context: Science and Technology in The Large Glass and Related Works, Princeton University Press, 1998.

8

Indeed this image would be taken for the linear arrangement, if the idea of

an hypercube were not raised. This leaves us with two possibilities for this

image, indicated with a slight rotation of the representation.

First, three cubes in a row:

Second, a hypercube:

9

Seeing four-dimensional shapes then becomes a matter of “getting the

knack” of seeing two-dimensional representations in a certain four-

dimensional way. Undoing the conventional reading of the diagram above to

reveal a hypercube is what Poincaré meant when he said that a discrepancy

between “muscular convergence of the eyes” and “accommodation” in

space would lead to seeing the fourth dimension. The failure to make a

correlation between the visual experience of space and the feeling that one

is in space is what Poincaré implies is the experience of the fourth dimension.

So here we have both Bradgon’s higher plane and Poincaré’s failure to see

three-dimensional “reality” and an ironic tension of opposites, just as

Duchamp often used in his art.22

The question then is why would we see one or the other of these geometric

arrangements in such an image. Poincaré argued that we choose to and do

so because of the perception of conventional consistency between physical

spatial concepts and sense experiences. If we add a shadow to the

perspective, the linear arrangement of cubes is either eliminated or

confirmed according to correlation between the shadow and the shape that

cast it.23

The Linear cube with a shadow:

Hypercube with a shadow:

22 See, Francis M Nauman, “Marcel Duchamp: A Reconciliation of Opposites” in Thierry de Duve(ed), The Definitely Unfinished Marcel Duchamp, MIT Press, 1992, pp. 41-67 23 This also means that the movement used earlier to reveal the geometry of the object is no longer necessary.

10

The shadow indicates the choice of figure one sees, linear cubes or

hypercube and, as such, three-dimensional or four-dimensional. This is

because the shadow provides a spatial consistency in which the image is

reduced to a single geometric concept and as such makes it seem more

“real.” This might lead to the suspicion that to see The Large Glass as four-

dimensional requires that it cast a shadow.

The  Four-­‐dimensional  Large  Glass  

11

Marcel Duchamp, The Bride Stripped Bare of Her Bachelors, Even, also known as

The Large Glass, 1915-23

Duchamp said, in an interview with Pierre Cabanne in 1966, “Simply, I thought

of the idea of a projection, of an invisible fourth dimension. Something you

couldn’t see with your eyes.”24 The “projection” was of a four-dimensional

object into three dimensions, just as a shadow of a three-dimensional object is

a two-dimensional shape, like a shadow.25

The argument by simile is based on extension in number; four is like three by

extension, much like inductive reasoning in formal logic. The commonly

expressed explanation is that Duchamp is arguing by simile, analogy or

metaphor. Strictly speaking, the argument appears to be by a simile because

four dimensions are likened to three, but it could equally imply actual

shadows.

It might also be possible that the explanation using “shadow” is describing

both a simile and an artistic effect and as such is a “shifter,” a concept from

linguist Roman Jakobson, as described by Rosalind Krauss.26

Krauss  and  the  shadowy  “Shifter”  

Rosalind Kraus writes of the “shifter,” which means a word that is “filled” with

meaning in specific situations, but otherwise empty and ready to be filled.27

Words such as “this” when attached to nouns, such as in “this table,” or “I” as

in “I am,” make the second term specific. In visual imagery, it should be

24 Pierre Cabanne (Ron Padgett trans), Dialogues with Marcel Duchamp, da Capo Press, 1971, p. 40. This same statement appears in a number of other places including interviews and texts and is often used to explain Duchamp’s “position” on regard to the fourth dimension in The Large Glass. It is often quoted in research about Duchamp and the fourth dimension. 25 Duchamp was also interested in mirrors and mirroring, especially the idea of reflection of four dimensions in a three-dimensional “surface.” However, this idea does not occur in regard to The Large Glass. 26 Rosalind E Krauss, The Originality of the Avant-garde and other Modernist Myths, MIT Press, 2002. 27 Rosalind E Krauss, The Originality of the Avant-garde and other Modernist Myths, MIT Press, 2002, pp. 196-209. From the linguist Roman Jakobson (1896-1982).

12

pointed out that this repeatability yet individuality is similar to icon, a point

used by Krauss later in the text.28

Shifters shift their position according to that which they are attached. For

example the shifter “I” shifts between the voices in a conversation according

to whoever is speaking. The “I” in “I am” is claimed by whoever utters the

phrase.

A shifter, such as “I,” is filled at any one time with a range of characteristics

that can be ordered using an “index.” Indexes provide actual frameworks in

which things leave traces, such as names, footprints, and shadows.29 It seems

that without the presence of indexes, shifting cannot properly occur.

