The Flatland Trap: Two Dimensional Thinking in a Three Dimensional World

A powerful lesson about the importance of contextual inquiry comes from one of my favorite books, Flatland: A Romance of Many Dimensions. Edwin Abbott wrote Flatland in 1884. In the book, Abbott created a two-dimensional world populated by various polygons as a satire of the social hierarchies of Victorian culture. In the world of Flatland, the more sides you have the higher your social status. In addition to a satirical commentary on nineteenth century social norms in England, Abbott provided an elegant analogy to explain how our frames of reference influence what we claim to know. The narrator of Flatland is a square, literally. In the first part of the book, the square describes for us life on a two-dimensional plane. One day the square is visited by a sphere, a three-dimensional occupant of Spaceland, and his life is turned inside out.

Imagine the experience of encountering a three-dimensional entity from the perspective of a sentient two-dimensional figure. When the sphere speaks, the square hears a voice that seems to come from all directions at once. From the vantage point of three dimensions, the sphere can see the inside of the buildings as well as insides of the inhabitants of Flatland. When the sphere gently touches the interior of the square, the square experiences the interaction as a sharp sudden pain in his gut. In an attempt to describe the nature of three-dimensional objects to the square, the sphere passes through the plane of Flatland. From a Flatland worldview, the intersection of the two worlds looks like a point that turns into a circle. The circle grows in size, then shrinks to a point and disappears.

For the square, knowledge of three dimensions gets constructed through interactions with the sphere. At first, the square ascribes unnatural powers to the sphere, “’Monster,’ I shrieked, ‘be thou juggler, enchanter, dream, or devil, no more will I endure thy mockeries’” (1884, p. 75). The square has no choice but to build an understanding of his encounter with a three dimensional object from his two-dimensional paradigm. Eventually the sphere disenthralls the square from both his plane and from his frame-of-reference by transporting him into space, “I shrieked aloud in agony, ‘Either this is madness or it is Hell.’ ‘It is neither,’ calmly replied the voice of the Sphere, ‘it is Knowledge…’” (pp. 77 – 78).

The square comes to a profound and novel understanding of observed reality that cannot be shared with the other inhabitants of Flatland. Had the square continued a dialogue with the sphere without the benefit of experiencing a three-dimensional world, the square might have acquired a new, but diminished – one might say, “two-dimensional” – understanding of three-dimensional reality. Without an experience of three dimensions, the square would be no better off than the inhabitants of Plato’s cave who strive to understand the world by seeing only shadows cast on the cave wall. Our quadrilateral hero must live with the burden of understanding that for all other inhabitants of Flatland, the existence of three-dimensional objects is both real and unknowable. Not surprisingly, the square has difficulty explaining his epiphany to other Flatlanders and is eventually jailed as a heretic.

We are trapped in our own versions of Flatland when we accept our daily experiences as the only reality that matters. The easiest way to miss something important is to keep our heads down and our attention narrow. Contextual inquiry is about widening our focus to avoid blind spots. When conditions are stable and predictable, we can operate on autopilot without getting stuck. In a volatile world where threats are unpredictable and opportunities arise from unexpected sources, we may need to expose and disassociate ourselves from the artificial constraints of our comfortable world-views.

 

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