The Mystery of Mazes and Length of Labyrinths

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The Mystery of Mazes and Length of Labyrinths

• By Derek Chase

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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.

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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.

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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.

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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).

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• 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.

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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.

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Cultural Meanings

• Christian churches used the labyrinth for prayer
  and meditation as early as the mid-fourth
  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.

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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.

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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

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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.

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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.
  

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Classical Labyrinths

• (Chakra-vyuha)
• A unique design based on the seven-circuit
  labyrinth it is based on a three-fold rather
  than four-fold seed pattern and is often drawn
  with a spiral at the centre.

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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.

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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.

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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.

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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.

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Medieval Labyrinths

• (St. Omer)
• Although its pathway seems to be random, the St.
  Omer labyrinth was developed from the standard
  medieval form.

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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.

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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.

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Mazes

• 1. Dimension
• 2. Hyperdimension
• 3. Topology
• 4. Tessellation
• 5. Routing
• 6. Texture
• 7. Focus

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• 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)

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• 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.

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• 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).

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• 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.

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• 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).

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• 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 non-elitists have longer paths.
• Symmetric Pathways and environment are symmetric.

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• 7. Focus Techniques to create mazes?Adding 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.

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