Through Mazes to Mathematics

## Simple, Alternating, Transit Mazes

The best known of these mazes is the Cretan maze. It can be drawn as a game.

Here is another example. This maze appears in several medieval hebrew manuscripts. Although this maze has a superficial resemblance to the Cretan maze, a close comparison shows they are quite different. The Jericho maze has 7 levels, whereas the Cretan maze has 8, and the sequence in which the levels are reached differs from one maze to the other. In both mazes the path goes directly to level 3 (counting the outside as 0) but in the Cretan maze it then doubles back through levels 2 and 1, whereas in the Jericho maze it continues on through levels 4 and 5 before returning to 2 and 1. The complete level sequences are

Cretan 032147658

Jericho 03452167.

The properties that these two mazes share, and that will serve to define a class of mazes with a simple mathematical description, may be summarized by calling them simple, alternating, transit mazes (s.a.t. mazes).

TRANSIT mazes because the path runs without bifurcation from the outside of the maze to the center. For example these mazes are not transit mazes: in one, the path comes out on the same side it entered; and in the other there are points where the maze-runner must choose which way to go.

ALTERNATING because the plan of the maze is laid out on a certain number of concentric or parallel levels, and the maze-path changes direction whenever it changes level. For example this maze, which represents the plan of Constantinople in a medieval Arabic geography book has distinct levels but is not alternating, since it spirals from level 10 to level 1 without changing direction. Its level sequence 0.3.4.5.6.7.8.9.10.1.2.11 cannot occur, as we shall see, as the level sequence of an alternating maze.

SIMPLE because the path makes essentially a complete circle at each level; in particular it travels on each level exactly once. For example, this alternating transit maze, which occurs as a pavement maze in Chartres (shown here) and in several other cathedrals, and carved on a pillar in the Cathedral in Lucca, is not simple: there are four different points at which the path can change levels. This maze-pattern seems to have occurred as a christian elaboration and amplification of the cretan design and its roman descendants; the association with the Theseus myth persisted. The new design was remarkably persistent itself. It is seen in medieval manuscripts dating from the 9th century; Chartres cathedral was built around the year 1200; and in the early 16th century the identical pattern is painted on a florentine cassone. It is now rendered 3-dimensionally, with an armored Theseus battling the Minotaur in the center.

The TOPOLOGY of a simple, alternating transit maze is completely determined by its level sequence.