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FLATLAND Imagine a universe of only two dimensions: a flat plane. This imaginary universe is inhabited only by two-dimensional beings—we'll call them Mr. and Mrs. Flat. They can only conceive of two dimensions since that is all they are capable of experiencing directly.'°° Suppose a three-dimensional being comes along and pokes his finger into their two- dimensional space. What would Mr. and Mrs. Flat see? The intersection of the three-dimensional finger into the plane of their existence would appear to them as a circle. That is all that would appear to them in the limitations of their two-dimensional world. The finger would seem to emerge from nowhere and would not be understood. And while intruding into their two-dimensional space, it could interfere wouter ema 1 This three-dimensional "super being," possessing simply an additional dimension beyond Mr. and Mrs. Flat's, would enjoy remarkable advantages. The being, for instance, could easily enjoy a proximity with them independent of their own geographic placement within their known universe. The being could also remove or relocate either of them beyond the rules of their own geometry. Such a being would have capabilities totally beyond their comprehension. Imagine a sphere of three dimensions passing through their two-dimensional universe. It would also seem to appear out of nowhere and then disappear in the same manner. Perhaps even more puzzling to Mr. and Mrs. Flat would be a more complex geometrical object passing through their two-dimensional universe. The object would also continually change shape as it entered or withdrew from their universe. Attempts to communicate the existence of a three-dimensional object to Mr. and Mrs. Flat would prove extremely difficult. One approach would be to "unravel" a template of the cube into a two-dimensional representation: 53 with their freedoms and perceptions.