Images: Soil ternary plot, Mike Norton via wikimedia.org, CC BY-SA 3.0. Chip ternary plots, S. Heard.
I’ve always been mystified by ternary plots – you know, those cool looking triangular ones. I shouldn’t be; they aren’t really that complicated. But while Cartesian plots (in two dimensions or three) speak to me easily and clearly, ternary plots remain stubbornly silent.
I’ve survived this cognitive failing for nearly 30 years by deploying a strategy based entirely on avoidance. Ternary plots just aren’t used that much, in my field, except with a couple of specific kinds of data that are conveniently treated as mixes of three components – soil composition (sand, silt, and clay; above) being perhaps the most common. But my avoidance strategy came crashing down around me last semester, when I taught part of second-year Ecology as a sabbatical fill-in. There is was, right there in the 4th week’s lecture outline: soils. Field capacity, available water capacity, wilting point, soil horizons, and – oh, the humanity – that conventional ternary plot of sand, silt, and clay. I had to teach it – and I didn’t understand it.
Something had to give, of course, and I knew it had to be me. I needed a ternary-style dataset I understood intuitively (and soil composition wasn’t working for me). The penny dropped when I realized I’d been ranting about potato chip flavours. Over and over again, I pick up a new and interesting-sounding flavour of chips (most recently, “Harissa hummus”), only to discover they taste pretty much like last week’s flavour. I figured out why, and in doing so I developed a Grand Unified Theory of Potato Chips (GUT-Chips)* – and it was exactly the kind of theory that called for a ternary plot.
GUT-Chips is simple: it holds that the dozens of flavours of potato chips on the market are all just simple variants of the three fundamental potato chip flavours: BBQ, salt & vinegar, and sour cream & onion**. (It’s tempting to describe the flavouring powders adhering to chips of these three flavours as elementary particles.) To illustrate the theory, we need a ternary plot (above). Each fundamental flavour occupies a vertex of the plot: Salt and vinegar chips, for example, are at the top, at (unsuprisingly) the 100% extreme of the salt-and-vinegar axis (the right-hand one). [Notice, just to keep you awake, that the chip plot reverses the axis directions from the soil one]. At 0% on the salt-and-vinegar axis we find BBQ (at right, 100% on the BBQ axis) and sour cream and onion (at left, 0% on the BBQ axis but 100% on the sour-cream-and-onion axis. Other flavours are described as mixtures of the three fundamental flavours. For example, dill pickle chips (yes, they exist, and if they aren’t marketed in your region you’re missing out) are simply 50% sour-cream-and-onion, 50% salt-and-vinegar – look for them half-way along the left axis. Ranch is a little more complicated: about 70% sour-cream-and-onion, 20% BBQ, and 10% salt-and-vinegar. Practice reading that off the axes. As you move along an axis, you follow the hatch lines that leave it at an obtuse (>90º) angle. For the salt-and-vinegar axis at right, those are the horizontal hatch lines (10%, for ranch); for the BBQ axis on the bottom, they’re the hatch lines that rise to the right (20%, for ranch); and so on.
If that isn’t enough to make ternary plots clear, I find it helps to think about vectors. Imagine that you held one component constant and varied the mix of the other two. Below, I’ve done that for a vector connecting ranch to salt-and-pepper chips. Both, according to GUT-Chips, are 20% BBQ. Possible flavours at 20% BBQ run from 20% BBQ, 80% sour-cream-and-onion (a hypothetical chip at the base of the vector) to 20% BBQ, 80% salt-and-vinegar (salt-and-pepper chips, at the tip of the vector). Each time this vector crosses a pair of intersecting hatch lines, that’s replacement of 10% sour-cream-and-onion (the left axis) by 10% salt-and-vinegar (the right axis). This vector exercise, actually, is what finally turned the ternary light on for me. I get it! I get ternary plots! And all thanks to GUT-Chips.
But as often happens, you work at learning one thing, and you get some serendipitous insight about something else. What’s with the pink region in my chip plots? This is the GUT-Chips 1st Corollary: there is a Region of Culinary Repulsion (RCR) in which combinations of the three fundamental flavours are not actually very good. Here’s the main GUT-Chips plot again:
What’s interesting is that the RCR sits near, and includes, the middle of the plot. The three fundamental flavours are all delicious, and we can tinker around their edges to good effect; but more equal mixtures are a bad idea. The RCR includes the Ur-Chip (centre; 33% salt-and-vinegar, 33% BBQ, and 33% sour-cream-and-onion); we could make such a chip, but we shouldn’t. Honey-mustard chips (just left and down from centre) flirt with the RCR; we do make these chips, but we probably ought to stop***. The vector from ranch to salt-and-pepper (both good) crosses the RCR, and so do lots of other interchip vectors. In life, as in chips, there are a lot of continua where either end is great, but the muddled middle is a bad idea. As an evolutionary ecologist, of course, I’m tempted to analogize this to hybrid disadvantage – but perhaps that’s pushing things too far. I would never do that.
© Stephen Heard (firstname.lastname@example.org) April 14, 2016
*^Yes, I am that much of a nerd. Enough of a nerd to develop such a theory; and enough of a nerd to call it that. But if you’ve been reading Scientist Sees Squirrel, you probably knew that.
**^GUT-Chips cannot accommodate a few very weird flavours, like blueberry, milk chocolate, and octopus. Perhaps I should specify that GUT-Chips applies to flavours with market share exceeding 0.001%. The fuller range of novelty flavours awaits the Grander and More Unified Theory of Potato Chips (GaMUT-Chips). Yes, I am uncommonly proud of that acronym.
***^Hey, it’s my blog. If I want to make unsubstantiated and arbitrary statements of opinion masquerading as fact, who’s going to stop me?