Following fractal maps


Nail-clippers, toasters and voyages of fractal dimension – Salesforce’s Peter Coffee has maps and metaphors on his mind.

Peter Coffee
Peter Coffee

Upon arriving at Heathrow airport, quite some years ago, I noticed that I had snagged a fingernail on something. It needed a quick trim to avoid further damage.

A Boots drug store presented itself, right there in the terminal, and there I found a clipper that was airport-security-friendly: rather than a sharp-pointed nail file, which someday might be classed as a dangerous weapon, it had an abrasive plate that was just as effective and clearly non-lethal. I don’t remember what I paid, at the time, but you could buy the identical item today from a U.S. provider for about £4.

We live in a world of such remarkable artifacts. For example, a typical smartphone checks at least seventy of the boxes in the periodic table of the elements (which you may not have had in front of you since you last took a chemistry class). Even so, for some reason, it’s those ordinary clippers that force me to think about the networks of materials and skills that are usually hidden behind the scenes of our lives.

If I try to imagine digging out the ores, smelting the base metals, combining them into stainless steel, forming it into those precisely shaped parts, and adding an edge where it’s needed, I wonder if it would take me years to produce a bad imitation of what can be bought (if you’ll settle for a minimal solution) for as little as £1.25. Further, as soon as you look at anything more complex, you quickly progress to book-length examinations and multi-year investigations.

For example, Henry Petroski’s nail-clipper moment was apparently triggered by an ordinary wooden pencil: he ultimately spent 448 pages exploring a path that began in the 1500s, when graphite was a remarkable discovery. Uncountable twists of both engineering and economics have led us to presume, today, that every student can be expected to arrive at a testing session with several #2 pencils at hand – a remarkable thing, really, considering that so-called “black lead” was once such a strategic material that its theft by closely supervised workers was classified in 1752 as a felony.

Thomas Thwaites didn’t settle for merely documenting the history of another everyday object, the electrical toaster. He set out to build one, “with the intention of only using pre-industrial tools and methods” – and realized, quite quickly, that “by taking things like trains, or using Wikipedia, or even not making my own shoes for walking to a mine, I was already in a sense ‘cheating’.” That’s exactly what I meant when I referred, above, to “networks of materials and skills”: once you start working your way from any single point toward the origins of anything, it becomes a voyage of fractal dimension. The more precisely you try to measure it, the longer the measured length becomes.

For this reason, I wince when politicians and regulators try to go beyond the already challenging task of such measurement: when they try, in effect, to redraw the economic map, though it defies any attempt to magnify even its current state enough to see every pathway. At best, command economies and prescriptive regulations may merely prevent us from finding better routes; at worst, they may steer us into pitfalls, because a map can be “accurate” in a formal sense while still being dangerously misleading.

Mapping thoughts

Metaphors of maps, and their errors, have been on my mind of late, as I’ve shared with several of the groups to whom I’ve spoken in the past several weeks. I started by thinking about the distortions of the maps most commonly seen on classroom walls, which typically break the Pacific Ocean into pieces at the left and right sides of the map and thereby disguise its true size. Speaking to an audience in Sydney, I noted that a different view makes the scope of the “Pacific Rim”—as a marketplace, as well as a geographical feature—far more apparent.

I object, in principle, to calling such alternative maps “upside down,” preferring the label “Corrective Map” used by Stuart McArthur. (Yes, he’s an Australian.) Merely panning and rotating the view, however, does not complete the needed adjustments if one is still using the common “Mercator” projection. A Mercator map does exactly one thing well: it allows a straight line (drawn with one of Petroski’s pencils?) to represent a course that can be steered with a constant magnetic-compass heading.

McArthur map

For voyagers who limit their journeys to the Earth’s middle latitudes, using the navigational tools of previous centuries, the Mercator map is a useful aid. As soon as one leaves the tropics, though, it becomes clear that (i) Mercator-map lines (also called “rhumb lines”) are not great-circle paths of minimum distance between two points, and (ii) the sizes of land masses near the poles are grossly exaggerated by this projection of a spherical surface onto a flat page.

In short, these maps that we know best are ideally suited for using obsolete methods, to steer inefficient pathways, to destinations that are actually much more limiting than they appear.

There’s an oft-repeated quotation, “If you don’t know where you’re going, any road will take you there”. It’s usually said to come from Alice’s Adventures in Wonderland, but that may be a mis-attribution of what appears to be a George Harrison lyric. If you’ve followed my road this far, I hope I’m offering you two useful warnings for the next time someone is proposing to give you directions.

First, the territory is much richer in vital details than any map can capture, so plan with humility: expect to be surprised, and be ready to reroute quickly rather than stubbornly following the first line drawn.

Second, every map has an agenda, in terms of what it makes most accurate at the expense of what it distorts: choosing a familiar map may minimize arguments at the beginning, but it may not take you where you want to go.

Image credit - Peter Coffee