
Over the years of wandering the starmaze I have developed my own private
language. Here are some of the words I use when I talk about the maze.
 Astrology
 The art of gaining insights into past, present, and future situations by considering
the room, house, passages,
and hexagrams associated with a particular time and date.
A time and day of the year is assigned to every passage in the starmaze by starting at midnight January 1st
in the starting room and following the grand passage tour
over the course of a year. As the tour progresses, a fixed departure time is established for one end of each passage, and an arrival
time for the other end. By consulting the Hexagram Concordance, an
astrologer can determine the latest arrival or departure before a given date and time, typically a person's birthdate,
and use that to calculate a precise position in the maze. The hexagram associated with the immediately preceding
passage can then provide insights into the forces which created or influenced the situation or person of interest,
the room and house of the maze can provide context for the current situation, and the next passage in the sequence
can suggest future trends. Alternate passages can suggest alternate outcomes or strategies, and cycles
can be explored to shed light on recurring patterns of behavior. The seasons,
elements, and forms associated with each room can provide additional
layers of meaning. For more information, see the Hexagram Concordance.
 Atlas
 A list available in the starmaze explorer which assigns a map location to each room in
the maze. Locations are in the form +Aa where the capital letter indicates microcosm (A to P),
the lowercase letter indicates room position (a to p), and the sign is used to differentiate between
the two inverse patterns sharing that position. A plus sign indicates a
sun room, a minus indicates a moon room. In 1998, the starmap
was revised, so the atlas was revised as well. Both versions have been recreated for this exhibit.
See Atlas and Revised Atlas.
 Base Hexagram
 That part of a hexagram which persists when moving lines are ignored.
There are 64 base hexagrams defined in the I Ching. But each of these base hexagrams supports
64 patterns of moving lines, for a total of 4096 hexagrams. Pairs of hexagrams sharing the same base hexagram
are used to form the chain of hexagram transformations in the grand passage tour.
 Bastion
 One of the four house types in a threedimensional starmaze.
Bastions are shaped like tenstory cubes topped with open plazas; they face outward at ground level.
There are four bastions, one for each of the four temples.
The east bastion, comprising the House of Words, contains a vast library and faces a forest
whose trees are occasionally consumed to create more books; its plaza features a treehouse
with a rope suspension bridge connecting to the Tower of Bones and a spiral staircase hidden in its trunk.
The south bastion, comprising the House of Sand, faces a desert and features an observatory atop its plaza.
The north bastion, comprising the House of Numbers, presses against the mountains with a view onto a
spectacular waterfall; its plaza features a reflecting pond and a slender bridge to a tunnel
behind the waterfall leading to the deep below.
The west bastion, comprising the House of Time, sits on a cliff overlooking an ocean;
its plaza is open and is sometimes used as a gathering place by monks of the Four Temples to watch the sunset.
 Binary Coordinates
 Each room (pattern) in the starmaze corresponds to one of the 512 corners of a ninedimensional hypercube.
Each of these corners can be given a coordinate in binary space. A binary coordinate or address consists of a nine
digit number where each digit (bit) is either a zero or a one. A coordinate of 001000100 would indicate the corner
which extends from the origin in the 3rd and 7th dimensions. By convention, the origin, 000000000, is assigned
to the pit (
room 0). Adjacent corners differ by a single bit, so every movement in the maze involves flipping
a single bit of a binary coordinate. The current form of the puzzle makes binary coordinates visible in each pattern
by assigning a circle to those cells corresponding to zero bits and squares to those cells corresponding to one bits
where cells/dimensions are numbered according to the Lo Shu. For example, the ending pattern (
room 495) consists of a circle surrounded by eight squares because its binary coordinate is 111101111.
 Binary Space
 In normal space, the sort we live in, each axis is a number line with an infinite number of x, y, and z coordinates.
In binary space, each axis has only two coordinates: zero and one. The starmaze completely fills ninedimensional binary space.
 Binary Trickle Chart
 A chart which illustrates the different paths by which a binary coordinate
can transform from all 1s to all 0s one bit at a time. The size of the chart depends on the number of
dimensions involved. For the coordinates of a tesseract,
the chart would consist of one node at the apex, 1111, leading to the four nodes with a single 0: 0111, 1011, 1101, and 1111.
