In his Pulitzer prize-winning book, Gödel, Escher, Bach: An Eternal Golden Braid, Douglas Hofstadter wove a network of connections linking the mathematics of Gödel, the art of Escher, and the music of Bach. According to Hofstadter, the common denominator shared by these three masters of contradictorily similar yet dissimilar arts can be identified in their application of the phenomenon known as the "Strange Loop." As Hofstadter defines it, a strange loop "occurs whenever, by movement upwards (or downwards) through the levels of some hierarchical system, we unexpectedly find ourselves right back where we started" (10). Although many of Escher's prints illustrate this concept, a particularly striking and familiar example is "Drawing Hands," in which two hands appear to be drawing each other.¹ We can solve this puzzle in different ways. One way to approach the paradox of "Drawing Hands" is to recognize that it is Escher, the artist, who is drawing both hands and who stands outside of this particular puzzle. Alternatively, we may adopt a Zen-inspired solution and let mystery be mystery by choosing to embrace a unity which contains oppositions. Jorge Luis Borges described a similar type of recognition in his poem, "Labyrinth," which begins with these words:

There'll never be a door. You're inside
and the keep encompasses the world
and has neither obverse nor reverse
nor circling nor secret center.
(In Praise of Darkness 39)

The word labyrinth, used here to describe the works of both Borges and Escher, is of Greek origin; the Greek labyrinthos and later Latin labyrinthus denote an intricate enclosure or structure containing a series of winding passages hard to follow without losing one's way. While labyrinth sometimes appears as a synonym for maze, the two are distinct: it is possible for a maze to have more than one entrance or exit, whereas a labyrinth has only one opening that is simultaneously entrance and exit. In traversing a labyrinth, the process of exploration leads into the center and then back out to where one started; thus, a labyrinth fits Hoftstadter's definition of a strange loop.

Escher constructs his visual labyrinths with images that lead us further into, and then back out of, a particular scene; Borges constructs his literary labyrinths with words that wind and unwind until we are somewhere neither inside or outside, but both. J.L. Locher's characterization of Escher's work as self-referential, "extremely compact" (19), and full of repetition is a description that can be readily applied to Borges' work. Further, as the complexity of Escher's visual labyrinths involves the viewer's ability (or inability) to read the structure of the composition in more than one way -- i.e., a single surface may be simultaneously floor, wall, and ceiling, as in "Concave and Convex" -- so also does Borges require the reader to enter the labyrinth of his fiction while maintaining enough perspective to allow for multiple readings and interpretations. Where Escher questions the unity of the spatial-visual world, Borges questions the unity of the spatial-temporal world (Barrenenchea 32), and the questions of both artists reflect major preoccupations of the human intellectual enterprise since we discovered the concept of relativity. Both artist and writer bring a unique, sometimes sarcastic, and often esoteric sense of humor to the construction of their labyrinths as well -- as humor that has been called, in Borges' case, "a new dimension [to South American writing] ... a new element of high playfulness" (Bell-Villada 40).

Although Escher's and Borges' labyrinths have been dismissed as intellectual illusions -- optical and literary -- overly dependent upon mathematical precision in their organization of shapes in space or portions of a narrative (Barrenechea 23), the element of play is present in both. Escher was often excluded from consideration as a serious artist because critics found his prints "too cerebral to be real art" (J.L. Locher 41); hence, his work still appears most often to illustrate concepts in mathematics and the natural sciences (Broos 29) or in popular commercial contexts. As C.H.A. Broos observes, the progression of Escher prints (including examples like "Day and Night" or "Metamorphose") illustrating everything from science textbooks to pop music albums is itself Escher-like (Broos 29, 36). In Escher's visual fantasy worlds, "even the wildest fantasy remains subject to the rules of the game ... higher laws of logic and mathematical laws that draw the universe and all its opposing elements together" (Broos 36). Borges might add that in a world marked by human insecurity, with conflicts and changes on a global scale, a writer can no longer rely on traditional realism to depict such a world; hence, aspects of unreality are vital to Borges' fiction as a means of "alluding to reality indirectly" (Bell-Villada 43).

