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.

Buckminster Fuller on the Moebius Strip

Möbius Strip II (Red Ants) M.C. Escher 

What is a Moebius Strip?

Möbius Strip

A Moebius Strip is a loop formed by taking a strip of paper and adhering the two ends together while twisting one of the ends a half turn.

This object has some rather unique properties. For instance, this loop has only one side. Imagine, every piece of paper has two sides, and yet, somehow, from a two-sided piece of paper an object possessing only one side is created.

Another unique feature of the Moebius Strip is that if you draw a line down its length from the center of the strip, you will have to go around the loop twice to return to your starting point. If you were to cut along this line you would end up with a single loop twice as long as the original with two full twists, rather than the original's half twist.

In his magnum opus, Synergetics (1975), Buckminster Fuller had this to say about the Moebius Strip:

 831.10   Moebius Strip
831.11   In the same operational piecrust-making strictness of observation, we realize that the phase of topology that Moebius employed in developing his famous strip mistakenly assumed that the strip of paper had two completely nonconnected faces of such thinness as to have no edge dimension whatsoever. When we study the Moebius strip of paper and the method of twisting one of its ends before fastening them together and scribing and cutting the central line of the strip only to find that it is still a single circle of twice the circumference and half the width of the strip, we realize that the strip was just a partially flattened section of our piecrust, which the baker would have produced by making a long hard roll, thinner than a breadstick and flattened out with his wooden roller. What Moebius really did was to take a flattened tube, twist one of its ends 180 degrees, and rejoin the tube ends to one another. The scribed line of cutting would simply be a spiral around the tube, which made it clear that the two alternate ends of the spirals were joined to one another before the knifing commenced.

What exactly is Fuller talking about in the above section of text? The point he is making is that the apparently amazing properties possessed by the Moebius Strip are the result of a semantic trick. Looking through the lens of Iain McGilchrist's The Master and His Emissary (2009), we can see this as an example of the left brain's tendency to be seduced by the idealised forms hidden behind the labels used to describe objects such as the Moebius Strip.

A sheet of paper is seen as a two-dimensional plane, having only width and height. The same, naturally with a strip of paper. The strip of paper has two sides and four edges. When the two ends are joined with a half twist, the resulting object has one side and one edge. However, the left brain's method of understanding is to focus on individual features while ignoring the whole.

The right brain doesn't have such a good eye for individual details, rather it sees the whole picture as though it were fuzzy or a bit out of focus. So where the left brain sees a one-sided loop, the right brain sees a loop made from a flattened tube of dough. Topologically there should be no real difference between a strip of paper and a strip of dough unless we took the strip as being two-dimensional and having a depth of zero. Yet that is exactly how we have been trained to look at it.

Fuller also considers what happens when a line is drawn down the center of the Moebius Strip and cut along with scissors, only he sees it as a tube of dough around which a helix is cut, not a spiral as he wrote, and though it is a common enough error to confuse the two, it seems out of character for Fuller who is usually so precise.

Still, his point is valid. The magic possessed by a two-dimensional object with only one side seems much greater than that of a three-dimensional object whose contour is molded with a decorative half-twist. The first seems an impossibility, while the second is simply an interesting choice of design. The difference in perception is due to the nature of the edge; a razor sharp, one-dimensional edge isolates each side of the strip from its opposite, while an edge with some thickness helps the observer to recognize the true nature of the object as a flattened torus whose bodywork has been modified with a simple cosmetic half-twist.

All of this is related to the examination of the fourth dimension which I described in my book The Gnostic Notebook: On Plato, the Fourth Dimension, and the Lost Philosophy as it was August Ferdinand Moebius, the discoverer of the Moebius Strip, who was the first to work out the effects of rotation in four-dimensional space. Indeed the two subjects are closely bound as we'll see in the next post.