Dual, Transpose, Complement

Overview

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).

Dual

The dual relationship reverses the rhythm of the base pattern, reversing the layering and walking order of the components without changing the inward or outward direction of each component. You start at the end of one pattern and make it the beginning of another pattern. There are a number of ways to find the dual pattern:

• rotate the pattern (the rectangular diagram) 180°, or

• subtract each circuit number from the total levels and reversing the order of the resulting numbers, or

• reverse the order of components, or

• reverse the level changes.

Self-dual patterns have palindromic rhythms. If a self-dual pattern has a transpose, it is also self-dual. This also means that in a self-dual Base-Transpose pair, each of the two labyrinths is both the transpose and complement of the other.

Transpose

The transpose relationship reverses the circuit order of the base pattern, reversing the walking order of the components and the inward or outward direction of each component without changing the layering order. This results in a similar, yet altered rhythm, as well as changes in the number of nested turns.

Another way to understand this relationship is to think of anchoring the segments, making the entrance and way into the center elastic, and pulling them to the opposite edge of the pattern. The entrance is flipped up to enter the center and the previous connection to the center is flipped down to become the entrance. The resulting pattern will be in the opposite chiral orientation, so it can also be flipped horizontally to end up in the same orientation as the original.

A few patterns are self-transpose and have palindromic rhythms. If a self-transpose pattern has a dual, it is also self-transpose. This also means that in a self-transpose Base-Dual pair, each of the two labyrinths is both the dual and complement of the other.

Complement

The dual of the transpose pattern is the complement of the original base pattern. In this relationship, the layering order of components is reversed as well as the inward or outward direction of each component.

Contrasting with the transpose relationship, here the entrance and way into the center are anchored (but elastic vertically) and the rest of the pattern is flipped vertically.