# Experiments with Ternary Plots in Tableau

A few months ago, I started
a conversation on Twitter where I asked for people’s thought’s on parallel
coordinates plots:

I don't have a lot of experience with parallel coordinate charts but I've encountered a couple recently. My general opinion of them, at this time, is not positive. Seems like there are much better options. So change my mind! Anyone have any good use cases or good examples?— Ken Flerlage (@flerlagekr) February 7, 2019

This led to some
fantastic conversation which, I’ll admit, definitely helped to change my
perspective on these charts. It also led to me attempting to build a number of
different charts to visualize multivariate numerical data, which I’ll share in
a future blog. As this conversation developed, we also began a discussion about
alternatives to parallel coordinates charts and RJ Andrews, author of the fantastic book Info We Trust, mentioned the Gibbs Triangle, also known as a Ternary
plot.

Did I hear someone mention the Gibbs triangle? (1873!) https://t.co/2HN6UrVSuz— RJ Andrews (@infowetrust) February 8, 2019

Trilinear/ternary plots aren't really a thing anymore. Has anyone ever tried one out? pic.twitter.com/1nPsRiT7ZF

I had never heard
of this chart type before so I read up on it a bit and decided to include it in
my project on visualizing multivariate numerical data. Shortly after this
conversation, the brilliant Adam McCann wowed us with this
beautiful and insightful Tableau visualization, which became Viz of the Day on March 1, 2019.

This visualization
used a ternary plot to show “the most unique words by character for each
chapter in the 5 Game of Thrones books.” After seeing this, I immediately
messaged Adam to tell him how fantastic it was. I also mentioned that I had
been planning to build a ternary plot for another project and he said to go for
it, so I began working on it.

Fast forward a few
days and I saw this tweet from Alex Selby-Boothroyd, Head of Journalism at The
Economist.

I'm always in awe of the great things my colleagues come up with, but this interactive by @JamesFransham, @martgnz and @futuraprime is exceptionally good https://t.co/MDIUOzSuE7 pic.twitter.com/CAYi5ErrPa— Alex Selby-Boothroyd (@AlexSelbyB) February 22, 2019

The tweet shows
another ternary plot, but what I loved about it was the tracking lines that
help provide context about the actual values of each of the three measures
being visualized. So, I decided to also build that feature into my own ternary
plot.

In this blog, I’ll
discuss the results of my experiments with ternary plots, discuss some of the
technical details of how it was built, what I learned, and what I like and
dislike about this chart type.

**Ternary Plots**

Let’s start out by
talking a bit about ternary plots. A fantastic blog on datavizcatalogue.com defines a ternary plot as a “triangular-shaped graph is used to plot a
dataset with three variables, where the sum of all three adds up to a constant
amount. Typically the data is in percentages or in an equivalent decimal form.
Ternary Graphs visualise the ratios between the three variables, by simply
positioning a dot in accordance with its position on each of the three axes
(using barycentric coordinates).”

Okay, so in some
ways, these plots are similar to a scatter plot, yet they plot 3 variables on a
triangular. As the blog does a fantastic job of explaining ternary plots and
their use cases, I won’t go into this much further here. Just go read the blog
and come back. I’ll wait…

**Percentage Ternary**

As described on the
blog, ternary plots are very useful in fields such as geology and chemistry as
you can use them to fairly easily plot the makeup of various samples into three
component parts. That being the case, my first attempt at building a ternary
plot in Tableau showed chemical compositions—carbon, nitrogen, and silicon—of
1000 samples (note: this data set is completely fictional and simply used to
show a sample of the chart).

My reason for using
chemical composition data is simply that I found it easier to understand when
the three variables add up to 100%. In such cases, no normalization of the
variables is required, which makes the chart easier to comprehend. In addition,
when all add up to 100, a large value in one variable will always mean a
smaller value in another variable, which means that placement on the triangle
will not only tell you about the relationship between the variables, but also
something about the raw magnitude of that variable (I’ll come back to this when
I discuss my second example).

