[Neri]

A Further Observation

Let us say that from an intuitive or sensible perspective the universe was a two-dimensional spatial reality. This would mean that if intelligent, perceptive creatures existed on a universal scale, the idea of three spatial dimensions would be insensible to them. They would nonetheless have a sense of time, which they could conceive of in a space-like way as one-dimensional and always pointing in the same direction. Suppose that these creatures were able to calculate that at the intersections of their two-dimensional spatial coordinates there did not exist perfect (infinitely small) points but rather incredibly small circles of expanse. Their calculations would reveal something that would seem quite strange indeed. They would have to allow for a “third dimension.” Such a thing would be unimaginable to them, yet without it their calculations would make no sense. They would be forced to conclude, therefore, that their world (the universe) was virtually two-dimensional yet not perfectly so.

Let us suppose that what we creatures call “the universe” is actually only one of these minute three-dimensional worlds. To us, of course, a world with more than or less than three spatial dimensions is counterintuitive. On the other hand, if we somehow knew that there was a “two-dimensional” universe of which we were but a part, we would conclude that there were actually five spatial dimensions.

Let us go a step further and suppose that at the intersections of any three spatial coordinates of our world, there do not exist perfect points but rather incredibly small spheres each containing its own separate three-dimensional realm. If our mathematics was able to reveal the existence of these minute spheres, we would be led to conclude that there were three additional dimensions in the universe. This would bring the total to eight spatial dimensions.

Let us make one final supposition and say that at the intersections of any three coordinates within any “minute sphere,” there again does not exist perfect points but rather lesser, even-more-minute spheres each with three dimensions of its own. Were we able to detect this condition, we would conclude with confidence that there were in reality eleven spatial dimensions and four separate realms of time.