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 12 and 36 unique forms.
The 25 forms in this composition are the complete set of unique forms that result from connecting the 8 vertexes of an octagon into 2 triangles and 1 line segment. Of the 25 forms, there are 14 types, defined by the kinds of triangles they possess. There are 5 "pure" types, which employ 2 triangles of the same kind: small isosceles, wide isosceles, tall isosceles, right, and obtuse. There are 9 "compound" types, which possess triangles derived from 2 of the pure types. 9 colored outlines join the types which share the same kind of triangles; that is, each colored outline groups the compound type with its 2 corresponding pure types. The result is a "map" of overlapping territories of forms that share kinds of triangles.
The 13 forms in this composition result from the question: how many distinct arrangements of 4 line segments are possible when connecting 4 pairs of points arrayed in 2 parallel rows? Although each is unique in its linear organization, the 13 forms can be grouped according to the sets of rotations and reflections that leave the forms apparently unchanged. From the center of the circle outward: 3 forms possess 1 axis of reflection (yellow), 2 forms remain apparently unchanged by no reflections or rotations (green), 2 forms are symmetric under rotation through 180 degrees only (red), and 6 forms are symmetric under reflection across 2 axes (blue), hence also under 180-degree rotations.