maze labyrinth

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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).
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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.
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
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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
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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