Flatland analogy for perceiving 4 dimensions



“In One Dimension, did not a moving Point produce a Line with TWO terminal points? In Two Dimensions, did not a moving Line produce a Square with FOUR terminal points? In Three Dimensions, did not a moving Square produce—did not this eye of mine behold it—that blessed Being, a Cube, with EIGHT terminal points? And in Four Dimensions shall not a moving Cube—alas, for Analogy, and alas for the Progress of Truth, if it be not so—shall not, I say, the motion of a divine Cube result in a still more divine Organization with SIXTEEN terminal points? Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this—if I might quote my Lord’s own words—‘strictly according to Analogy’?


Again, was I not taught by my Lord that as in a Line there are TWO bounding Points, and in a Square there are FOUR bounding Lines, so in a Cube there must be SIX bounding Squares? Behold once more the confirming Series, 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, ‘strictly according to Analogy’?”


Edwin Abbott, Flatland


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