F. DAVID PEAT

In 1993-94 I acted as a consultant to the National Gallery of Canada for an exhibition of the work of Escher. Although Escher’s status amongst critics and art curators is somewhat ambiguous, the work of this Dutch artist has attracted a considerable public, particularly for his play with visual and mathematical paradoxes. Escher’s interest in tiling, figure-ground ambiguities and the illusions of recession in three dimensions are particularly attractive to scientists and mathematicians and his work was exhibited at the 1964 International Mathematical Congress in Amsterdam.

The mathematician Roger Penrose also saw some of Escher’s work, in the company of his father, Lionel Penrose, a psychiatrist. Intrigued by Escher’s approach Roger Penrose sketched an impossible figure—the Penrose triangle—which Escher later incorporated into several of his works. After Escher’s death, the mathematician regretted that he had not lived long enough to take advantage of the discovery of Penrose tiling. Another mathematician who took a professional interest in Escher’s work was H.S.M. Coxeter.

The Ottawa exhibition came about as the result of a generous donation of works by Escher’s son who had emigrated to Canada. In preparation for the selection of works, the catalogue, and educational activities at the gallery, Brydon Smith from the National Gallery gathered us together for a highly stimulating series of meetings and discussions. Below are some notes arising out of these discussions. They are relatively rough and incomplete but, in view of the interest Escher continues to excite, they may be of interest to someone.

Escher: Visual Codes, Ambiguity and Perception

I want to discuss how Escher questioned and explored the visual codes and conventions of 19th century graphic art in a way which also connects, to some extent, to the approaches and sensibilities of scientists/mathematicians/ philosophers.

Escher’s art historical inheritance represents only one out of a wide set of possible conventions and areas of interest—visual codes, conventions and short-hand signs employed within a particular historical period to depict and ‘read’ an image.

In questioning our perception, experience and depiction of space Escher was sensitive to contemporary scientific concerns about the structure of space and the subjective role of observation. His visual experimentation echoes the mathematician’s concern with the way abstract formal structures are employed to represent reality.

In order to question the assumption—that may still be held by some—that the standard of art is an illusionistic mimesis of the real world one should point out the many possibilities open to the artist and endorsed by his or her culture—these can variously be decorative, ceremonial, magical, representational, narrative, abstract, aesthetic, etc. Each of these employs a variety of codes and paradigms that are understood by members of the society of the time and which enable them to ‘read’the image. Illusionistic realism is only one out of many artistic approaches.

1. Ritualistic and ceremonial art in which the artist and society is less concerned with naturalistic depiction than with the numinous qualities of an image and direct experience of a transcendental world of energies, powers and spirits;
2. Chinese scrolls in which the viewer’s eye can journey through a landscape, revisiting particular locations;
3. Medieval European art codes where space and time together are represented so that the various stages in the life of a saint can be visited;
4. Islamic art which emphasizes pattern, decoration and the relational properties of two-dimensional space rather than narration and illusionistic depiction.

There are also the various naturalistic conventions of the Renaissance for the depiction of solidity, form, distance and space. In particular the convention, or visual code, of perspective. Perspective, or to the mathematician projective geometry, is an excellent device for depicting the spatial structure and relationships of buildings, streets and cities. Yet this striking spatial illusion is achieved at the expense of the distortion of the actual size and shapes of three-dimensional objects and the subjugation of other pictorial conventions to a single overarching scheme. The invention of photography, in which a three-dimensional scene is projected by a lens onto a flat plane, happened to employ the same mathematical system of representation and led to a sort of confusion because it was originally a set of conventions for depicting certain aspects of reality became the touchstone for naturalistic representation.

Towards the end of the nineteenth century, the limitations of these particular conventions became increasingly apparent and led to a rapid sequence of movements in art. At the same time, similar questions were being posed in science about the nature of objective reality and the structure of space and time. In both cases the distinction was being made between reality itself and its formal conventions of representation. David Hockney, another artist more popular with the general public than the critics, has also questioned the conventions of perspective and photography, arguing that they are both very far from reproducing the actual experience of vision.

