This is about how to calculate the recipes for each of the glazes in a triaxial blend, with a view to implementing this as a function in glaze software.

Suppose we want to use three glazes, *A*, *B* and *C*, to make a triaxial blend. It’s useful to visualise the blends as being arranged in a triangle, with *A* at the top, *B* at bottom left, and *C* at bottom right. We’ll call the number of glazes along each side of the triangle *d*. Note that *d* must be at least 2.

For indexing purposes, it’s convenient to modify this picture so that A lies directly above B:

The relative proportions into which A must be divided are:

while the relative proportions into which C must be divided are:

So if the index *i* counts the rows from the bottom to the top, starting at 0, and the index *j* counts the columns from left to right, starting at 0, then the glaze in row *i*, column *j* has *i* parts *A*, *j* parts *C*, and a yet-to-be-determined number *k* parts *B*. The sum of the parts for each glaze must be the same, and since *A* has *d* - 1 parts A, plus 0 parts B, plus 0 parts C, each glaze must be made of *d* - 1 parts. Therefore

*i* + *j* + *k* = *d* - 1,

so the glaze in row *i*, column *j* has

*k* = *d* - 1 - *i* - *j*

parts *B*.

To summarize: The glazes in a triaxial blend can be indexed by the coordinates of a triangular array, where the entry at position (*i*, *j*) corresponds to the blend which consists of *i* parts A, *d* - 1 - *i* - *j* parts B, and *j* parts C. The glazes A, B and C therefore make up the following proportions of the blend: