Synergetics Coordinates

Synergetics coordinates are a set of triangular coordinates in their plane (or their generalization to tetrahedral coordinates in space, or the analogs in higher dimensions). In the plane, coordinates are measured along three axes , , and , with the -axis oriented downward and the and axes oriented at angles to each other as illustrated above (left figure). Interpreting , , and as points on the sides of an equilateral triangle obtained by parallel-displacing from the origin three pairs of lines oriented at angles with respect to one another, the coordinates can be interpreted as specifying a given equilateral triangle (right figure).

A nice property of these coordinates is that the vertices of the triangle obtained by parallel-displacing by are given by , , and (see above figure), so that the sums of the coordinates of the vertices are always zero. This property also holds when the coordinates are generalized to three and higher dimensions.

The synergetics coordinates also have the property that the edge lengths of the equilateral triangle described by is precisely , which again generalizes to higher dimensions.

Synergetics coordinate provide a convenient way to construct regular circle and sphere packings. For example, the ring of circles illustrated above at left can be generated by picking all sets of integer synergetics coordinates that sum to zero and such that the sum of the absolute values of the three coordinates divided by two equals one (Nelson). Similarly, the second ring of circles can be obtained from all sets of integer coordinates that sum to zero and such that the sum of the absolute values of the three coordinates divided by two equals one. The zeroth, first, and second rings are illustrated above at right.

Similar properties hold in three dimensions, where appropriate sets of synergetics coordinates give successive shells in a regular sphere packing (Nelson), illustrated above.