(Note: It would be important to know if Escher knew the work of the scientist, philosopher and linguist Alfred Korzybski (1879-1950). His book Science and Sanity (1933) had an enormous influence on many scientists and thinkers as well as the American Constructivist movement in art. In particular Korzybski coined the phrase ‘the map is not the territory’ and stressed that one should not confuse the physical nature of the signs, codes and conventions for what they represent.)

While, in his earliest works, Escher accepted the conventions and visual codes of the graphic arts he was also willing to push some of them to their limit by exploring sweeping panoramas, highly elevated vantage points, contrasts of extreme close-up and far distance, dualities of foreground and background, and the perspectival problems of depicting tall towers.

It goes without saying that the  major figures of twentieth century art have been more far-reaching in experimenting and creating new visual codes and conventions. Escher, by contrast, was content to explore in the manner of a scientist, mathematician or philosopher, within the general conventions of graphic art. His method was to work within art historical conventions for the depiction of space, solidity and dimensionality but by using such approaches as paradox and complementarity to explore one set of limits as to how we all read a visual scene. In so doing Escher was calling into question the whole meaning of codes and conventions, the participatory nature of our visual apparatus and philosophical questions about the nature of reality.

Below are a number of different ways in which Escher explored and exploited the conventions of graphic depiction.

The Plane

From the very beginning Escher was always concerned with the plane itself as the ground for his graphic work.

1. His experiences of Moorish art and his contact with mathematicians like Coxeter led him to explore the way repetitive shapes can be used to tile the plane. This approach allowed him to explore ideas of duality and transformation. While at heart a graphic artist, Escher also employed imagination and a dry sense of humor in ways that are not unlike those of a mathematician.
2. Working on the surface of the plane allowed Escher to come to terms with mathematical notions of limits and infinity.
3. Escher exploited the tension between the flatness of the plane and the codes of graphic art for depicting three-dimensional solidity by portraying intersecting planes and flat figures escaping from the plane.

View Points and Vanishing Points
In his earliest work Escher favoured striking viewpoints; later he began to explore the implications of different vanishing points within the same scene. (In this he echoes David Hockney’s recent preoccupation with multiple vanishing points and reverse perspective.)

Solidity and Illusion
His exploration of perception and visual codes included the tension between a shape outlined on a flat plane and, through the addition of graphic marks, the illusion of a solid object.

Dualities
Escher’s preoccupation with dualities is a constant presence in his work. (It would be interesting to know of Escher’s philosophical reading in this field). Dualities are expressed in foreground/background; light and dark; flatness and dimensionality; representation and decoration; frame and scene; large and small; viewpoint and vanishing point; form and negative space; positive and negative; light and dark; observer and observed, as well as the metaphysical aspects of good and evil.

Some Visual Themes in Escher’s Work

In approaching Escher I am looking for some integrating vision, an overarching concern or impulse that is common to his work. Escher clearly had a technical dexterity but this was combined with painstaking observation and execution that at times seems almost obsessive. While his interests seem to anticipate issues at the forefront of the sciences he appeared indifferent to the currents of art that surrounded him. Thus, he seems to have accepted his medium and made no attempt to extend or experiment with its physical techniques.

His early works are the result of careful observation, and an attempt to approach the actuality of the surface appearances of the world. Yet he does not attempt to enter into their inscape, their inner worlds. Neither does he attempt to rearrange, invent, abstract, extend or to color work by his own emotional reactions. The only major elements of subjective response appear to be his interest in extreme contrasts of light and dark, of dualities and of high vantage points and sweeping perspectives. In this he has the sort of controlled and disciplined romanticism that is more characteristic of the scientist.

