Recycling Symbol

The Möbius strip can be found in many places in the wild, often with lay people being unaware of the background, notably of which is the oft-seen recycling symbol, commonly found on paper and cardboard products (Fig. 1). This consists of three chasing arrows folded in a Möbius strip. Although not impinging on the main study (above) in any way, in the spirit of a 'fun' inclusion, I have decided to look into its instigation. A few obvious questions include whom the designer is and when the symbol was introduced. Upon research, this is relatively recent history, with the beginnings not lost in the mists of time and so pleasingly can be fully answered. In short, it was designed by Gary Anderson in 1969. The background is that Container Corporation of America, a large producer of recycled paperboard, sponsored a contest for art and design students at high schools and colleges across the country in 1969 to raise awareness of environmental issues. It was won by Anderson, then a 23-year-old college student at the University of Southern California. It was immediately used at the first Earth Day on 22 April 1970 (an annual event to demonstrate support for environmental protection), albeit it was not immediately widely adopted. Indeed, Anderson himself had rarely seen the symbol in the US, and it was not until some ten years later that he noted it widely displayed prominently on recycle bins in Amsterdam. To some extent, the man himself had also largely been forgotten, with no known interviews up to 1999. This then changed somewhat, when Penny Jones and Jerry Powell (with an interest in recycling) hunted him down in 1999 and got his story. From this, there was a renewed surge of interest in him, mostly from the environmental community, rather than the mathematical. Even so, interviews are relatively rare, albeit more frequent in recent times. Historical pictures, and accounts, are also rare. Indeed, there appears to be only one picture with Anderson in association with the competition (Fig. 2).

This shows Gary Anderson and Hans Buehler of the CCA with the winning design, in 1970. Further, the competition poster advertising the competition appears to have been lost; there is no known photograph of it. (Does anyone know of it? Do let me know! Anderson, in a video interview with Nicole Robertson, at 29.50, also states that he has not seen the poster since.) Another slightly unsatisfactory element for recreation purposes of the symbol is that the only photo is taken at an angle, and so recreating the design exactly i.e. (its proportions) is not straightforward. However, there is a contemporary orthogonal drawing that shows the design in this sense 'better', in American Home Magazine, 1971, and a large-scale sketch, from Anderson himself.

Largely out of personal satisfaction, I decided to try and recreate the exact proportions (Fig. 4). Surprisingly, for such an obvious idea, I have not seen this elsewhere. For this, I have drawn on isometric paper, given the 60° angles of the design. The key to understanding is not of determining the overall outline, an obvious first thought (and as tried), of a broad irregular hexagon with rotational symmetry (negating the interior detail), but rather that is based of the interior, an equilateral triangle, with the arrows on the exterior. From this, the intricacies then fall swiftly into place. It will be seen that the outline remains the same, while the interior is subtly different in one of the arrows, albeit easily missed, Therefore, it is essentially one drawing (of a folded arrow), repeated three times.

This gives an exact recreation or is as near as practical as it can be. As related above, I am working from a sketch, without an isometric grid, and so there may be very small differences (such as the gaps between the arrows). However, if so, these are very minor and do not materially affect the core premise.

I have purposefully retained the (pencilled) construction/registration lines, albeit they are hard to see in this drawing. The curves are made up of two small and large circles. As a reminder, the arrows are not all alike in interior. All is well and good. However, the question remains as to the constituent of the Möbius strip - is it of one or three half-twists? I now also look into this matter. To better understand the background, i.e. as a 'pure' Möbius strip, I also drew the same strip without the individual arrows (Fig. 5).

Again, I have purposefully retained the (pencilled) construction/registration lines. To this end, I then made paper models, with a half twist and then flattened, all the while keeping the orientation of the Anderson drawing (Fig. 6).

Fig. 6. Top, two Möbius strips, half twists, folded towards and away from me (to give the two 'variations'), lower, flattened.

So what have we here? The left column matches the recreation of the Anderson diagram, whilst the right column does not. It will be seen that the half-twist model 'naturally' (with minor assistance) folds flat into the 'rotational hexagon' outline.

Continuing in the same vein, I now recreate the 'other' (truly rotational) recycling symbol, based on the Anderson model in terms of proportions (Fig. 7).

As can be seen, this subtly differs, in that both the interior and outline remains possess order 3 rotational symmetry.

And likewise, I now show it as a Möbius strip without the arrows (Fig. 8)

As above, I now move on to showing these as physical models. As can be seen, this necessitates three half-twists. For this model, it will be seen that the strip must be relatively long in proportion to width e.g. 15:1 (even a 10:1 strip will not fold). A short, stubby strip will not fold easily (if at all), whereas a long, narrow one will.

Again, these fold as a truncated triangle.

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The first account I have seen that discusses this topology curiosity was by Cliff Long, a mathematics professor, in 'Möbius or Almost Möbius', in 1996 (Fig. 10), a single page discussion. I repeat the text immediately below:

Have you noticed that there is more than one version of the ubiquitous recycling symbol? There is a distinct topological difference between these two versions, which frequently appear in newspapers and on envelopes, cardboard cartons, bottles, and recycling containers:

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Their arrows suggest forming strips with three folds, as shown in the corresponding figures below. Note that in the strip on the left, the two lower folds twist the strip in opposite senses, undoing each other, while in the right-hand strip, all three folds produce twists in the same sense (each a counterclockwise half-twist, if one follows the direction originally indicated by the arrow.

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The strip that corresponds to the leftmost symbol has a single half-twist - it is the famous nonorientable Mobius strip. The other strip has three half-twists. A string that follows the edge of this strip around two circuits, ending where it began, forms a trefoil knot in space.

Created 26 April 2024. Last Updated 26 April 2024