The Mystery of Mazes and Length of Labyrinths By Derek Chase Differences among Mazes and Labyrinths • Unlike mazes, which create challenges through forked routes and dead ends (among many other deceptions), labyrinths consist of a single route which winds its way to the goal at the center; to get back out you simply retrace your path. Differences among Mazes and Labyrinths • In contrast, a maze is a puzzling structure having many choices and dead ends that cause one to get lost and confused. Differences among Mazes and Labyrinths • But the idea that labyrinths have a single path arose only in the last century. In ancient times, a labyrinth was a perplexing construction that was chaotic and confusing for those trapped within it. Today, we would not hesitate to call this depiction a maze. History of the Labyrinth • About 3,500 years ago, the first datable example of labyrinths were created in Knossos, Greece, in the form of a scribes etchings on a clay tablet. The labyrinths that this scribe drew is now known as the classical seven-circuit labyrinth, or Cretan labyrinth (referring to the design found on ancient coins on the Island of Crete). • • • • Greek mythology gives explanation for the creation of labyrinths. Before Minos ascended the throne of Crete, he struggled with his brothers for the right to rule. Minos prayed to Poseidon to send him a snow-white bull, as a sign of approval by the gods for his reign. He promised to sacrifice the bull as an offering, and as a symbol of subservience. A beautiful white bull rose from the sea, but when Minos saw it, he coveted it for himself. He assumed that Poseidon would not mind, so he kept it and sacrificed the best specimen from his herd instead. When Poseidon learned about the deceit, he made Minos' wife fall madly in love with the bull. She had Daedalus, the famous architect, make a wooden cow for her. She climbed into the decoy and fooled the white bull. The offspring of their lovemaking was a monster called the Minotaur. The creature had the head and tail of a bull on the body of a man. It caused such terror and destruction on Crete that Daedalus was summoned again, but this time by Minos himself. He ordered the architect to build a gigantic, intricate labyrinth from which escape would be impossible. The Minotaur was captured and locked in the labyrinth. Every year for nine years, seven youths and maidens came as tribute from Athens. These young people were also locked in the labyrinth for the Minotaur to feast upon. When the Greek hero Theseus reached Athens, he learned of the Minotaur and the sacrifices, and wanted to end this. He volunteered to go to Crete as one of the victims. Upon his arrival in Crete, he met Ariadne, Minos's daughter, who fell in love with him. She promised she would provide the means to escape from the maze if he agreed to marry her. When Theseus did, she gave him a simple ball of thread, which he was to fasten close to the entrance of the maze. He made his way through the maze, while unwinding the thread, and he stumbled upon the sleeping Minotaur. He beat it to death and led the others back to the entrance by following the thread. Cultural Meanings • Labyrinths are a widespread symbol in many other cultures as well. A traditional European story explains that the Labyrinth evolved from observations of the recursive swings of the planet Mercury. In one solar year, Mercury swings backwards three times and forwards four times in the sky, symbolizing the seven-circuits of the classical labyrinth. Cultural Meanings • Christian churches used the labyrinth for prayer and meditation as early as the midfourth century. • In medieval Europe, the eleven-circuit labyrinth served as an alternate path for pilgrimage after Jerusalem fell to the Moslems, symbolizing the one true path to Christian salvation. Cultural Meanings • The design most popular within Christianity comes from Notre-Dame de Chartres Cathedral in Chartres, France; it contains four 11-cuircuit labyrinth. • This labyrinth was built in the early 13th century and is meant to be walked as a pilgrimage or for repentance. • In walking the Chartres style labyrinth, one meanders through each of the four quadrants several times before reaching the goal at the center. • At the center is a rosette design which has a symbolic value of enlightenment. The four arms of the cross are readily visible and provide significant Christian symbolism. Uses • While there has been a vast amount of uses for labyrinths, the most common uses in history are: - Ceremonial pathways - Protective sigils - Traps for unwelcome spirits - Games - Dancing Classical Labyrinths (Cretan, seven-circuit) • The classical labyrinth is a drawing of lines delineating a unique path from the exterior to the center and covering all available surface. It forms seven circuits, bounded by eight walls surrounding the center point. The construction stems from a four-fold seed pattern which is the most prominent reason for its passage through history. Classical Labyrinths (Baltic Wheel) • The “Baltic Wheel” style labyrinth is a relatively simple reconstruction of the Cretan labyrinth, with exception to the double spiral at the center with separate entrance and exit paths. Classical Labyrinths (Chakra-vyuha) • A unique design based on the sevencircuit labyrinth; it