Groups, Series, Sets

Overview

Many labyrinths can be understood as part of groups, series, and sets. Recurring components and segment patterns help to compare labyrinths as well as to translate graphical representations to the walked experience.

• Groups: Each labyrinth type is part of one symmetry group of one to four labyrinths based on Base-Dual-Transpose-Complement relationships. When there is an odd number of circuits (even number of levels) the group has two (each self-dual or self-transpose) or four labyrinths (BDTC), with the exception of the 1-circuit SAT labyrinth which is both self-dual and self-transpose. When there is an even number of circuits (odd number of levels) the group has one (self-dual) or two labyrinths (BD, no viable transpose or complement). 

• Series: Component patterns with at least one expandable part result in a series of related labyrinths such as Nested Meanders (Single Meander, Double Meander, Triple Meander, etc.). Some labyrinths are part of more than one series depending on which component is being considered the expandable part. The smallest labyrinths can be at the beginning of multiple series, due to the underlying mathematical principles, even though they may not seem related. This builds upon and expands the work done by Tony Phillips and Richard Myers Shelton, describes more SAT series, and incorporates non-SAT labyrinths.

• Sets: Labyrinths that use the same components, although in a different order, are part of the same set. Components can be complete bands or quasi-bands (nested bands) vertically stacked on top of each other in uniaxial and some multiaxial labyrinths, or the components may fit together horizontally and/or diagonally in the rectangular diagram of the labyrinth pattern.

See Groups, Series, and Sets document for more examples.