Krauss’ index is material, actual and in the realm of the real. By using the

distinction between “Symbolic and Imaginary,” such as the difference

between language as a past and its use, Krauss provides the index of physical

presence, or the trace of things. The trace of things, such as actual cast

shadows “…serve as the indexical signs of objects.”30

Krauss then uses Duchamp’s Tu m’ to illustrate the point. Shadows are cast to

indicate the presence of objects, as their index. Krauss then tells us that the

title of the work, Tu m’, or “You me,” presents us with two shifters, which also

form a kind of “index.” Shadows and specifically the shadows in Tu m, are also

shifters because they can be filled with various purposes, either aesthetic or

linguistic, which is as metaphors or as artistic effects by contributing to

representations of objects.31

28 This idea is the reverse of Walter Benjamin’s aura of the individual artwork, as describe in “The Work of Art in the Age of Mechanical Reproduction,” which can be found in Richard Kearney and David Rasmussen, Continental Aesthetics: Romanticism to Postmodernism, An Anthology, Blackwell Press, 2001, pp. 166-180. 29 Krauss provides examples of the breakdown of shifters in descriptions of psychological dysfunctions within which the claim for a shifter is not grasped according to indexical conventions. For instance, “I” is confused with “him” in autistic children and so the shifting of the term is disrupted and meaning begins to become confused, which Krauss understands a failure of identity. 30 Rosalind E Krauss, The Originality of the Avant-garde and other Modernist Myths, MIT Press, 2002, p. 198. 31 A number of authors have noted contradictions in Duchamp’s explanations of his works. One of the clearest examples of this is the contradiction between early explanations of the strong relationship between the Readymades and The Large Glass against the almost non-relationship that Duchamp later claimed, as described in, William Camfield, “Marcel

13

Marcel Duchamp, Tu m’, 1918

But the shadows in Tu m’ are representations, not actual cast shadows. This

means that the objects are indexical only within the representation. To move

beyond the limitation of representational space, an actual object is attached

to the canvas creating a contradiction between painted and actual

shadows, if the light is right. Irony persists when the difference between an

image and reality is structurally challenged in the work, which is to say when

the index appears as actual in relation to representational. For example,

when the index appears as an actual shadow cast in the installation space in

relation to the representation space within the art object.

The  Shadows  of  the  Large  Glass  

Shadows are not shown in any of the photographs of The Large Glass that I

have found, so I have made a three-dimensional computer model, with

precisely calculated shadows from which I have made images.32

Duchamp’s Fountain, Aesthetic Object, Icon, or Anti-Art?” in Thierry de Duve (ed), The Definitively Unfinished Marcel Duchamp, MIT Press, 1992, pp. 133-171. The “shifter” provides a way to understand Duchamp’s mode of thinking in explaining his various works and parts of works. 32 Nor were they shown in the exhibition “Duchamp, Picabia and Man Ray” at the Tate Modern in London, April 2008. The Richard Hamilton reproduction of The Large Glass was lit to minimise crisp shadows.

14

Naturally, this image alone reveals no hidden secret. However, it does

demonstrate the spatial relationship between a representation and the

shadow of the object that does the representing. The shadow is of the

representing object and not what is represented. This means that there is a

spatial discrepancy between the shadow and the “content” of the work, of

the kind that Poincaré says leads to seeing the fourth dimension.

 

15

Conclusion  

It may well have been that because the fourth dimension was relegated to

kitsch popular culture, Duchamp felt that it had lost its seriousness and its

place in the intellectual world with which his art played. As such, the fourth

dimension could well have become an embarrassment, one that needed the

dismissal that his tone in responding to questions suggested.

Hyperlink to interview sound-clip.

There is possibly no conclusion to this discussion. Instead there might be

investigation in which The Large Glass is placed under lighting that casts

shadows to see what the effect would be.

I shall finish with the following quote from the late Jean Suquet (1928-2007)

who no doubt would have been amused by my peculiar twists and turns:

If the fourth dimension is to be treated only as a mathematical question, if one

forgets to color it with the sentiments at the heart of Duchamp’s endeavor, then it

becomes a Byzantine speculation indeed. But is sex is involved, if not an angel’s

at least a bride’s, a little flesh is returned to the skeleton of cosa mentale.33

33 Jean Suquet finishing the discussion after a paper by Craig Adcock, “Twisting our Memory of the Past “for the fun of it,” in Thierry de Duve(ed), The Definitely Unfinished Marcel Duchamp, MIT Press, 1992,p. 348.