A corollary arising out of the previous post on ƒ(u) ∼ dt u , this is an abstract musing in the tradition of many popular science abstractions. One such imaginative piece of writing, Flatland: A Romance of Many Dimensions by Edwin Abbott Abbott (1884), came to my attention many years ago. A satirical novella set in a world with only two dimensions.
Your are looking down a cosmic microscope where an infinitesimal point fills your field of vision. Sight becomes infinitely resolved.
A. Imagine a dot, an infinitesimal point in three-dimensional space and time does not exist.
Now imagine this point repeated infinitely so each point touches the next along a single axis. The point is now a line.
At any point in space, the line will appear as a single point. You cannot see the line because time does not exist and therefore you cannot move from one point to the next.
B. Now imagine an infinitessimal point in time without spatial dimensions.
The point is now extended in time to form a line.
Remembering there is no space, you cannot perceive the timeline, only a point in time.
C. Now take a point in a world where space and time exist together. Extend it to form a line. You can now see the line because it extends both in time and space. You are able to move through both space and time simultaneously. You can now perceive the line as a continuum of infinitesimal points extending in space and moving through time.
This thought experiment has many caveats and appears reductive: an infinitesimal point seems counterintuitive as does a dimension devoid of physical extensions with only time as a parameter and conversely a space without time. In addition, a line itself without thickness or breadth, only length is also a pure abstraction. The whole thing is counterintuitive. These things are hard to imagine and can only be spoken of in metaphor or using mathematics because to our brains that are seated and immersed in a spacetime world these things do not make sense in view of experience. Experience tells us that it cannot be so. We are made and exist in this world. Our perceptions and minds have been formed as fractals or reflections of ‘real world’ phenomena and the laws that govern it. Such things cannot exist in our Universe.
My simplistic thought experiment is a way of imagining space and time as inextricably linked to form the conceptual fabric containing all becoming, existence and change in our universe. Strip one from the other, and the impossibility to experience existence becomes self evident. This was one of Einstein’s insights following on from Maxwell.
Add another spatial dimension and we enter a world which is alien and again counterintuitive. We can only construct projected shadows cast from such a world onto ours by the imaginative means of strange solid forms. Likewise, our-world solid objects project shadows onto flat surfaces, as an infinitesimal slice of the object that projected it. An idea Abbott made use of in Flatland with the passing sphere.
We have no problem in perceiving and conceiving of shadows as projections of a higher (our) dimension because they exist in a world of lower order than ours and one in which we experience shadows everyday. However, when confronted with a world containing more than three linear dimensions it becomes impossible to imagine such a world and we make recourse to geometric shadows in the form of strange solids and mathematical means to describe them. It is only possible to hint at what a world with four spatial dimensions might be like using animations. It is indeed a strange world.
When I think of my work in three dimensions, I perceive it in time too as my mind traces the surfaces and contours. When I see shadows projected by the work, I see something else, a journey through space riding on beams of light and reforming the world. A world that exist in three dimensions at a subatomic level, but appears flat, in two dimensions. I then recreate that universe in my mind to one that is congruent with an intuitive mind formed in this universe of spacetime.
Looking into the past and future is also a work of shadows: shadows of ideas and events that do not fully form into rounded experience but play themselves on the screen of the mind as words, pictures and imputed movement.