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. Arguably more so than any other representation it has acquired a degree of fame (albeit not readily realised to the uninitiated), seen around the world, so much so that it has a dedicated, in-depth subpage of the logo section.
A few obvious initial questions include who the designer was and when the symbol was introduced, both of which are not generally known. The design is invariably shown uncredited. Upon research, this is a 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. Three designs were submitted, of minor variations of each other.

This shows Gary Anderson and Hans Buehler of the CCA with the winning design, in 1970, left; with a preliminary sketch, right.

The original intention of Anderson was to have the design to be shown 'top heavy' as in the sketch. However, William Lloyd, of the CCA, suggested rotating it 180, for the sake of better balance, and this is how it is normally portrayed. 

Much of the artwork and imagery in general has apparently of the competition has been lost. Indeed, there appears to be only one picture with Anderson in association with the competition (Fig. *). Additionally, there is only a concept sketch known. 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 recent video interview with Nicole Robertson, at 29.50, also states that he has not seen the poster since.) 

The symbol 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. I have not been able to find pictures of such use here.

It should be noted that the symbol took a while to be commonly adopted. Indeed, Anderson largely put it aside whilst he carried on with his career. 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. As a broad statememt, it only began to proliferate in the 1980s. 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. 

A matter of some importance is the exact proportions of the design itself. Amazingly, this does not seem to have been discussed previously. Therefore I thus address the matter. However, with a paucity of detail, and one concept sketch, this is not as straightforward as may be thought. The concept sketch does not show the underlying isometric grid. The only known 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, From this one can extend lines that form the underlying structure, and so recreate, if not exactly, then to a very close degree

Largely out of personal satisfaction, I decided to try and recreate the exact proportions (Fig. *). 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). 

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

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

Created 3 July 2024 (from existing text of April 2024). Last Updated 3 July 2024