# How Do I Use Ternary Diagrams?

*Depicting Three-Component Systems in the Earth Sciences*

**Ternary Diagrams** *are also known as:*

*triangular diagrams*

ternary plots

three-component diagrams

## An Introduction to Ternary Diagrams

Ternary diagrams are graphical representations used to visualize systems with three components. Ternary diagrams allow geoscientists to represent the relative proportions of three components in a system. In order to plot data on a ternary diagram, the components are normalized to 100% (i.e., you must determine the proportions of each component relative only to the other components represented on the diagram; more on that later). Ternary diagrams are what we call "field diagrams." That is, once you've plotted your data, you can interpret something about your data by determining the field in which the data plot. Many types of ternary diagrams exist in the geosciences, and they are used in many applications. Often, the fields on a ternary diagram are used to classify earth materials or landforms, such as rock, soil, sediment, or mineral compositions; lithofacies; sediment size; dunes or deltas; among many others.

**A Sweet Example:**

Let's consider a non-geoscience three-component system, one with the components chocolate, milk, and sugar, like the image to the right. First, notice how the diagram is labeled: **Chocolate** is on the top **apex**Each corner of the triangle is an apex. *Apices* is the plural form of *apex* (purple), **Milk** is on the left apex (red), and **Sugar** is on the right apex (blue). Also note that there are some colorful "fields" inside the triangle. Fields on ternary diagrams help us to interpret the data and are usually based on empirical data/observations. Several different combinations of chocolate/milk/sugar are plotted (as points) in this ternary diagram and represent different ways that a given data point can be plotted.

- Points that plot at one apex of a triangle, like the blue point labeled "Pixie Sticks," indicate that the sample is made up of 100% of a single component (in the case of the blue dot, 100% sugar).
- Points that plot along one side of the diagram, like the black points labeled
**Semi-sweet**and**Dark Chocolate,**indicate that the sample only contains two components (in this case of the black dots, only sugar and chocolate; semi-sweet chocolate = 50% chocolate/50% sugar, dark chocolate = 72% chocolate/28% sugar; neither contain any milk). - Points that plot inside the triangle, like the pink points on the plot, indicate that the sample contains some proportion of all three components, and the "field" that contains the point gives you information about how to interpret your sample. In this case, the pink dot in the green field would be classified as
**Milk Chocolate**(and contains about 31% milk, 41% chocolate, and 28% sugar); the pink dot in the yellow field would be classified as**Ice Cream**(and contains about 62% milk, 10% chocolate, and 28% sugar).

Where do your tastes lie? Are you a sweet tooth who likes 100% sugar, a chocolate and sugar person, or some combination of all three?

## How Are Ternary Diagrams Constructed?

You may be wondering how I got the relative percentages for the milk chocolate and ice cream dots above. Let's consider a slightly modified diagram like the one in the "Sweet Example" (milk/chocolate/sugar ternary diagram).

**apex**of the triangle is labeled as

**100% of one end-member**

**component**and there are colored numbers along each side corresponding to percentages of each. We can overlay lines that connect the percentages and form a

**grid**(just like any plot, except this grid, it is a bunch of triangles instead of rectangles/squares; see figure at left). Note: Sometimes the grid is provided, and sometimes you will need to construct it (see Step 0 in examples below for more information).

Let's start by breaking down how we made the grid on the ternary diagram to the left. Each end-member component (plotted at each apex) has a set of lines associated with it. **The set of lines associated with an end-member component are** **parallel****to** (do not intersect) **the side of the triangle opposite the end-member's apex**. In the figure to the left, 10 lines (color coded) and 1 apex are associated with each component. All the purple lines are for component A (at left, A = chocolate), red lines are for component B (at left, B = milk), and blue lines for component C (at left, C = sugar). Each line is labeled (on one side of the triangle at left) decreasing from the apex toward the opposite side of the triangle. If we separate each component and the associated lines, they would look like the image below:

The image to the right shows **three triangles**, color coded for components A (purple), B (red) and C (blue), and lines are labeled on both sides; that is, a sample with 70% chocolate could plot anywhere along the purple line labeled 70. To know where it plots along that line, you need to know the proportions of the other components. For example, note that the Milk Chocolate, Ice Cream, and Dark Chocolate described in the "Sweet Example" have 27% sugar (if you draw a line from Dark Chocolate and parallel to the blue lines, it will pass through all three of the dots on the diagram). However, the proportions of other components determine where the points are plotted and how they are classified.

## How Do I Plot Data Points and Interpret Them Using a Ternary Diagram?

*This section describes plotting and interpreting data on ternary diagrams. If you are looking for how to interpret a point that is already plotted,* *skip to the next section*.

