Other distortions appear in the “distant zone.” Rather than appearing to extend infinitely, space seems “finite, shallow in depth, and slightly concave,” and “distant phenomena are experienced visually as if seen through a telephoto lens”; that is, they appear to be “closer, flatter, and with their surface planes turned to face the viewer.” In addition, parallel lines “bend upward and come together to meet at a point in front of the viewer on the horizon and at a finite distance” (29). Looking at an extended horizon below eye level “such as the sea seen from the top of a cliff,” the viewer “seems to be at the center of a great bowl with its rim on the horizon.” An extended horizon above eye level, such as the sky, is experienced as “a vaulted structure.” Finally, the “apparent size of very distant objects” in hyperbolic space is mutable and “depends on whether there are local cues and how these are construed” (31).9
Because Euclidean visual space is culturally normative, the terms used to describe hyperbolic space (“distortion,” “optical illusion”) connote aberrance from the norm—yet it is hyperbolic visual space that is grounded in the human body, its phenomeno-logic informed not only by external material forces but also by the intentional directedness of consciousness toward its objects. As Heelan puts it: “A Body defines the human subject functionally in relation to a World as the ground for an interlocking set of environing horizons. Being-in-the-World implies being now related to one horizon, now to another” (13). Which horizon, which system of orientation and coordination one lives, depends ultimately on what “makes sense” in a specific context. For a situation to provide “a Euclidean perceptual opportunity,…it must…be virtually populated with familiar (stationary) standards of length and distance, and be equipped with instantaneous means for communicating information about coincidences from all parts of space to the localized visual observer, wherever he/she happens to be” (51). A situation that provides “a hyperbolic perceptual opportunity” is incommensurable with the Euclidean situation in that its sense emerges precisely from the localized visual observer, wherever he/she happens to be. The visual observer making sense in hyperbolic space, rather than relying on abstract, standardized, and stationary measures, “must…use the rule of congruence which…is embodied in the capacity of the unaided visual system to order the sizes, depths and distances of all objects in the unified spatial field of vision.” What is involved on these perceptual occasions is a “purely visual estimation” of size and distance and a reliance on “a significant local standard of length relative to which the surrounding environment could be spatially structured” (51).
Without either an abstract or local standard of measure, worldly space and the objects within it lose their meaning and become hermeneutically ambiguous, indeterminate, and disorienting. Furthermore, one begins to doubt one's own body. Phenomenological geographer Yi-Fu Tuan describes the spatial and bodily effects of one such situation of “being lost” when neither Euclidean nor hyperbolic standards of measure are at first available:
What does it mean to be lost? I follow a path into the forest, stray from the path, and all of a sudden feel completely disoriented. Space is still organized in conformity with the sides of my body. There are regions to my front and back, to my right and left, but they are not geared to any external reference points and hence are quite useless. Front and back regions suddenly feel arbitrary, since I have no better reason to go forward than to go back. Let a flickering light appear behind a distant clump of trees. I remain lost in the sense that I still do not know where I am in the forest, but space has dramatically regained its structure. The flickering light has established a goal. As I move toward that goal, front and back, right and left, have resumed their meaning: I stride forward, am glad to have left dark space, and make sure that I do not veer to the right or left.10
Reading this passage, making sense of it with our bodies and recalling some similarly anxious disorientation, we can understand quite carnally how Hansel and Gretel, lost in the forest and darkness, must have hurried ahead—eagerly, indeed gratefully—toward the light shining from the window of the house of the wicked witch.
Similar spatial ambiguity and its permutations and resolutions are dramatically recounted by Michael Asher, a Westerner and travel writer, who became briefly lost with companions in the Sahara desert. In response to the problem of people becoming spatially disoriented and dying in the desert, he tells us that “the government had put up a series of markers” without which “it was almost impossible to travel in a straight line.” And he continues:
I soon understood the need for markers. The desert we walked out into the next day was utterly featureless…. There was nothing at all to attract the eye but the metal flags spaced out every kilometre. It was like walking on a cloud, an unreal nebula that might cave in at any moment. Sometimes its dappling ripples looked like water, a still, untided ocean undulating to every horizon. In all that vastness there was not a tree, not a rock, not a single blade of grass.11
For a solitary human being (like Tuan in the forest before he saw the flickering light), the space of this featureless desert without objects would be neither hyperbolic (with some known thing or someone else to provide local measure in terms of one's own body) nor Euclidean (with given objects known to be spaced, as were the markers, at an abstract measure of one kilometer apart). In such a contextless context “one” (the pronoun chosen precisely here) would be truly “lost in space.”
Asher is not solitary, however; his companions provide him “local measure” relative to his own body, and, suddenly lost and without markers in the desert, he and they live the Sahara hyperbolically. That is, close to him, others have “intelligible” shapes and sizes, but objects, shapes, distances, and motion that are not in the “near zone” are grossly distorted:
In the afternoon we passed [a] caravan…. From afar the columns of [camels] seemed to stand still. They appeared to remain motionless until we came abreast of them, then they sprang out suddenly into three dimensions. It was a strange phenomenon caused by the lack of anything to mark the distance between us…. Then we heard the boom of engines and pinpointed two trucks in the sand. Like the…caravan earlier, they appeared not to be moving. Not until we passed them did they seem to accelerate into action, roaring by a mile away. Or was it 2 miles? Or even 10? There was no way to judge distance or scale in Ténéré.12
Asher also remarks on the difficulties of orienting oneself and moving against the featureless landscape:
I watched Marinetta once as she ran away from our caravan…. She zig-zagged crazily over the sand…. When I tried it myself I realized that without anything to fix on, it was impossible to run in a direct line. Any ripples or shadows on the surface gave the impression of relief. We found ourselves moving towards what appeared to be a mass of dunes only to find them dissolving into sandy waves a few inches high. A piece of discarded firewood could be mistaken for a camel or a tent, a blackened sardine can for an abandoned car.13
Everything in Asher's vision is measurable only locally, in terms of the human body and the meaningful size and order it confers on known things. Hyperbolic space, then, is primordial and subjectively lived—and, in terms of human sense-making, it precedes Euclidean abstraction and Cartesian objectivity. As Dorothea Olkowski puts it: “Lived space is not linear, it is a field and an environment…. [T]he primordial space of our existence is ‘topological'; it corresponds to the diacritical oppositions of our perception.…[I]t is a ‘milieu in which are circumscribed relations of proximity, of envelopment,'…[relations] which are not merely geometrical or cultural but are lived.”.14
Indeed, this topological space is precisely the space of a child's world before it and the child have been properly “disciplined” and “sized.” Here it is illuminating to point to the lived difference between Euclidean and hyperbolic geometries by contrasting the model of Renaissance perspective with a child's survey of the subject/horizon/world relationship. As those of us in film studies know, much has been made of the subject's “mastery” of the world according to Renaissance perspective:
