In 1884 an English schoolmaster penned a groundbreaking novella called Flatland. The book’s protagonist and narrator is a square living on a vast two-dimensional plane inhabited by a plethora of geometric shapes. As Mr. Square walks the reader through what life in his dimension is like, it's easy to be fascinated with how foreign their universe seems when viewed from the “inside”. For instance, to us a square is easily distinguishable from a circle, but to a Flatlander both look like straight lines. In fact, to a Flatlander, everything looks like a straight line! One evening, the square was visited by a being with extraordinary powers. The guest appeared straight out of nowhere and was at first as small as a point, but grew quickly into a magnificent circle. When the awe-struck square demanded to to be told what he was looking at, the mysterious being obliged by introducing himself as: The Sphere.
In the beginning, the square could not fathom a dimension which rose “out of” his space. It would have to be an infinitely small dimension he decreed. Well, the sphere explained, for Flatlanders it might seem that way, but to us, the third dimension is infinitely big. Still not convinced, the square asked which direction the third dimension was in. Not knowing what words to use, the sphere could only explain that it points orthogonally from the square’s insides! Realizing that words would not suffice to explain the true nature of flatland, the sphere tore the protagonist out of his universe and took him on an extraordinary journey into the 3rd dimension, pushing the limits of his imagination.
The second half of this volume is a sequel called Sphereland where the square’s grandson continues exploring intriguing spatial concepts. The main characters inadvertently discovered non-Euclidean geometry when they found that the sum of angles on very large triangles amounted to more than 180 degrees. After working their grey matter for a long time they postulated that Flatland might be bent in an unseen direction. This turned out to be the case as Flatland was in fact stretched out along the surface of a giant sphere.
A fun thing for me to do while reading the book, was trying to imagine the equivalent of what the square was experiencing except in our dimension. Which direction is the fourth dimension pointing in? I suppose, just like for the square, it points out orthogonally from our own insides but in a direction that we can’t see. What happens when we move around in this new direction? If flipping a Flatlander results in him being “reversed”, then flipping a right-shoe 180 degrees in the fourth dimension should result in a left-shoe. Consequently, if a person were spun the same way, the individual would feel unaltered but everything from books to cars would be backward. A fourth dimensional being could do things like remove objects from boxes without opening them, and untie knots without touching the ends of the rope. And maybe our world, just like Flatland, is folded into a 4th dimensional sphere so that if we had a big enough telescope, we could see ourselves from behind.
While reading Flatland, I couldn’t help but feel sorry for the inhabitants who could only “see” straight lines in every direction, but upon further reflection, the same limitation applies to us. We can only see a thin two-dimensional surface in every direction, and just like Flatlanders, we have to infer the thickness of a solid. When I see another person standing a bit away, I can’t be one hundred percent certain that they are not just a very well made cardboard cut-out. A fourth-dimensional being however, could see all of a person, including his insides and must view our human existence as infinitely mundane. What would the eye-ball of such a being look like!? I suppose that if our iris is a concave plane in the back of our eye, the iris of a 4D being would have to be a solid, except curved in the 4th dimension. Pretty spaced out!