Other artists may be concerned with exploring and depicting the essential uniqueness of an individual object—Cézanne’s apples is a noted example—both in its appearance and its underlying life and inscape. Escher seems less concerned with the particular object before him than with what it represents, how it is a manifestation of underlying laws and principles of nature. Again, in this, Escher seems to me closer to the scientist who seeks the general in the particular, than to the artist.

Throughout his work Escher appears concerned with conventions, visual codes and schema that can be used to depict solid objects and three-dimensional spaces on the flat surface of the paper. In this sense he can be placed within an Art Historical context for Escher is constantly aware of the limits of the conventions of depiction. As an example, a solid object that can be touched, explored and walked around must be distorted in order to be portrayed illusionistically (on paper or canvas) as inhabiting three-dimensional space. This problem of the dominance of depiction from a single viewpoint was also the concern of Picasso and Braque who explore the alternative Cubist convention involving a superposition of multiple viewpoints.

Although he was not a professional mathematician Escher appears to have been interested in the laws relating to the processes of depiction, as with the actual structure and potential of the plane on which his images were made. It is as if Escher felt that by uncovering these laws he was also coming to understand something of the inner structure of the universe. Thus Escher constantly returns to questions of transformation and distortion, limitations of perspective, impossible objects, solid and flat, the question of a plane being in front or behind an object, etc.

Some of his explorations include:

1. Reflections
Reflections in Plane Surfaces—water and mirrors
Escher explores the dualities of positive and negative space; of light and dark; foreground and background. Is the distant space a magical world deep down in the water or the reflection of a world behind the viewer.
Escher also explores Reflections in plane mirrors and their relationship to objects placed around the mirror; the paradoxes of mirror worlds; of the worlds in front of and behind the mirror. He suggests that mirrors are windows and the mirror frame can be a play on the picture frame. The problem of the frame is also explored by using books at a window as a frame for the street outside.
There are several technical studies of distortion in a convex reflecting surface. These include the artist contemplating his image. This self referential concern relates the act of the artist looking at himself and drawing himself and to pictures of hands drawing themselves. (Reflections in convex surfaces—droplets, eyes, polished spheres, eyes.)

2. Self-reference
The question of self-referential images resonates throughout Escher’s works—reflections of the artist, hands that draw themselves, the visitor in a picture gallery who looks at a print that contains himself, It is one of those elements in Escher’s work that attracts people from outside the visual arts for it relates to preoccupations of Hofstadter and questions of self-reference in logic, mathematics, biology, cognitive science and A.I.

3. Grids and marks on spheres and ellipsoids
Paradoxes relating to Art History and the depiction of solid forms on the plane. For example, are we looking at a regular grid drawn on a solid object, or a deformed grid drawn on a flat object?

4. Patterns of marks as an optical code
In some of his earlier works Escher employs regular patterns of marks, lines or scratchings to suggest a sort of vertical plane that divides the immediate foreground from background or middle ground.
In one work he uses an ingenious curvilinear grid around a vertical tower. This has an interesting optical effect for it plays on the paradox of single point perspective—that a high tower should appear to narrow, yet our eyes tend to compensate for this and we continue to ‘see’ the walls as parallel.

