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.