Thirdly, there must be a certain degree of complexity in the design, a degree which it is manifestly impossible to define as it must be considered in conjunction with other characteristics in any particular case. In the case of a unicursal labyrinth, _i.e._, one in which there is only one path, the complexity lies in the multiplicity of turnings and the extent of the departure from pure geometrical figures such as the meander, the zigzag, and the spiral; in the case of a puzzle-figure it lies partly in this but partly also in the number and disposition of branch-paths. It naturally follows that in a unicursal design there cannot be absolute symmetry, although, with a little ingenuity, a very pleasing _appearance_ of symmetry may be obtained.
Fourthly, there must be communication between the component parts of the design; in other words, the path must be continuous. This does not preclude the occurrence in the design of closed "islands," but only makes it clear that such inclusions do not form part of the labyrinth proper.
Fifthly, there must be communication between the interior and the exterior. We might not altogether withhold the application of the term "labyrinth" or "maze" in the case of a closed design, but we should have to qualify it, _e.g._, by prefixing the word "closed." In the case of the beautiful and intricate mosaic pavement found in the _Casa del Labirinto_ at Pompeii mentioned on page 46, for example, although we know that the pattern was intended to convey an allusion to the Cretan labyrinth, we cannot look upon it as a true labyrinth design; not only is there no communication with the exterior, but by its repetition of purely geometrical design it fails to satisfy our third condition.
If the reader chooses to formulate for himself a working definition based on the above remarks he is at liberty to do so, but he may take for granted that nobody else will accept it. However, he will have gained, at any rate, a clearer conception of the matter than he would perhaps have gathered from any dictionary.
We have seen that mazes and labyrinths may be roughly divided into two types as regards the principle of their design, namely, into _unicursal_ and _multicursal_ types, or, as some say, into "non-puzzle" and "puzzle" types respectively. The word "unicursal" has hitherto been chiefly used by mathematicians to describe a class of problems dealing with the investigation of the shortest route between two given points or of the method of tracing a route between two points in a given figure without covering any part of the ground more or less than once (_e.g._, the well-known "bridge" problems), but there is no reason why we should not apply the adjective "unicursal" (= "single course" or "once run") to denote those figures which consist of a single unbranched path, using the term "multicursal" as its complement, or antonym. We must not draw too hard a line between these two types; for instance, we could not reasonably insist that the turf maze at Wing (Fig. 60) is multicursal simply on account of the dichotomy of its path to form the central loop. Where the loop is itself relatively large and complex, as in the Poitiers example (Fig. 55), there are better grounds for doing so, but it is plain that in such cases the point is one to be decided by common-sense.