Imagine for a moment that you live in only two dimensions. You are like a coin on a countertop, able to move around and across the space but not able to lift yourself up, out, or down through the flat plane of the countertop. You’d see only straight lines as your co-inhabitants of the space, as if you lived among shadows, unable to view your world in its entirety, from above or below. This situation is described in Flatland, a book from 1884 that is considered early science fiction and satire of Victorian society and social heirarchy – and a fun way to consider a two-dimensional world.
Intersecting a sphere with a flat plane can form a circle shape or an oval.
Now imagine a sphere cutting through the countertop. If the sphere approached the countertop at differing angles, the intersection of the plane (countertop) and the sphere would be either a circle or an oval. Imagine the Flatalnders’ surprise when this happens to their world. One inhabitant named A Square (a clever pun of the author’s name) , noticed a Presence one day. The Presence was a circle that got bigger and bigger as it approached the 2-dimensional world. The circle increased in size because it was a sphere from Spaceland who intersected Flatland. The sphere announces, “I am many circles in one” and dislodges A Square to demonstrate what living within a third dimension is like. Denial of other dimensions beyond the third provokes scorn from the sphere.