The element of play can also include the dreamworld, and dreams are useful to both Escher and Borges as one way of linking different aspects of reality and unreality. However, dreams are not always necessary for different spatial experiences to coincide in the same place (Escher) or for different temporal experiences to coincide -- or at least overlap (Borges). As in Escher's prints, the mix of reality and unreality in Borges' fictions captures our attention and draws us into the labyrinth -- often, a labyrinth already occupied by characters preoccupied with the unreal realities of their own existence and multiple repetitions of the same themes. "Space can be filled to infinity with contiguous prints," wrote Escher (Graphic Work 4); it is easy to imagine Borges revising this statement for his own purposes and writing something like this: Time can be filled to infinity with contiguous dreams or narratives. Part of the challenge of the game is to see how far infinity can be pushed while finding some delight in the process. For both Borges and Escher, play is part of the game -- sometimes fun, but always to be taken seriously.

Both artists blend detailed observations of the world around them with invented details supplied by their own imaginations to arrive at their unique presentations of impossible/possible subjects. Like many of Escher's prints, such as "Verbum," the effectiveness of Borges' strange loops depends on a combination of precise details that appear to match our sense of what is real juxtaposed with elements of the unreal or impossible. Both artists understood the "Blind Men and the Elephant" paradox: namely, that looking closer to see more details can also mean seeing fewer details, or missing the "whole picture." Sometimes, as is the case with the figures on the desert floor of the South American Plain of Nazca, which can only be seen in their entirety from the air, great distance is required to see the whole in the proper perspective. Such a perspective makes the difference between a perception on the viewer's or reader's part of chaos or some larger all-encompassing order. Escher wrote that he had "a deep aversion to disorder" (qtd. in J.L. Locher 47), a characteristic which may be observed in all his prints, even those like "Liberation" that also appear to represent some aspect of disorder. Presumably, Borges, in his role as librarian, might echo Escher's words.

Although these are only a few of the similarities between the works of Borges and Escher, it has been useful to chart their labyrinths in very broad terms before playing more specific examples against each other in contrapuntal harmony. Escher's print "Concentric Rinds" is a good visual beginning for a discussion of Borges's story, "The Circular Ruins," in which explicit references to dreaming lead into a labyrinth characterized by shadows and unreality. As Borges scholars and critics have been noting for years, the "strange loops" in Borges' fictions are often constructed around the themes of dreams and infinity. "The Circular Ruins" exemplifies both themes. Dream or otherwise, the gray man who is the Dreamer requires certain minimal conditions in which to exist: rice, fruit, and later, a son. These necessary conditions place the Dreamer in a position similar to the positions of the white and black human figures in Escher's print "Encounter" who meet and shake hands with each other. Escher wrote of this print that beings "who desire to live need at least a floor to walk on" (Graphic Work 11). By providing a floor, Escher continues, the "circular gap in the middle ... [forces them] not only to walk in a ring, but also to meet each other in the foreground" (Graphic Work 11).

As Borges' story progresses, we discover that the dreamer/creator in the story is himself the product of another mind's dreaming. It does not require the addition of mirrors to imagine the potentially infinite regression of dreamers dreaming in ruined temples, of worlds within worlds, as may be illustrated by another Escher print, "Puddle". Escher described this print as a puddle on a woodland path which reflects trees and a clear sky a short time after a recent shower created the puddle. In the mud surrounding the puddle, tracks made by two sets of car wheels, bicycle wheels, and pedestrians travelling in opposite directions may be seen. But where exactly are the trees and sky? What is inside the reflection, or dream image? Perhaps the puddle is a rift in the ground's surface -- a keyhole through which we may glimpse another reality operating on a different level.

