# James Mai

I employ combinatorial and permutational procedures to produce sets of related but unique visual forms: related in that all forms of a given set share common features of line or shape, and unique in that each form is a distinct arrangement of those features. Further, I aim to produce form-sets that are both non-redundant and complete: non-redundant in that no two forms are alike by symmetric transformation, and complete in that every unique combination or permutation permissible by the given parameters is present. I also intend that these relationships be visually discernible in the artwork, so I choose parameters that will yield a modest number of forms, usually between about 12 and 36 unique forms.

The central tree-like form shows bifurcation branching (1, 2, 4, 8, 16). The twisting and overlapping of each pair of branches (V-pairs of same color) results from arranging the branch-nodes (white dots) such that each pair is separated by a different number of intervening nodes---for example, from left to right, the 8 topmost pairs are separated by 5, 0, 7, 3, 6, 2, 4, and 1 branch-nodes (rearranged, the intervening numbers are consecutive integers). Similarly, the level below has 4 node-pairs separated by 3, 1, 2, 0; the level below that, by 2, 0. Around the edges of the diamond are 24 branches of another kind, each figure a different permutation of branch-lengths of 1, 2, 3, and 4 units, in bottom-to-top order on a central “trunk".

The 22 forms in this composition are the complete set of figures that result from connecting the 8 vertexes of a regular octagon to yield 1 quadrangle + 2 line segments. Those forms with the same kind of quadrangle (but different arrangements of line segments) share the same colors. The forms are also grouped by scale and distance from center according to the number of internal axes of reflective symmetry: 1 central form possesses 4 axes of symmetry; 4 forms in the ring surrounding the center possess 2 axes; 10 forms in the next ring possess 1 axis; 7 forms in the outer ring possess 0 axes (asymmetrical).

These forms result from the question: how many distinct arrangements of 4 line segments are possible to connect 4 pairs of points arrayed on opposite edges of a square? Of the 13 possible ways, 2 forms are asymmetrical (green), 2 forms are rotationally symmetric (red), 3 forms possess 1 axis of reflection (yellow), and 6 forms possess 2 axes of reflection (blue).