### Example 1: Plotting and classifying soil textures

**You are tasked with classifying the texture of a soil sample that you are studying.**After collecting the sample, you weigh the sample, dry it (reweighing it to determine the amount of water), and perform a textural analysis. You determine that your soil contains

**21 g water, 4 g organic material, 18.5 g sand, 21.5 g silt, and 10 g clay.**You have access to the soil texture triangle shown below to use for classifying your soil.

**Use the soil textural triangle below to plot and classify the texture of your soil sample.**

**Here are some steps to follow when** **plotting data****on a ternary diagram. Below each step (initially hidden from view) you will find an example of how to do that step for the following problem. **

**Step 1: Determine** the components shown on the diagram (there will be three) and which apex reflects 100% of each component (and which sides of the diagram show proportions of each component. If the sides of the diagram are not labeled, you will need to decide which side of the triangle will show percentages for each component.

**Step 2.** If the problem gives you *more than three* components, **determine** which components in the problem are relevant to the ternary diagram of interest. If the problem only lists the three components shown on the diagram, you can skip to step 3.

**Normalizing**data means determining the relative proportions of the components to one another.

**Step 3.** **Normalize** the data into percentages of each component. To normalize, divide the amount (weight, fraction, value, etc.) of each component by the total amount (weight, fraction, value, etc.) of all three components. Equations to normalize components:

A: `A/((A+B+C)) " * " 100 = A%`

B: `B/((A+B+C)) " * " 100 = B%`

C: `C/((A+B+C)) " * " 100 = C%`

When you have done the calculations, percentages must sum to 100%: `A% + B% + C% = 100%`

**Step 4: Draw **a line that represents the proportion of **one end-member component** at the appropriate percentage and so that it is parallel to the side opposite the apex representing 100% of that end-member. In some cases, you may have to approximate (interpolate) the location of the line for percentages between those that are labeled.

**Step 4a: Repeat the above process **for the other two components. Your lines should **intersect at a single point **representing the sample composition.

**Step 5. Plot** the normalized data as a *single point.* Determine where your lines cross and place the point there.

**Step 6: Interpret the data plotted on the diagram.** As discussed above, each ternary diagram has fields that allow you to categorize the sample based on the field into which it plots.

## How Do I Interpret Points Already Plotted on a Ternary Diagram?

This section describes how to read and interpret ternary diagrams that already have points plotted on them. Please visit the previous section for information about how to plot points.

### Example 2: Reading mineral compositions for a given ultramafic rock

**This sample plots in a field labeled lherzolite, so, as you learned above, a rock with the components represented by the red dot is classified as a lherzolite** (see **Step 6** above for information about interpreting ternary diagrams**). Use the phase diagram to the right to determine how much of each of the end member components is present in the lherzolite plotted as the red circle so that you can determine which of your rocks is lherzolite.**

**Step 1: Determine** the components shown on the diagram (there will be three) and which apex reflects 100% of each component.

**Step 2: Draw three lines** through the point and parallel to grid lines **to determine** the relative abundances of each end-member**.**

**Step 3: Record the percentages** of each end member component, confirming that they sum to 100%.

**Step 4: Interpret your findings.** Use what you have learned to answer the question posed.

## Where Are Ternary Diagrams Used For in the Earth Sciences?

Many Earth materials and systems can be represented by three components, creating an easy way to classify minerals, sediments, rocks, and landforms. Many sub-disciplines in Earth science have developed standardized ternary plots that were created from observational or empirical data, including:

**Geochemistry:**trends in chemical components with respect to space or time**Igneous petrology:**classification of rock types; phase diagrams**Metamorphic petrology:**phase relations; facies; reactions**Sedimentology:**classification of carbonate and clastic rock types**Mineralogy:**mineral classification (e.g., feldspar, pyroxene)**Water Quality:**cation and anion variation diagrams**Soils:**soil texture classification**Geomorphology:**dune or delta morphology; hydraulic geometry

The figure above shows some examples from the fields of soil science (colorful version of the soil texture triangle in example 1), petrology (coarse grained igneous rocks), and geomorphology (deltas).

## Next Steps

## More Help (Resources for Students)

- Wolfram MathWorld: Ternary Diagrams
- Interactive: How to use a ternary diagram (Geology is the Way)
- Blank Ternary 2%.PDF (Acrobat (PDF) 275kB Jun6 23) Blank ternary diagram with gridlines every 2%
- Blank Ternary 10%.PDF (Acrobat (PDF) 163kB Jun6 23) Blank ternary diagram with gridlines every 10%

*Pages written by Kelly Deuerling (University of Nebraska, Omaha) and Ryan Kerrigan (University of Pittsburgh at Johnstown). Edited by Jennifer M. Wenner (University of Wisconsin, Oshkosh)*