"The Circular Ruins" has been described by Gene Bell-Villada as "a parable not just of procreation but of creation in general" (87); it poses the problem of where boundaries may be drawn between our individual wills and the wills of others, or even an Other beyond the human sphere. We can never be sure of where those boundaries are, or even if they exist at all beyond the realm of our own limited frame of reference. Escher wrote that he used his "imaginary images to approach infinity" (qtd. in J.L. Locher 37), and Borges is concerned repeatedly with the same theme. Just as "The Circular Ruins" approaches infinity through dream images, the final stanza of Borges' poem, "The Game of Chess," uses a chess game as a metaphor for considering the same problem:

God moves the player and he, the piece.
What god behind God originates the scheme
Of dust and time and dream and agony? (Dreamtigers 59)

Dreams appear also in "The Secret Miracle," a story for which Escher's haunting print "Eye" is an instructive illustration. In the story, dreams provide insights into the nature of Hladik, the protagonist, before he experiences his private miracle just prior to, or perhaps simultaneously with, his execution by a Nazi firing squad. The strange loop in this story puzzles the reader by raising questions about the nature of time. Does time really stop? Is time merely a matter of perspective? And it also dances around an interesting theological puzzle as well. If a miracle happens, but it is secret and there is only one witness, is it really a miracle? After all, the Nazis still shoot Hladik. However, from Hladik's perspective, he has succeeded in making a pact with God and accomplishing his major goal in life before he is shot to death. "Eye" provides a similar challenge to perspective, as Escher draws not only his own eye as observed in a mirror, but also, reflected in the center of the pupil is the "one who watches us all" (Graphic Work 13), namely death. Who is watching whom? Is it the artist, death, or an Other who watches everyone and everything from a greater distance? Are three perspectives, and possibly more, presented simultaneously in "Eye"?

Psychologists have observed that human ability to think complex thoughts can be measured at least in part by whether or not an individual is able to consider both sides of a paradox, or multiple viewpoints to a complex problem or issue, without becoming insane, that is, losing the capacity to think altogether (Peck). Both Escher and Borges challenge our commonplace notions of thinking about the world; they invite us to expand our thought horizons by contemplating the labyrinth without escape into the realm of insanity.

Within the narrative boundaries of "The Secret Miracle," time is free to alter its movements and its speed as events require, just as musical time changes pace during the course of a symphony; the writer, like the conductor, determines how, when and where the pace changes. The passing or non-passing of time also means different things to the different players in the game -- in this case, to Hladik and his executioners. In whatever time is actually involved between the order to "Fire!" and its consequences, Hladik is able to complete his drama in his own mind: "Meticulously, motionlessly, secretly, he wrought in time his lofty, invisible labyrinth" (Borges, Labyrinths 94). Since Hladik's purpose in life has been accomplished, he is free to meet death whenever it arrives and in whatever form it takes. As Hladik's drama is a labyrinth of his own mental creation, the larger narrative is also a labyrinth carefully designed for a particular purpose -- a purpose which may be read metaphorically as a search for order in the universe.

One Christian theological concept associated with this same search for order is predestination, and it is a concept Escher explored in his print with the same title ("Predestination"). If one accepts the notion of predestination, then all events are to be understood as part of a greater divine plan, even if the events are beyond human comprehension or our ability to rationalize the inexplicable. Escher's print illustrates this idea as a bird and a fish evolve only to meet in a fateful, and fatal, encounter. As Escher described this print, the meeting of the two represents doomed innocence that is no match for aggression (Graphic Work 11). We could call the bird "Hladik" and the fish "the Nazi firing squad," but whatever we call them, we are made to see the logic problem inherent in predestination: theodicy -- who or what is responsible for evil in the universe? If God is responsible for evil, then God is not entirely good -- a solution which contradicts other basic notions of Christian theology. If humans are responsible for evil because God creates us with free will to choose either good or evil, then isn't God still responsible for turning morally defective creatures loose upon the rest of creation? Such questions have baffled great minds throughout the centuries and I do not propose to attempt answers here. However, Borges does raise such questions in "The Secret Miracle," just as Escher illustrates these questions in "Predestination."

