Conceptual Model

The three states conceptual model (current draft 2025), part of my doctoral research, provides a theoretical framework for understanding the generative processes underlying labyrinth patterns.

Each circuit permutation that produces a viable SAT type can be depicted as an open meander—an infinite non-self-intersecting curve in a plane that crosses a fixed line a finite number of times—which is oriented horizontally (with flow direction variations used by different mathematicians).

• All open meanders (even and odd n) can be developed into an SAT labyrinth by making the curve finite, orienting it vertically, and stretching it around a central point (i.e., developed expression). This vertical orientation before development corresponds to the rectangular thread diagrams (with flow direction variations used by different typologists).

• Some even n open meanders work as processional labyrinths without any development. Those that have less nesting may not look much like a labyrinth, but with increased nesting it becomes more obvious that they directly express an SYP labyrinth (i.e., direct expression).

The source set S_n contains the quantum potential state of all Simple Labyrinths. It is the complete symmetric set of permutations given a particular number of circuits (n). The numbers are rearranged without consideration of any connections to other circuits or to the additional levels (0 and in some cases n + 1) that give context for the subsets.*

For example, when n = 3, S_3 contains six permutations: 123, 132, 213, 231, 312, and 321. We can take these permutations and apply the conditions of each subset to move from the quantum potential state to the dynamic thread state and identify the viable types:

• SAT_3: 0|123|4, 0|321|4

• SXT_3: 0|1x23|4, 0|1x2x3|4, 0|12x3|4

• SZT_3: ∅

• SYP_3: 0|12(3)|0, 0|1y(3)2|0, 0|2(3)y1|0, 0|(3)21|0

Processional types are a special case. These four SYP_3 level sequences represent only two types which each have a base approach and a transpose approach:

• B: 0|12(3)|0, T: 0|(3)21|0

• B: 0|1y(3)2|0, T: 0|2(3)y1|0

Continuing on to the static expression state, we can choose one of these seven viable 3-circuit types and make decisions about size, shape, style, orientation, chiral reflection (right- or left-hand entrance), materials, and decorative embellishments to create a realized labyrinth.

* The Simple subsets are Simple Alternating Transit (SAT), Simple Spiral Crossing Transit (SXT), Simple Bayonet Crossing Transit (SZT), and Simple Crossing Processional (SYP).