Flatlanders

“Just as a being may be imagined as having only two dimensions, and living always on a plane surface (in a space of two dimensions), and having no conception of a space of three dimensions, so we may think of ourselves as living in a space of three dimensions but surrounded by a space of four dimensions.  The flat being could not point to a third dimension because he could not get out of his plane, and we cannot point to a fourth dimension because we cannot get out of our space.  Now what the flat being thinks is his plane may be the surface of an enormous sphere in our three dimensions; in other words, the space he lives in may curve through some higher space without his being conscious of it.  So our space may also curve through some higher space without our being conscious of it.” – David Eugene Smith, The Teaching of Geometry (1911)