Before leaving this story, a further connection between the two artists is worth noting. Time, at the very least, is relative and not absolute in "The Secret Miracle." Another Escher print, "Relativity," offers further visual parallels with this story. We could label any one of the figures "Hladik" and some or all of the rest of them "Nazis"; if we do so, it is easy to see that, while these figures may literally occupy the same space, they do not occupy the same plane or dimension in time. The nature of time -- like infinity, another recurring theme for Borges -- is questioned from the very beginning of the story, as is clear in the epigraph from the Koran:

And God caused him to die for one hundred years
and then brought him back to life and said to him:
"How long have you been here?"
"A day or part of a day," he responded. (Koran II.261)
(qtd. in Crow and Dudley 184)

In the story, "The Library of Babel," the first strange loop we encounter is the architecture of the library itself, with its octagonal outer perimeter and inward labyrinth of infinite floors with uncountable books, not to mention mirrors which "promise the infinite" (Borges, Labyrinths 51). Within the library, which we know from the opening sentence of the narrative is a synonym for the universe (51), the ongoing quest for knowledge continues, even if the outcome of the quest leads to depression and suicide rather than hope. Again, depending upon one's perspective, the quest in the library is similar to the Biblical quest alluded to in the title of the story, the unsuccessful human attempt to build a tower to the heavens; Escher illustrates this subject in his print "Tower of Babel." In the Biblical story, a confusion of languages among the workers (the result of divine sabotage) prevents the tower's completion; thus, humans are unable to reach the heavenly realm of God and achieve complete understanding of the universe in the process. In Borges' library, no additional divine intervention is required to produce confusion. Besides the presence of books in diverse known and unknown languages, confusion is already present in the structure of the library, with its mirrors and potentially infinite repetitions of rows of books on each floor multiplied by unknown numbers of floors.

There is a myth circulating in this library, and some people believe it: a Book exists that can explain everything if it can only be found. An optimist might argue that such a Book remains to be found, so the search is meaningful, whereas a pessimist might counter that the Book does not and never did exist. A parallel to the myth of the Book is another myth about a Chief Librarian; however, like the Book, if God (or some Other) is the Chief Librarian, it is not clear to patrons of the library exactly where the Librarian may be found. Even if the human library assistants or patrons or fellow travellers within the library labyrinth fail to arrive at a point where they began to consider such larger cosmic questions as the existence of a Book or a Chief Librarian, there is much -- far too much, in fact -- to keep them occupied forever or for a lifetime, whichever comes first. The ingenuity of the human mind is no match for the sheer volume of the library which surrounds and engulfs the individual patrons, especially when we recall that each volume on the library's shelves can itself be a labyrinth. Information overload, indeed!

What is the point of having such a library if it is constructed to be impenetrable? Borges, the quintessential librarian, might answer that the arrangement of a library is in itself an art ("June 1968" in In Praise of Darkness 71) or that pleasure may be found in keeping such a library even if, as in the poem "The Keeper of Books," the keeper has never learned to read books "secret and visible ... like the stars" (In Praise of Darkness 72-75). Perhaps God the Keeper/Librarian cannot read or is otherwise beyond the human fascination with accumulating knowledge; in that case, humans appear to be engaged in an endless, yet fruitless task. Escher used the figures of monks, long associated with scholarly pursuits in various religious traditions, to make a similar point in his print "Ascending and Descending." J.L. Locher observes that the sarcasm present in "Ascending and Descending" will be partially lost on the viewer unless she or he understands the inside joke: the monks climb up and down stairs into infinity because "the Dutch term for useless labor is 'monk's work'" (21). In this print, two monks stand out because they are not participating; Escher commented that although these two recognize the uselessness of the stair-climbing exercise, "sooner or later they will be brought to see the error of their nonconformity" (Graphic Work 15).

Borges would seem to agree with Escher: if the Library is the universe and vice versa, then there is no alternative but to keep travelling through the rows of books, just as the monks keep on climbing stairs, or, in another Escher print, "Moebius Band II," the nine red ants keep crawling around the two surfaces of the band that is also only one surface. One solution to the puzzle of the library is the "hope" that the narrator of "The Library of Babel" leaves with us, namely that an eternal traveler might eventually recognize that in the repetition of disorder, an Order might also be observed (Labyrinths 58). Another possible solution would be the recognition that the library, like the moebius band, is a closed cycle. Closed cycles offer both Borges and Escher myriad opportunites as well as limits, for these cycles, labyrinths, or strange loops may be constructed to encompass contrasting realities or unrealities. Further, not only do these realities/unrealities co-exist, they may also be perpetually engaged in changing places with one another, impacting "other" realities/unrealities not their own. All too often, however, antlike humans lack both the eternal perspective and the scope of thought required to embrace paradox. Thus, we must persevere and march on.

