What would a world look like if it had only length and breadth, but no height? In 1884, William Abbott Abbott answered this question by creating a strange fictional world through his satirical novel "Flatland: A Romance of Many Dimensions" where strange two dimensional inhabitants opened new vistas of thinking in several disciplines not limited to physics, maths and literature . The book, is essence, is a commentary on Victorian culture , but is equally appealing to scientists, mathematics and explorers of dimensions alike. In our geometry classes, we have heard about a line, square and a cube. But what lies beyond the known realm of space?
Flatland, gives us an unique perspective as to the complexity of perceiving a higher dimensional object such as a tesseract or hypersphere from our own. In order to understand what a tesseract is, let us take a step back and understand a cube in terms of a square.
A cube is in essence an infinite number of squares stacked one upon another in a direction that simply doesn't exist in its frame of reference. A tesseract is a theoretical fourth dimensional analogue of a cube, stacked infinitely many times in an imaginary direction, which is a little difficult to imagine.
In the world of flatland, a circle and square interacts with each other just like we do, but their visually perceive each other a little differently. It is important to understand that despite being three dimensional, we view objects as tesselations of two dimensional images and re-create the entire shape inside our heads. Our stereoscopic vision enables us to perceive depth, which is also aided by how light affects different surfaces. Flatlanders perceive their fellow inhabitants similarly, except as straight lines of varying sizes, with light affecting them depending on the angle of incidence and also the properties of the surface.
But suppose, if a 3d sphere were to visit their universe? How would it look like? Since flatland is a plane, the sphere would exist as circles of different sizes in different position. It would first appear as a point, grow into a large circle of its own radius, and then disappear into a point.
Well, as an analogy, a fourth dimensional hypersphere would first appear as a point, then grow to a sphere, before disappearing back into a point on the other end. Einstein's concept of space-time changed the way we view the world. Time is indeed the fourth dimension of our existence, but what if there exists a higher space totally shut away from the five sense? There is no way to tell, is there? Perhaps, we are what we are, truly limited by the dimension we perceive or a projection of some higher dimension, but that doesn't stop us from questioning reality .
