The Endless Ribbon

Are you familiar with the Möbius strip? A strip with half a twist in it, which at the first glance seems to have two sides, but it’s actually one-sided? To get familiar with it, check our article “Dimensions – A Weird Fourth-Dimensional Bottle”. Now, what if I told you that over the course of history, it was independently rediscovered again and again… and again, but each time from a completely different perspective? Let’s check it out:

A Philosophical Framework

Max Bill discovered something new – an idea in the form of a shape. Bill, a Swiss artist (1908 – 1994), was working in 1935 on a sculpture. While doing so, he experimented with paper strips by twisting them into various forms. Then, he saw it; years later, in 1972, he recalled that moment:

“I was fascinated by a new discovery of mine, a loop with only one edge and one surface. I soon had a chance to make use of it myself. In the winter of 1935-36, I was assembling the Swiss contribution to the Milan Triennale, and there was able to set up three sculptures to characterize and accentuate the individuality of the three sections of the exhibit. One of these was the Endless Ribbon, which I thought I had invented myself. It was not long before someone congratulated me on my fresh and original reinterpretation of the Egyptian symbol of infinity and of the Möbius ribbon.”

Over the years, Bill created many variations of endless ribbons and lots of other math-inspired works. Although he wasn’t the first one to discover the shape, he did discover it in a new light: as a philosophical framework. He wrote: “What I could not find in Möbius’ explanation [of the band] is of primary importance to me: the philosophical aspects of these surfaces as symbols of infinity.” and “The mystery enveloping all mathematical problems… [including] space that can stagger us by beginning on one side and ending in a completely changed aspect on the other, which somehow manages to remain that selfsame side… can yet be fraught with the greatest moment.”

A Mathematical Object

Johann Benedict Listing (1808 – 1882) and August Ferdinand Möbius (1790 – 1868) discovered something new – a mathematical object. Both were German mathematicians that worked independently, and in 1858 simultaneously discovered what would later become the “Möbius strip”.

Since its discovery, the Möbius strip has inspired artists, graphic designers, architects, writers, magicians, and more. One notable example is two woodcuts by M. C. Escher. One of them, the Möbius Band I (1961), depicts three creatures biting each other’s tails. But the image contains more than that: it also shows what happens in you cut a Möbius strip in half along its center line. Instead of getting two smaller one-sided Möbius strips, you get one long two-sided loop. In the same manner, the three creatures are actually connected, assembling together a single two-sided loop; the transparent space between what seems to be two sections of the woodcut is the cut in the middle, and if you ignore it, you get a single Möbius strip (a similar concept was applied in the featured image as well, by Paula Bassi). The second work, Möbius Band II (1963), depicts ants crawling the Möbius strip ladder, which is arranged in the shape of the infinity symbol.

A Mechanical Solution

Mechanical engineers discovered something new – a solution for various mechanical problems. In 1871, a few years after Möbius’ and Listing’s discovery was published in Germany, an article was published in Scientific American by an anonymous mechanic which described an approach for working with mechanical belts: “a rule for putting on a quarter twist belt to make it stretch alike on both edges, and do their work well, no matter what the width of the belt. The belt is to be put on in the usual way, and the ends brought together ready for lacing. Then turn one piece the opposite side (or inside) out, and lace. The belt will run, it will be found, first one side out, and then the other and will draw alike on both sides.”

The above makes clear that the mechanic not only made a Möbius strip out of a mechanical belt, but was also aware of its main property of having only a single side. This property allowed the “Möbius belt” to wear evenly on its entire surface (a small note here: mechanical belts are usually made of laminated layers with a strengthened core covered by a carrying surface layer. Hence, it’s beneficial to prolong the life of the surface layer by spreading the load evenly on “both sides”, or on one side in the case of a “Möbius belt”).

Is it possible that in a such short time the knowledge of the mathematical characteristics of the Möbius strip not only covered such a great distance but was also applied in a completely different area of knowledge? Possible, but unlikely. It makes more sense that it was discovered independently, possibly even based on some earlier tradition.

But wearing evenly the entire surface, i.e., having a single side, isn’t the only exceptional thing mentioned by the mechanic. He mentions another curious property of such “Möbius belts”: the ‘quarter twist belt’ part indicates that the belt was used to connect two pieces of machinery that are placed at ninety degrees (a right angle) between them. And indeed, it’s a common need to shift the motion by ninety degrees. So common, that it’s documented many years before Möbius, during the golden age of Islamic science, in the Book of Knowledge of Ingenious Mechanical Devices written by al-Jazari in 1206.

Al-Jazari, from Jazirat ibn Umar, present-day Cizre, Turkey, shows a diagram of a chain pump, made of a chain, or a rope, linking the buckets for the pump arranged as a Möbius strip.

A Mosaic

In ancient Rome, mosaicists discovered something new – a coiled ribbon with an uneven number of twists. In that period, it was common for mosaics to have a central emblem (usually a mythological figure), surrounded by several panels with geometric designs, and oftentimes a coiled ribbon as a border, with two colors to indicate the two sides of the ribbon. Sometimes, mosaicists had to discover an unpleasant truth: having two colors won’t be possible if the coiled ribbon is twisted an uneven number of times.

The Möbius strip can be done in a more sophisticated manner than with a single half-twist: having any uneven number of half-twists creates a one-sided Möbius strip, that cannot be colored with two colors (and having an even number of half-twists creates a standard two-sided ribbon). For mosaicists that portray long coiled ribbons, it’s inevitable to discover, sooner or later, this property.

It didn’t always go smoothly: in one Roman mosaic from Ostia containing a coiled ribbon, it’s clear that the artist was unaware of that property. He made a two-colored ribbon, twisting it an uneven number of times, just to discover at the end that it cannot be done, causing the artists to cover it with a clumsy fix.

Another mosaic that appears in a surrounding panel of the central emblem has a striking resemblance to a five-twisted Möbius strip, but whether it’s indeed a Möbius strip, or just a closely resembling shape, remains unclear.

But there’s one mosaic that is extraordinary. Dated to 200–250 CE, the mosaic portrays Aion, the god of eternal time, cyclic ages, and the cycle of the year and the zodiac. Aio’s time is perpetual and cyclic: the future is a returning version of the past, in contrast to Chronos which represents an empirical, linear, progressive, and historical time, which divides into past, present, and future. In the mosaic, Aion stands between a green and a bare tree (summer and winter). Before him is the mother-earth Tellus, the Roman Gaia, with four children, the four seasons personified. He holds a celestial sphere decorated with zodiac signs; however, the “sphere” isn’t a cycle but a Möbius strip.

Portraying the zodiacal signs on a Möbius strip allowed the artists to show all the signs, without hiding any of them on the “other” side of a circle, as it was usually done in other mosaics portraying Aion with the celestial constellations. The cost, however, is that some of the symbols of the constellations are not all in their natural places in the band of the ecliptic.

Was the use of a Möbius strip just a smart device to allow the representation of all zodiacal signs, or did the artists use it to strengthen the idea of cyclicity of time and its return to the original starting point, doing it more elegantly and leaving more powerful impression than using a cycle? One can only guess.

A Natural Phenomenon

But, without any doubt, long before the artists, mathematicians, mechanics, and mosaicists, the “Möbius” strip was invented by nature:

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Featured image by paulabassi2, Pixabay.

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