"Tlön, Uqbar, Orbis Tertius" has been characterized as a masterpiece of complexity, the representative par excellence of Borges' "best-known genre -- the discussion of an invented book or author, usually in relation to larger human concerns" (Bell-Villada 115). In this story, multiple strange loops operate in the manner of a Bach fugue. These strange loops are: 1) the existence of a four-page entry on Uqbar which is found in only one volume of an encyclopedia set; 2) a later discovery of one volume of an encyclopedia documenting an unknown planet, Tlön; 3) a postscript -- dated in the future when the story was published (Bell-Villada 129) -- which points out that the invention of Tlön is an elaborate hoax perpetrated by a secret society; 4) the fact that artifacts from Tlön began to appear on this planet, including one found by someone named "Borges"; and 5) the discovery of all the lost volumes of the Encyclopedia of Tlön, which precipitates human frenzy to the point that "the world will be Tlön (Borges, Labyrinths 18) -- thus, the world becomes antiworld.

Escher's print, "Double Planetoid," illustrates two worlds sharing a single planet; the darker world is constructed and inhabited by humans while the lighter world has remained in its natural state, complete with rocks, plants, and prehistoric animals. Escher notes: "The two bodies fit together to make a whole but they have no knowledge of each other" (Graphic Work 13). Conceivably, that chaos that might result on Escher's imaginary planetoid if the two bodies were to find out about each other might parallel what happens in Borges' story once Earth, a planet inhabited by materialistic humans, begins to appropriate Tlön, a planet inhabited by a race of idealists, for itself. Humans become enamoured of Tlön and dream of its unknown landscapes. Just as it is not clear in Escher's impossible landscape, "Belvedere," whether the figures are inside or outside, or in the back or the front of the bell tower, or in a valley looking up or on a mountain looking down at the world around them, the popular mind becomes crazed with Tlön. Like a sailor who dies of thirst on a voyage while surrounded by water, the human desire for all things Tlönian (no doubt fueled considerably by advertising!) becomes all encompassing and destructive. This obsession with Tlön is also quite possibly ridiculous, placing humans in a position similar to that of the person sitting on the bench in "Belvedere" -- oblivious to the fact that he is absorbed in the microcosm when the macrocosm, an entire architecture constructed according to what Escher termed the "same impossible style" (Graphic Work 15), is just behind his back. Regarding the preoccupation of the inhabitants of "Belvedere," Escher asks: "Is it any wonder that nobody in this company can be bothered about the fate of the prisoner in the dungeon who sticks his head through the bars and bemoans his fate?" (Graphic Work 15). Likewise, when the world has become Tlön, who will be bothered about the fate of a narrator who remains in his hotel room, revising a "translation (which [he or she does] not intend to publish) of Browning's Urn Burial" (Borges, Labyrinths 18)?

The fascination with Tlön that Borges explores in his story is at least, in part, the natural result of human curiosity about what is mysterious and unknown; when we encounter something that we do not understand, we return to it over and over again in an effort to discover more about it. What else is the human mind supposed to do when it encounters the mysterious: the "leaf of silk paper ... [over] one of the color plates ... [stamped with] a blue oval ... [inscribed] Orbis Tertius" and the puzzling, yet compelling "fragment[s] of an unknown planet's entire history" (Borges, Labyrinths 7)? We seek to transform mystery into understanding.

