Ternary plot

Flammability diagram for methane

A ternary plot, ternary graph, triangle plot, simplex plot, or de Finetti diagram is a barycentric plot on three variables which sum to a constant. It graphically depicts the ratios of the three variables as positions in an equilateral triangle. It is used in petrology, mineralogy, metallurgy, and other physical sciences to show the compositions of systems composed of three species. In population genetics, it is often called a de Finetti diagram. In game theory, it is often called a simplex plot.

Example QFL triangle for sandstone provenance

In a ternary plot, the proportions of the three variables a, b, and c must sum to some constant, K. Usually, this constant is represented as 1.0 or 100%. Because a + b + c = K for all substances being graphed, any one variable is not independent of the others, so only two variables must be known to find a sample's point on the graph: for instance, c must be equal to K − a − b. Because the three proportions cannot vary independently, it is possible to graph the intersection of all three variables in only two dimensions.

Reading values on the triangular plot

A ternary plot is drawn as triangle. Each base, or side, of the triangle represents a proportion of 0%, with the point of the triangle opposite that base representing a proportion of 100%. As a proportion increases in any one sample, the point representing that sample moves from the base to the opposite point of the triangle.

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  An example ternary diagram, without any points plotted.
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  An example ternary diagram, showing increments along the first axis.
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  An example ternary diagram, showing increments along the second axis.
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  An example ternary diagram, showing increments along the third axis.
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  Empty diagram
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  Empty diagram (alternative axis)

Using Cartesian coordinates

Cartesian coordinates are useful for plotting points in the triangle. Consider an equilateral ternary plot where $a=100\%$ is placed at $(x,y)=(0,0)$ and $b=100\%$ at $(1,0)$. Then $c=100\%$ is $({\frac {1}{2}},{\frac {\sqrt {3}}{2}})$, and the triple $(a,b,c)$ is $(b+{\frac {c}{2}},{\frac {\sqrt {3}}{2}}c)$

Example

This example shows how this works for a hypothetical set of three soil samples:

Sample #	Organic matter	Clay	Sand	Notes
Sample 1	80%	10%	10%	Because organic matter and clay make up 90% of this sample, the proportion of sand must be 10%.
Sample 2	50%	40%	10%	The proportion of sand is 10% in this sample too, but the proportions of organic matter and clay are different.
Sample 3	10%	40%	50%	This sample has the same proportion of clay as in Sample 2 does, but because it has a smaller proportion of organic matter, the proportion of sand must be larger, because all samples' proportions must sum to 100%.

Plotting the points

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  Plotting a point: finding the first intersection.
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  Plotting a point: finding the second intersection.
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  Plotting a point: the "third" intersection is already found, as it is mathematically dependent on the first two.
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  Showing points and intersection lines.
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  Showing only the points.

See also

• Barycentric coordinates (mathematics)
• Flammability diagram
• QFL diagram