is based on a three-fold rather than four-fold seed pattern and is often drawn with a spiral at the centre. Classical Labyrinths (Otfrid Labyrinth) • The Otfrid is based on the classical seed pattern, but is drawn concentrically with an additional set of turns added to create an eleven-circuit labyrinth. Roman Labyrinths • While the classical labyrinth was known throughout the Roman Empire, the popular use of the labyrinth as a design element in mosaic flooring resulted in a number of interesting developments, all conveniently classifiable as "Roman" varieties. Several researchers have attempted further classification of Roman designs, based on mathematical or geometrical properties, which allow the majority of the sixty or so Roman mosaic labyrinths to be designated as meander, serpentine, or spiral types. Medieval Labyrinths (Chartres, or eleven-circuit) • First developed during the ninth and tenth centuries, the medieval labyrinth combined the eleven circuits of the Otfrid labyrinth with the four-fold symmetry of the Roman labyrinths. Medieval Labyrinths (Varieties) • Much like the classical labyrinth, many variations have been made upon the basic theme of the medieval labyrinth. Circular, square, and polygonal forms of the basic medieval form have been created and do not need separate classifications. Medieval Labyrinths (St. Omer) • Although its pathway seems to be random, the St. Omer labyrinth was developed from the standard medieval form. Labyrinths Today • Labyrinths are still widely used today. • The World Peace Labyrinth was developed for the Salt Lake City Olympic Games in 2002. It is an elliptical seven-circuit labyrinth measuring 24 x 35 feet. • A central theme of the labyrinth is the Tree of Life. In the book of Genesis within the Christian Bible, the tree is in the center of the Garden of Eden, and four rivers flow from its roots in four directions. • The design consists of seven paths in concentric ellipses with proportions set by Sacred Geometric ratios. The lines are painted in Green, Brown & Blue to represent harmony with the Earth. The labyrinth is divided in quadrants, which represents the four winds. The material is white, a color representing peace. Now to Mazes • As stated previously, mazes differ from labyrinths in the fact that they evoke confusion and chaos for the individual who is attempting to solve it. Moreover, there are seven different methods that can be combined to create and complicate mazes. Mazes • • • • • • • 1. Dimension 2. Hyperdimension 3. Topology 4. Tessellation 5. Routing 6. Texture 7. Focus 1. Dimension: How many dimensions the Maze covers. • 2-Dimensional: Pathways may go in all four compass directions • 3-Dimensional: Pathways may go up and down in addition to the four compass directions. • Weave: Pathways can overlap each other (ex. Bridges) 2. Hyperdimension: The dimension of the pathways (not the environment). • Hypermaze: 3-dimentional or higher state of environment. Many choices at each divergence. • Non-hypermaze: 2-dimentional and has relatively limited amount of choices to make at each divergence. 3. Topology: Geometry of the space in which the maze exists. • Normal: Standard maze in 2-dimentional form. • Planair: Connecting the edges of the Maze so that they intersect (ex. cube). 4. Tessellation: Geometry of the units that make up the Maze. – Orthogonal: Rectangular grid cells have pathways that intersect at right angles only. – Delta: Made up of triangles cells have up to three pathways. – Sigma: Made up of hexagons cells have up to six pathways. – Theta: Made up of concentric circles either the start or finish is in the center. – Upsilon: Made up of octagons or squares cells can have 8/4 pathways. – Zeta: Laid out on a rectangular grid have 45° as well as right angle pathways – Omega: Non-orthogonal tessellation. – Crack: No consistent tessellation – Fractal: A maze made up of nested mazes of any tessellation. 5. Routing: Types of passages within maze environment. – Perfect: Pathways never re-connect with one another, so every path you choose either leads to another path or a dead end. There is only one solution, which can be found by placing either the left or right hand on the wall at the beginning and continuing through the maze without removing your hand. – Braid: No dead ends. Contains one or more passages that loop back into other passages, causing you to spend time going in circles. – Unicursal: No intersections. One single path leads you to the desired end point (ex. labyrinth). 6. Texture: Style of pathways. – Bias: Maze environment tends to favor one direction more than another (horizontal vs. vertical). In a vertically biased maze, there is much more terrain covered from moving up and down through the pathways rather than left to right. – Run: The amount of cells that exist between pathways. The more cells without a pathway, the longer the run of the maze. – Elite: Length of the (shortest) path from start to finish in a maze. Elitist mazes have short paths while nonelitists have longer paths. – Symmetric: Pathways and environment are symmetric. 7. Focus: Techniques to create mazesAdding walls and cutting paths. • Adding walls: Begin with an outside wall and add boundaries to create maze paths. • Cutting paths: Cut paths through a solid figure to create maze.