We can look at this chemical composition ternary and immediately get some good insights. The color of each dot is a specific category of samples, so we can quickly see that the blue category are more heavily made up of carbon than nitrogen and silicon, but there are a number of outliers in this category (in the upper right hand side of the chart) that have very low amounts of carbon. And, there are some in the middle area of the plot that contain a much more balanced amount of each element.

**The Technical Stuff**

Having discussed my
first attempt briefly, let’s go back and discuss some of the technical aspects
of building this in Tableau. Fortunately, the basic ternary plot is relatively straightforward—we
just need the math for finding the x and y coordinates so we can plot them on a
scatter plot. Luckily, the calculations are pretty well documented on Wikipedia. Essentially, our three
measures are labeled A, B, and C. The maximum A value will be in the lower left
hand corner of the triangle (coordinates 0, 0), the max B will be at the bottom
right (1, 0), and the max C will be at the top (0.5, √3/2). We can then take
the values of A, B, and C and plug them into the following formulas to get the
x and y coordinates of each of those points:

__X__

__Y__
From there, we just
plot the x and y coordinates on a scatter plot. To get the triangle in the
background, I simply used an image that I created in PowerPoint.

That’s the easy
part. The most complicated aspect of this implementation of ternary plots is
the tracking lines. The math here is relatively complex, so I am not going to
go into a lot of detail. I will, however, give you the basic idea of the
methodology, at a high level. First of all, we need to use some data
scaffolding to create the extra data needed to draw each of the three lines.
Next, we’ll need some geometry to figure out how to draw the lines. The key is
to find the intersection (shown in blue below) between the lines radiating from
each point (orange) and the lines that make up the sides of the triangle (grey).

To do this, we need
to put each of the six lines into equations in slope/intercept form. Then, we’ll use some math to find the intersections. The basic idea of the
math is that we’re finding a point on each line where both x and y are the
same. Because these are different lines, there will only ever be one such
point. So, since we have both equations in slope/intercept form, which describe
the line using x and y variables, we can essentially set the equations equal to
each other and solve. Finally, having found that intersection point, we can end
our radial lines at that point.

Okay, I know that
doesn’t do a great job of explaining it, but I’m not going to go into any
further detail about that here. If you’re interested in the math, I’d recommend
that you read the following, which proved to be a good refresher for me: intersection of two lines. Also, feel free to download my workbook and explore my calculated
fields. I’ve tried to do my best to clearly organize them into folders and to
include comments on each field so that they are as understandable as possible.

Finally, we need to
set it up so that the connecting lines will only be drawn for the highlighted
point. To do this, I used one of my favorite features—set actions!! (Note: I
created this before parameter actions were a thing. Had I created this now, I’d
probably use parameter actions instead). I created a set action that would, on
hover, add the point to a set. From there, I was able to create some calculated
measures to determine the end points of the tracking lines, then use those
measures as a dual axis to actually draw them. Again, feel free to download the
workbook if you’d like more details.

**Non-Percentage Data**

Okay, with the
technical stuff out of the way, let’s build a slightly different ternary plot
using non-percentage data. I dug up some data I had previously visualized which
rates various comic book superheroes and villains in a number of different
categories, rating them on a scale from 0 to 100 (source: www.superherodb.com). Specifically, I chose to visualize each character’s strength, speed,
and intelligence.

Fortunately, these
three measures use a shared scale. As mentioned in the math section, I believe
this aids understanding tremendously, as it does not require the data to be
normalized, the act of which takes an already difficult-to-understand chart and
further obscures the data.

However, even with
a shared scale, the numbers are not percentages—they do not always add up to a
single value. One character may have strength of 10, power of 20, and
intelligence of 30, while another character may have 90 for all three. So, from
that standpoint, the chart can still be a bit difficult to understand. For
example, Solomon Grundy, appears in the bottom left hand corner, indicating
that he is very strong (a rating of 93).

