Ian Stewart*

The catapult that Archimedes built, the gambling-houses that Des- cartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions--not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our sub- ject's glorious and inglorious past: statues, plaques, graves, the card where the famous conjecture was made, the desk where the famous ini-

tials are scratched, birthplaces, houses, memorials. Does your home- town have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical signifi- cance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the Mathematical Tourist Editor, Ian Stewart.

The Jubilee Maze

Klaus Treitz

Labyrinths have been the object of great fascination for thousands of years. One reason for their long-lasting popularity is that they may be a metaphor for the com- plexity of life with its roundabout ways, blind alleys, and alternations of failure and success.

But surely there is also quite a different reason for the continuing interest in labyrinths: their challenging mathematical character.

The oldest form, which is spread all over the world, is the Cretan type. It is found on many Cretan coins, in many stone structures in Sweden, and in wall carv- ings. The adjective Cretan calls to mind the legend of Theseus and the Minotaur. The Minotaur was a crea- ture with the head of a bull and the body of a man. He was caged in a complicated building on the isle of Crete in the Mediterranean Sea. Annually, he demanded seven Greek maidens and youths for sacrificial food. Theseus, a prince of Athens, found and conquered the Minotaur. He himself escaped the Labyrinth with the aid of the clue of thread provided by Ariadne. If the prison of the Mino- taur was as intricate as the legend says, the preserved pictures must be only ideograms for it (Fig. 1). Never- theless, they represent the classic type of labyrinth. This type is called "unicursal," having no branches.

A similar sort of labyrinth is the Chartres type. In the Middle Ages, it was used in the pavements of impor- tant churches and was traced by pilgrims fulfilling vows. These labyrinths also belong to the unicursal type. Ac-

Figure 1. Graffito on a pillar at Pompeii (about A.D. 79): "Here l ives Minotaur." (From alpha, Mathematische Schiilerzeitschrifl, Berlin 17 (1983), 4.

* Column Editor's address: Mathematics Institute, University of War- wick, Coventry, CV4 7AL England. (Fig. 2).

5 4 THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4 (~ 1993 Springer-Verlag New York

cording to Christian philosophy, the journey through life may be difficult and troublesome, but in any case can lead to the goal. Theseus became an allegory for Christ

Figure 2. Cathedral of Chartres. (From Notre-Dame de Chartres, page 13, 15 ANNEE, Chartres 1984.) Photograph by B. Deffontaines, 1983.

Then a striking change takes place: With the Reforma- tion and the beginning of modern times, a new type of labyrinth replaced the unicursal. Human life no longer appeared as well programmed as it did in medieval times. Garden labyrinths appeared in which it is really possible to find the way to the center and the way out only by trial and error ("Irr-g~rten"). The walls consist of hedges.

Although the old unicursal labyrinths are of some mathematical interest (it is possible to look for a clas- sification, or to ask about methods of construction), the mazes are, of course, more interesting. Questions about solving algorithms which they raise connect to modern and important questions of graph theory and informatics (computer science).

For the mathematician on vacation, seeking recreation and inspiration in visits to mazes, the place to go is Eng- land. Many castles have not only an old history and spirit of their own, but also a marvelous garden with a maze [1]. For instance, Leeds Castle, halfway between Dover and London: Its maze has a subterranean exit

Figure 3. Maze at Hampton Court Palace. Photograph by G. Gerster, 1990. ([2], p. 92)

through a romantic grotto. Thirty kilometers away is Hever Castle, the childhood home of Anne Boleyn, of- ten visited by Henry VIII. In the garden, you find a pretty maze. On the Thames not far from London lies the royal palace of Hampton Court. In the gardens, you can walk through the oldest and most famous maze of all, known as "Wilderness" since the 16th century (Figs. 3 and 5). Models of it are favored for tests of new labyrinth- solving computer mice (and for real mice in ethology). On the top of Catherine's Hill near Winchester is a turf labyrinth cut into the ground. It dates back to the dawn of history.

The labyrinths of Rocky Valley date from between the 18th and 14th century B.C. You reach them from Tin- tagel after an hour 's walk along the breathtaking Cor- nish coast. It is a unique experience to follow, with your fingertips, the geometric lines, which someone carved into the stone nearly 4000 years ago (Fig. 4).

Figure 4. Mazes at Rocky Valley near Tintagel/Cornwall (with my wife). Photograph by K. Treitz, 1992.

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993 5 5

Figure 5. Layout of the Hampton Court Maze.

M e t h o d s of Maze S o l v i n g

It is fun to walk about in a maze, and usually in 15 or 20 minutes you will reach the center. But as a mathemati- cian you will feel challenged to find it systematically. Usually you do not have a map and can gather only lo- cal information about the structure of the maze; then, in many cases, the rule of right-hand-on-right-wall will work. In a simple maze, you will then go along all the alleys twice, including the blind alleys. You will traverse the center and in the end you will return to the entrance.

But if the maze is more complicated, this rule will fail. If the center is-- l ike the asterisk in Figure 5-- located between walls of islands (between walls that are not connected to the remaining walls), then you will only be led through some parts of the maze and back to the exit, without having reached the center. Moreover, if you make a mistake and cross over to the wall of an island, you would be condemned to walking around it for the rest of the day without finding the way out.