5. Landscape and Perspective

1. 1. Art historical problems of using a visual code to depict and differentiate foreground, middle ground and background. This is complicated by Escher’s addition of extreme close-up i.e. leaves.
   2. Explores problems of depicting a large area of landscape. Escher sought out high places in order to portray his landscapes—cliffs, an aircraft, top of a ship, etc.
      In this sense he wished to look down upon a landscape and portray it as a plane with less distortion. That is, look down on a square field from and aircraft and it still appears square. But seen from the ground and in perspective the sides of the field are no longer parallel and the square becomes a trapezoid. Thus Escher plays with viewpoints in which the vanishing point is very high in the air or beneath your feet. Note that in Up and Down Escher creates an ‘impossible figure’—a print in which both vanishing points are used.
      Escher therefore appears aware of the duality between the ‘real’ shape of an object on the ground and its depiction in perspective. In a sense perspective distorts and is not ‘naturalistic.’ I wonder to what extent Escher was concerned with what could be called the ‘ethical’ problem of perspective and this may be why he explored, through paradoxical figures, the way perspective can distort reality.
      Escher is also aware of the art historical tension between visual schema used to depict a landscape—i.e. the early cartographic convention was to depict land and rivers as if seen from above, while houses, churches and streets were seen face on. Thus two viewpoints and two conventions were adapted within the one map/landscape. Escher is aware of these possibilities and plays with them.
      It is interesting that these problems of perspective were the concern of other artists. The convention of single-point perspective was satirized by Hogarth (I think it was an illustration he did for a book on perspective by John Kerby) who invented an impossible world. The ethics of the distortion of the world through perspective has been of concern to David Hockney with his use of reverse perspective, multiple viewpoints in his ‘joiners’, and with his V(ery) N(ew) series of paintings in the early 1990s.
   3. The Vertical in Single Point Perspective
      Escher was aware of the problem of depicting a vertical column or wall that faces you. There is a tension between the rules of perspective and the eye’s tendency to compensate for what it thinks it should be seeing. So should a wall or tower bulge, taper or appear regular? Escher plays with this in a variety of ways.
      The Church Tower, I find a particularly interesting example—although it is not visually striking as an image. Escher uses a curved grid of faint marks as a background to a high church tower which almost appears to twist as it rises. These marks play an interesting visual game against the strict perspective of the tower. Erase the grids and it would be interesting to see how one’s perception of the ‘naturalness’ of the tower changes.
      The centre of a building that ‘bulges’ as if seen through a magnifying glass may be related to this and to other concerns.

6. Escher’s long scrolls
These scrolls whereby repetitive dualities are in a process of constant transformation evoke interesting resonances. On the one level they are purely technical exercises of covering the plane and in the use of various symmetry transformations. But one can also speculate on its connection to the scroll in Chinese and Japanese art—I don’t know if Escher knew of such a tradition. In Oriental scrolls single point perspective is abandoned in favour of a journey through time as one moves along the scroll and reads it. In a similar way one abandons single-point perspective and moves in time along Escher’s scrolls—scrolls in which the end point is also the starting point.
Note how excessively static is Escher’s work, yet these scrolls demands that one views them ‘in time.’ This aspect of ‘reading in time’ as the eye moves along a changing landscape is also present in Escher’s portrayal of landscapes from a high vantage point.

7. Dualities
Dualities occur throughout Escher’s work—dualities of light and dark, positive and negative space, convex and concave, up and down, the foreground and a reflection of what is behind the artist, dualities of shape. In addition there are the metaphysical dualities of ‘angel’ and ‘devil,’ good and bad.
The duality between Frame and object—i.e. extreme foreground and background in perspective. And the sort of duality/self reference present in Picture Gallery.

8. Solid objects depicted on a flat surface
Escher explores and is aware of the art historical process of devising ways of depicting solid objects on a flat canvas through codes, schema in such a way that the viewer will decode these marks in unambiguous ways.
Just as Escher wishes to expose the ‘unreal’ nature of perspective so too he makes visual jokes and plays upon these codes by:

1. 1. Depicting ‘impossible objects’ that nonetheless look regular
   2. Objects that emerge out of the plane and return to it
   3. Using the convention of light and shade, Escher plays with the illusions of convex and concave, floor and ceiling

9. Iteration
Escher often uses single elements, shapes and motives that are repeated throughout the picture. In this way a complex totality can be built out of the iteration of single elements. Today this can be related to such fashionable topics as the generation of chaos and of complexity through iteration; the generation of fractals; the generation of complex behaviors in computer and brains—Artificial Intelligence and Cognitive Science; the sort of questions discussed by Hofstader; the conjecture by Heinz Pagels that the universe it built out of iterative procedures even on the quantum level—i.e. David Finkelstein’s notion of a space-time code.
Escher also appears to have been interested in the notion of infinity. Of course any graphic artist who employs perspective must think about the idea of limits and infinity. This interest was already present in the 1950s but seems to have been sparked off again by his contact with the mathematician Coxeter. It led Escher to the idea of covering the plane through the repetition of shapes and smaller and smaller scales. Further interest in this seems to have been sparked off. The result seems, to me at least, to be a remarkable anticipation of Mandelbrot’s fractals which were developed in the 1960s and 1970s—although I would like to see Coxeter’s book which may have sent Escher on this trail.