Two further illustrations may be borrowed from Escher. In "Waterfall," it is impossible to discover any fault in the construction; however, as Escher points out, "changes suddenly occur in the interpretation of distance between our eye and the object" (Graphic Work 16), and suddenly the water in the conduit is flowing uphill toward the waterfall, while the entire scene appears to be submerged under water. Stairs lead down (or up), laundry (perpetually wet?) hangs on a clothesline, and corals, sponges, and other lifeforms appear to move gently in an invisible current which propels them upward. Another print, "Three Worlds," is also a reminder of how different worlds may impinge on one another: leaves float on a surface of water reflecting trees above it; however, the water remains transparent enough to see the fish swimming beneath the surface. Whether or not these three worlds co-exist in harmony remains a question of interpretation; potentially, the trees or the fish or even the dead leaves could be hostile agents. Further, the elemental nature of water itself is questioned, as it appears simultaneously "as a surface, as depth, and as reflector of the world above it" (J.L. Locher 18).

In "Tlön, Uqbar, Orbis Tertius," Borges stretches the thin boundaries between reality and unreality to the point where they cease to exist -- that is, if they ever really existed. Once again, he questions the purpose and limits of human knowledge and, via another strange loop, leads us back into the Library of Babel, since we will recognize (if we have been paying close attention) that a mysterious quotation which appeared in reference to the Library is actually a linguistic artifact from Tlön (Bell-Villada 129). The progress of human knowledge cannot be championed as either an end in itself or a means to an end when that end entails greater exposure of our ignorance and, consequently, increased discontent with our fate precisely because we know that we do not and cannot possibly know everything. As is the case with the other three stories discussed here, Borges raises many questions in "Tlön, Uqbar, Orbis Tertius" and leaves them unanswered, with an implied multiplicity of possible answers. Still, the labyrinths again reveal some truths about the nature of human existence. What those truths are will depend on how we read and see things, and how we decide to "solve" the puzzles presented to us.

Where Escher uses the repetition of forms and the interplay of light, dark, and shadow combined with a multiplicity of perspectives both to explore and illustrate underlying mathematical principles, Borges invents his fictions to explore underlying human concerns, including large metaphysical questions about the mysteries of our existence and, as Borges himself said, "what is most important ... [the mysterious nature] of ourselves" (qtd. in Bell-Villada 114). The visual and temporal labyrinths of Escher's art and Borges' fictions may be described as a set of Chinese boxes (Barrenechea 39), an elaborate puzzle, a moebius strip, a circle, or a strange loop; however, underlying them all is an attempt to explore the workings of the universe and where we as humans fit into the picture. The explorations of Borges and Escher are just two particular examples of the creative potential of the human brain -- that internal labyrinth common to each of us, close at hand, yet mysterious and largely unknown.

What is it about the workings of the human mind that lead a Borges or an Escher to construct their own labyrinths out of words and images? Further, if we find ourselves fascinated by such labyrinths, why are we willing to play these intricate games where, as in Escher's print, "Circle Limit IV: Heaven and Hell," angels and demons are partners in a complex dance which involves, perhaps, occasionally switching roles or costumes? To the extent that one can answer such questions, I venture to say that whether we are engaged in creating labyrinths or our own or exploring those created by others, we play these games because they have something to teach us: valuable information about our world at large or about ourselves. The discovery of the labyrinth's secrets has the potential to amaze us and leave us in a state of wonder that is, paradoxically, a state of confusion and clarity. If we recall the myth of the first labyrinth, built by Daedalus to house the Minotaur, and then think in psychomythic terms borrowed from Joseph Campbell, the journey into the labyrinth is the hero's or heroine's journey to the underworld and back. It does not matter that we end up where we started, because we will return changed if we complete the journey. No matter what our adventures, each one will teach us something valuable. Perhaps we too will arrive at a place, time, or space where we can echo Borges' concluding words from the "Epilogue" of Dreamtigers: "A man sets himself the task of portraying the world.... Shortly before his death, he discovers that the patient labyrinth of lines traces the image of his own face" (93).

¹ Since it is not feasible to reproduce illustrations of the Escher prints mentioned in this paper, the reader is strongly encouraged to consult a collection such as M.C. Escher: The Graphic Work.

² While Hofstadter does use Lewis Carroll's characters throughout his discussion of Gödel, Escher, and Bach, he does not argue that Carroll is a fourth master to be included specifically in his study.

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