There are better rules known, which allow you to thread any maze: the methods of Tarry and Tremaux. Let us represent the maze as a graph (Fig. 6). The junc- tions become the nodes and the paths the edges; we get a finite, connected graph. Then the method of Tremaux consists of the following rules ([2], p. 74).

1. No edge may be traversed more than twice. 2. When you come to a new knot, take any edge you like. 3. When you come along a new edge to an old knot,

return along the edge by which you came. 4. When you come along an old edge to an old knot, take

a new edge if possible, otherwise an old.

Figure 6. Tremaux's algorithm.

Figure 7. Center of Jubilee Maze. Photograph by K. Treitz, 1992.

56 THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 4, 1993

The Jubi l ee Maze

There are about 80 mazes in England, but one of the most attractive for the mathematical tourist is surely the Jubilee Maze at Symmonds Yat. It is situated 40 km north of Bristol on the border with Wales, in the beautiful valley of the river Wye. The maze has the octagonal form of a Labyrinth of Love, popular around 1600. It was set out by two young people, the brothers Edward and Lindsay Heyes, to celebrate the Silver Jubilee of Queen Elizabeth II in 1977. The cypress hedges are 2 m high. In the center stands a graceful stone pavillion (Fig. 7). The Heyes are there all day and are conspicuous in their Victorian white flannels, blazers, and straw hats.

The second attraction in the Jubilee Park is a museum. In a very appealing exhibition "Mystery and Mazes," the history and the mathematics of different labyrinth types are presented. This is a "hands-on" museum. Visitors are invited to do several experiments and to construct various forms of labyrinth.

The third attraction of the Jubilee Park is the show "Mazes and Micros," where you can see maze-solving

computers at work. The highlight of this exhibition is a maze-solving robot (Fig. 8). It uses an algorithm in- vented by N. Lee at the Bell Corporation ([3]). Adults and children are invited to build a maze from small boards on a table divided into squares. Then the tech- nical function of the mouse is explained. It feels the walls around its present position with infrared sensors. The location of the walls is recorded in the "memory- map" in a Random-Access Memory (RAM) which can be updated. The maze-solving algorithm is stored in a Read-Only Memory (ROM). The program is carried out by a microprocessor, which controls the step-motors. At the beginning, the micromouse knows from its program that it points "South" in the Northeast corner (Fig. 9). Diagonal moves are not allowed.

Finding the Way. The robot uses a readily understand- able and efficient method.

Imagine that the squares of the maze are like steps of a staircase: Each square is assigned a number which is the number of steps from that point to the goal . . . . One step for each square. If there is a path from start to goal and if the robot has a map of this path in its memory, it has simply to step down the steps following this map.

Figure 8. Robot in the exhibition "Mazes and Micros." Pho- tograph by K. Treitz, 1992.

4 5 6 7 8

+

3 4-~ 5 6 7

2 3 4 5 6

Updating the Step-Map. But how can the robot get a map? To begin with, it has a step-map in its RAM without any walls. Running through the maze, it checks the edges of ceils for walls. When a wall is found, the robot waits and the step-numbers of all the squares are recalculated to update the memory-map: This is done by testing all of the 25 squares in turn against squares accessible from them. If no step down is accessible from a square, then the robot's microprocessor adds one to the step-number for that square, until the step-number exceeds that of an adjacent square. For this operation, it needs only a frac- tion of a second. Then also the position of the discovered wall is recorded in the RAM-map: South, West, North, East of the square the robot is in at the time. Then it will go on to the next cell. Figure 9b shows the state of the step-map after the robot has reached the square (*) and the updating is finished.

The method always results in a shortest route; and it is possible for the robot to skip over the investigation of whole parts of the maze. For instance, in our example it will never be concerned with the wall at (+); it will not even detect it!

This algorithm is important for transport problems, for layouts of electronic circuits, and for pattern recognition.

Mr. Heyes is mathematically knowledgeable and will also answer specialized questions.

If you are accompanied by your family, it will be pleas- ant to find them sharing enthusiasm for your science.

It should be mentioned that there is also an amazing puzzle shop, a pleasant restaurant, and a picnic area at the Jubilee Park.

1 2 3 4 5

0 1 2 3 4

4 5 6 7 10

3 4 5 8 9

2 3 4 7 10 I t

1 2 5 6 7

1 6 7 8

Figure 9. (a) Robot at start position. (b) Step-map after the eighth step of the robot.

References

1. Fisher, Adrian, and Gerster, Georg, The Art of the Maze, Wei- denfeld and Nicolson, London 1990.

2. Hemmerling, A., "Labyrinth-Problem," alpha, Mathem. Schiller- zeitschrift, Berlin 17 (1983), pp. 73-75.

3. Lee, C.Y., "An Algorithm for Path Connections and its Ap- plications," I.R.E. Transactions on Electronic Computers, Sept. 1961.

Gymnasium Rheinfelden 7888 Rheinfelden Federal Republic of Germany

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