10. Crystallographic References
In some images Escher depicts the ways repetitive units can arrange themselves in three-dimensional space. This is the subject of crystallography and its mathematical expression is ‘group theory’ or ‘symmetry theory.’ While Escher appears to have had contact with crystallographers he also developed his own independent approach to classification.
Of course in Escher’s case his three-dimensional researches are depicted on the plane. Escher sometimes adds another level to this paradox by threading ‘flat’ worms through a three-dimensional grid—yet in the end everything is really a set of marks on a flat surface.

11. Covering the Plane
Along with his ‘impossible objects’ Escher became most famous for his covering of the plane. This topic combined many of his obsessions—dualities, iteration, symmetry and the two-dimensional space that is the arena through which all space is portrayed in art.
Escher’s interest in the plane is hinted at through his earlier landscapes drawn from a high vantage point. It may also have been stimulated by his experience of Arabic art. Out of this must have grown his interest in tiling the plane.
His many examples of covering the plane have attracted the attention of mathematicians and there is a great deal written on the subject. In addition I have noted above his interest in repetition at smaller and smaller scales to convey the sense of infinity on the plane.

1. 1. Sphere
      Note that Escher also looked at the question of covering the sphere and seems to have been aware of the fact that there must be a mathematical singularity involved in the covering of the sphere.
      Escher also played with a covering of the sphere by means of straight lines in his ‘blowballs.’
   2. Penetrating the plane, interlocking planes
   3. Figures that enter and leave the plane

12. Order and Chaos
Escher has a few meditations on Order and Chaos but these do not seem to be too profound.
However his meditation upon infinity and limits does seem to be more interesting for here an infinite complexity is created in an ordered way. Thus what could be mistaken for chaos—infinite complexity—is in fact order.

13. Light
Escher does not seem to be interested in light as would be a painter who uses gradations of light, tones and shadows to generate space, separate figures, create drama, etc. Escher seems more concerned with light as a duality of ‘light and dark’ as in fireworks, reflections off water, candles in crypt, immediate foreground and background. He also explored the paradox of using light and shade as ways of depicting the convex and concave.

14. Evolution/Metamorphosis

Escher played with the concept of evolution as a sort of visual joke as when objects ‘evolve’ out of the plane into three dimensions or ‘evolve’ up a scroll.
Additional preoccupations —shells, animal forms, curious planets, etc.

A Caution—the Mathematical approach

Escher’s work has excited mathematicians and crystallographers because of resonances to their own work. I also believe that, in his researches, Escher came close to a scientific approach to inquiry. It is therefore seductive to analyze Escher’s works in mathematical terms—group theoretical, symmetry theory, etc. Now while this may help our understanding and appreciation, I feel that there is a subtle danger in this approach. Escher was not a mathematician, he did not really understand their formal explanations and, I would guess, felt them to be of interest only up to a point. In his researches he preferred to go his own way which, I assume, was a sort of hands-on research involving sketching out tentative solutions.

It seems important, therefore, that Escher’s work should be approached in visual terms, in terms of the graphic arts. The works are visual experiments, explorations using line, and the marriage of tactile exploration and of an almost obsessive perception. In this way Escher was groping towards some understanding of the underlying rules and structure of the universe. After the fact it may be possible to describe the results in mathematical terms but this may not account for the way in which they were created. Scientists and mathematicians work in the same spirit but their means of expression are quite different. So I think Escher does evoke some very interesting questions about the similarities and differences between the arts and the sciences.