The theory of the description of mazes is included in Euler’s theorems given above. The paths in the maze are what previously we have termed branches, and the places where two or more paths meet are nodes. The entrance to the maze, the end of a blind alley, and the centre of the maze are free ends and therefore odd nodes.
If the only odd nodes are the entrance to the maze and the centre of it–which will necessitate the absence of all blind alleys–the maze can be described unicursally. This follows from Euler’s third proposition.
Again, no matter how many odd nodes there may be in a maze, we can always ﬁnd a route which will take us from the entrance to the centre without retracing our steps, though such a route will take us through only a part of the maze. But in neither of the cases mentioned in this paragraph can the route be determined without a plan of the maze.
A plan is not necessary, however, if we make use of Euler’s suggestion, and suppose that every path in the maze is duplicated. In this case we can give deﬁnite rules for the complete description of the whole of any maze, even if we are entirely ignorant of its plan. Of course to walk twice over every path in a labyrinth is not the shortest way of arriving at the centre, but, if it is performed correctly, the whole maze is traversed, the arrival at the centre at some point in the course of the route is certain, and it is impossible to lose one’s way.
I need hardly explain why the complete description of such a duplicated maze is possible, for now every node is even, and hence, by Euler’s second proposition, if we begin at the entrance we can traverse the whole maze; in so doing we shall at some point arrive at the centre, and ﬁnally shall emerge at the point from which we started. This description will require us to go over every path in the maze twice, and as a matter of fact the two passages along any path will be always made in opposite directions.
If a maze is traced on paper, the way to the centre is generally obvious, but in an actual labyrinth it is not so easy to ﬁnd the correct route unless the plan is known. In order to make sure of describing a maze without knowing its plan it is necessary to have some means of marking the paths which we traverse and the direction in which we have traversed them—for example, by drawing an arrow at the entrance and end of every path traversed, or better perhaps by marking the wall on the right-hand side, in which case a path may not be entered when there is a mark on each side of it. If we can do this, and if when a node is reached, we take, if it be possible, some path not previously used, or, if no other path is available, we enter on a path already traversed once only, we shall completely traverse any maze in two dimensions.
Of course a path must not be traversed twice in the same direction, a path already traversed twice (namely, once in each direction) must not be entered, and at the end of a blind alley it is necessary to turn back along the path by which it was reached.
I think most people would understand by a maze a series of interlacing paths through which some route can be obtained leading to a space or building at the centre of the maze. I believe that few, if any, mazes of this type existed in classical or medieval times.
One class of what the ancients called mazes or labyrinths seems to have comprised any complicated building with numerous vaults and passages.
Such a building might be termed a labyrinth, but it is notwhat is usually understood by the word. The above rules would enable anyone to traverse the whole of any structure of this kind. I do not know if there are any accounts or descriptions of Rosamund’s Bower other than those by Drayton, Bromton, and Knyghton: in the opinion of some, these imply that the bower was merely a house, the passages in which were confusing and ill-arranged.
Another class of ancient mazes consisted of a tortuous path conﬁned to a small area of ground and leading to a place or shrine in the centre.
This is a maze in which there is no chance of taking a wrong turning; but, as the whole area can be occupied by the windings of one path, the distance to be traversed from the entrance to the centre may be considerable, even though the piece of ground covered by the maze is but small.
The traditional form of the labyrinth constructed for the Minotaur is a specimen of this class. It was delineated on the reverses of the coins of Cnossus, specimens of which are not uncommon; one form of it is indicated in the accompanying diagram. The design really is the same as that drawn in ﬁgure ii, as can be easily seen by bending round a circle the rectangular ﬁgure there given.
Mr Inwards has suggested that this design on the coins of Cnossus may be a survival from that on a token given by the priests as a clue tothe right path in the labyrinth there. Taking the circular form of the design shown above he supposed each circular wall to be replaced by two equidistant walls separated by a path, and thus obtained a mazeto which the original design would serve as the key. The route thus indicated may be at once obtained by noticing that when a node is reached (i.e. a point where there is a choice of paths) the path to be taken is that which is next but one to that by which the node was approached. This maze may be also threaded by the simple rule of always following the wall on the right-hand side or always that on the left-hand side. The labyrinth may be somewhat improved by erecting a few additional barriers, without aﬀecting the applicability of the above rules, but it cannot be made really diﬃcult. This makes a pretty toy, but though the conjecture on which it is founded is ingenious it must be regarded as exceedingly improbable. Another suggestion is that the curved line on the reverse of the coins indicated the form of the rope held by those taking part in some rhythmic dance; while others consider that the form was gradually evolved from the widely prevalent svastika.
Copies of the maze of Cnossus were frequently engraved on Greek and Roman gems; similar but more elaborate designs are found in numerous Roman mosaic pavements. A copy of the Cretan labyrinth was embroidered on many of the state robes of the later Emperors, and, apparently thence, was copied on to the walls and ﬂoors of various churches. At a later time in Italy and in France these mural and pavement decorations were developed into scrolls of great complexity, but consisting, as far as I know, always of a single line. Some of the best specimens now extant are on the walls of the cathedrals at Lucca, Aix in Provence, and Poitiers; and on the ﬂoors of the churches of Santa Maria in Trastevere at Rome, San Vitale at Ravenna, Notre Dame at St Omer, and the cathedral at Chartres. It is possible that they were used to represent the journey through life as a kind of pilgrim’s progress.
In England these mazes were usually, perhaps always, cut in the turf adjacent to some religious house or hermitage: and there are some slight reasons for thinking that, when traversed as a religious exercise, a pater or ave had to be repeated at every turning. After the Renaissance, such labyrinths were frequently termed Troy-towns or Julian’s bowers. Some of the best specimens, which are still extant, are those at Rockliﬀ Marshes, Cumberland; Asenby, Yorkshire; Alkborough, Lincolnshire; Wing, Rutlandshire; Boughton-Green, Northamptonshire; Comberton, Cambridgeshire; Saﬀron Walden, Essex; and Chilcombe, near Winchester.
The modern maze seems to have been introduced—probably from Italy—during the Renaissance, and many of the palaces and large houses built in England during the Tudor and the Stuart periods had labyrinths attached to them. Those adjoining the royal palaces at Southwark, Greenwich, and Hampton Court were particularly well known from their vicinity to the capital. The last of these was designed by London and Wise in 1690, for William III, who had a fancy for such conceits: a plan of it is given in various guide-books. For the majority of the sight-seers who enter, it is suﬃciently elaborate; but it is an indiﬀerent construction, for it can be described completely by always following the hedge on one side (either the right hand or the left hand), and no node is of an order higher than three.
Maze at Hampton Court.
Unless at some point the route to the centre forks and subsequently the two forks reunite, forming a loop in which the centre of the maze is situated, the centre can be reached by the rule just given, namely, by following the wall on one side—either on the right hand or on the left hand. No labyrinth is worthy of the name of a puzzle which can be threaded in this way. Assuming that the path forks as described above, the more numerous the nodes and the higher their order the more diﬃcult will be the maze, and the diﬃculty might be increased considerably by using bridges and tunnels so as to construct a labyrinth in three dimensions. In an ordinary garden and on a small piece of ground, often of an inconvenient shape, it is not easy to make a maze which fulﬁls these conditions. Here on the following page is a plan of one which I put up in my own garden on a plot of ground which would not allow of more than 36 by 23 paths, but it will be noticed that none of the nodes are of a high order.
Garden plot by the author, Rouse Ball.
From Mathematical Recreations and Essays. W.W. Rouse Ball. 1892