From the Mobius Strip to the Fourth Dimension

A mannequin hand rotated in the fourth dimension a half-turn at a time (Lambert, 2016).

In 1827, as I noted in Volume Three of The Gnostic Notebook: On Plato, The Fourth Dimension and the Lost Philosophy, August Ferdinand Möbius wrote, "… two equal and similar solid figures, which are however mirror images of each other, could be made to coincide if one were 'able to let one system make a half revolution in a space of four dimensions. But since such a space cannot be conceived, this coincidence is impossible in this case.'"

Movement along the Moebius Strip flips object to its mirror opposite (Rucker, 1974).

To get a clearer picture of what exactly Moebius was getting at, we must return to his strip and consider what happens to a two-dimensional citizen of Flatland as he moves around the band. Still, to be absolutely precise, the Flatlander doesn't actually move along the surface of the strip, rather the strip is an invisible membrane through which the Flatlander flows. So he is not so much on the strip and present on one side or other of the piece of paper, as he is moving within the invisible membrane of the strip which to him is the entire Universe. Anyway, when this Flatlander finally makes his way around the strip he returns to the point where he began but now he is upside-down.

Rotation around the Z-axis.

The state of being upside-down, while probably not even relevant to a Flatlander, can be easily rectified by rotation around the Z-axis. However, even after being returned to its upside-up position, it becomes obvious that the being's left- and right-sides have switched sides as well. The only forms of rotation it has available are clockwise or counter-clockwise rotation around the Z-axis. Rotation around either the X- or Y-axis would require the Flatlander to pull itself up off of the Moebius Strip which is an impossibility as the transparent membrane of the strip is the Flatlander's entire Universe.

So the one-half twist in the Moebius Strip allows the Flatlander to become his mirror opposite. This allows us to reason by analogy and take the entire process one dimension up. Instead of a two-dimensional Flatlander, we can imagine the unattached left hand of a shop mannequin. Here, in our three-dimensional world, it will always remain a left hand. However, if a four-dimensional being were to reach into our world and rotate the mannequin hand a single half-turn around the fourth dimension, we would see the left hand transformed into a right.

And yet, Buckminster Fuller would not be impressed. It all begins with reasoning by analogy. "Suppose you were a two-dimensional being, how would you perceive a three-dimensional being moving through your world?" This is generally how the higher dimensions are approached, and yet, all too often, the premises are never adequately examined, which is to say the fact that two-dimensional lifeforms do not exist is conveniently ignored. And why is it that two-dimensional creatures do not exist? For the simple reason that we exist in a material Universe made up of pieces of matter and not an idealized Universe made out of weightless, two-dimensional scales which explode and billow like leaves in the wind before reorganising themselves into the surface features of all that we perceive. After all, regardless of what we may have been taught, we are not actually trapped within a computer simulation.



In my next post, we will examine the various types of dimensions beyond the usual three.