Were this a
percentage, it would indicate Solomon Grundy is stronger than all other
characters. But, because we’re not dealing with percentages here, that is not
the case. Galactus, for example, appears near the center.

Yet, Galactus has a
strength rating of 100, seven points higher than Solomon Grundy. Thus, in this
case, proximity to the corners do not necessarily mean higher ratings for that
specific variable. Rather, it means that the character’s ratings are more
heavily

*weighted towards*that variable. Solomon Grundy appears far into the bottom left corner because he is strong, yes, but also because he is slow and not very intelligent. Galactus, on the other hand, is strong, fast,*and*intelligent—he is not weighted towards any of the three characteristics. In other words, Galactus is much more balanced.
In my ternary based
on percentage data, I had labeled each axis to show where there were low and
high levels of a certain element, but because of what we’ve just discussed,
that will not work in this case. So, instead, I chose to create a separate key
which includes three triangular “quadrants.” The three quadrants in the corners
of the triangle indicate characters weighted more heavily towards one of the
three qualities, while the fourth, in the middle, shows those characters who
are more balanced.

With this key, I
think the chart becomes much less confusing and we can easily gain insight
quickly. We can, for example, see that most of the characters are fairly
well-balanced. Those who are not well-balanced tend to be more intelligent than
strong or fast, such as the Joker or Lex Luthor.

The tracking lines are also a significant problem. I wanted to draw them perpendicular to
the axes, as was done on the chart by the Economist, but, like the issues we’ve
discussed, these tracking lines do not work properly when the measures do not
add up to a standard total, such as 100%. Thus the parallel tracking lines do not connect to the correct position on each axis. At best, this makes them useless. At worst, they can be misleading.

**Final Thoughts**

After creating a
couple of ternary plots, I have to admit that I have mixed feelings about them.
There are some obvious problems. First and foremost, they are confined to just
three variables. This, of course, limits the potential use cases quite a bit. My
superhero data set, for example, actually has three other variables—Durability,
Power, and Combat. What if we wanted to visualize all six at once? We could use
a ternary and allow the user to select the three they wish to compare, but if
the goal is to visualize all six at once, that may not be the best option.

Additionally,
despite having spent quite a bit of time with them, I still find them quite
difficult to read. At their best—as in the case of comparing chemical
compositions—I think they can be quite effective as a point’s placement on each
axis is quite meaningful. At their worst—such as a case where we are comparing
measures with different scales—I feel like they can be very tricky to fully
grasp. That being said, the chart is ultimately more about getting a macro
understanding of your data than understanding each individual data point. And,
in that sense, they can be a valuable tool in our toolbox, especially when you
provide visual aids such as the key I share earlier. So, as I always say,
whether or not you use a ternary plot will be largely dependent on your data,
your audience, and the story you’re trying to tell.

Ken Flerlage, July 3 1, 2019

Hi Ken, I try to regenerate the dashboard in my own template. But when drag all of the dimension and measures to the worksheet as you did, i can only get the points in the worksheet. Is there any way to connect the intercept, and generate the line? I upload the dashboard into my public account. if you have time, please help to check that. Thanks https://public.tableau.com/profile/robert.wang#!/vizhome/Ternaryplots/Sheet2

ReplyDeleteThe tracking lines are triggered from the "Highlighted Record" Set. Something needs to be added to that set for the lines to be drawn. Take a look at the actions in my workbook and you'll see that I've created a set action on hover. Set up that same action and you should see the tracking lines being drawn.

DeleteThanks Ken. The track line show up with the action set. Have another question for the line connected each points (line AB,AC, BC, and the small triangle line in the big triangle ABC). How to get these lines show up in the worksheet? When I drag your pointX and point Y, the line automatically generated. But in my worksheet, I can only see the points, instead the points with line.

DeleteI'm not sure I follow. Perhaps it would be best to handle this offline. Would you be able to send me an email at flerlagekr@gmail.com?

Delete