These four nodes each lead to three of the six nodes with two 0s. Each of those six lead to two of the four nodes with three 0s.
And each of those four nodes leads to the node at the bottom: 0000.
I call it a trickle chart because you could imagine water in a fountain trickling from the top to the bottom via the
various channels shown. The number of nodes at each level, in this case 14641 always sum to the value of two
raised to the dimension of the chart (1+4+6+4+1 = 16 = 2^{4}). These numbers correspond to the coefficients of a
binomial expansion like (x+y)^{4}, which equals
1x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} +1y^{4},
which, in turn, appear on row N of Pascal's Triangle.
The pattern of movement depicted in the binary trickle chart recurs throughout the starmaze. The rooms of each
microcosm in the starmap can be mapped to a 4D binary trickle chart
with the dominant room at the apex. Rooms within each house in
the threedimensional starmaze are arranged on levels
and are given position numbers
according to the trickle chart. And the entire starmaze can be mapped to a 9D trickle chart with
room 0 (the pit) at the base and either
room 511 (the source) or
room 186 at the apex, depending on whether you are mapping movement between patterns or movement between coordinates.
In either case, the distribution of rooms per level is given by row 9 of Pascal's Triangle:
193684126126843691.
Binary trickles are also associated with Neutral Key puzzles.
 Cartan's Triangle
 A variant of Pascal's Triangle in which each number is the sum
of the number above it and to the left, and two times the number above it and to the right.
For example, the third number in row four (the fifth row since the first row is row zero), 24,
is the sum of 12 (above, left) and two times 6 (above, right).
The numbers in each row sum to powers of three (1, 3, 9, 27, ...).
Cartan's Triangle can be used to discover the number of components of an Ndimensional hypercube.
In row four we discover that a tesseract has 16 corners, 32 edges, 24 squares, and 8 cubes.
Row nine gives us the components of the starmaze hypercube: 512 corners (or rooms),
2304 edges (or passages), 4608 squares (512 of which form cycles),
5376 cubes, 4032 tesseracts, 2016 5cubes, 672 6cubes, 144 7cubes, and 18 8cubes (two of which comprise the
inner rooms and outer rooms).
For much more information, see Cartan's Triangle.
 Cell
 One of the nine positions in a starmaze pattern.
Each cell can be in one of two states: "on" (open) or "off" (closed).
An open cell is sometimes referred to as a star (hence the term "starmaze").
Movement to other patterns is only permitted via an open cell.
The cells are arranged in a threebythree array. Open cells are represented by
filled shapes, closed cells by empty shapes. The cells are numbered according to the
Lo Shu. There are three types of cells, yin cells
which are located in the four corners, yang cells, located between the
yin cells, and the center cell.
 CenterPipes
 The pipes in the starmap which represent centercell passages.
All the centercell passages from
one microcosm lead to corresponding rooms in another microcosm. These two microcosms are connected by
a single centerpipe. Unlike macropipes, centerpipes are not colored
to indicate direction. Instead, a colored dot appears in the center of each room of the map. If the color
of the center dot differs from the color of the pipe used to enter the room, it is permissible to leave
through the centerpipe and travel to the corresponding room in the microcosm at the other end of the centerpipe.
So each centerpipe contains 16 centercell passages, half of one color, half of the other.
 Chirality
 Left or righthandedness, a property of certain forms of starmaze patterns.
Most forms contain their own horizontal or vertical reflections. That is, any one of their patterns can be reflected
by either rotating or inverting. For this reason, these other forms are said to be achiral.
But the patterns in two pairs of forms, forms 7 and 8, and forms 9 and 10, have an asymmetric distribution of open cells
that cannot be reflected by rotation or inversion. Instead, each pattern in form 7 has its reflection in form 8 and vice versa.
These four forms are said to exhibit the property of chirality.
 Circuit
 Any path within the maze in which the last room in the path leads back to the first room.
A cycle is a special type of circuit. Because of parity,
all circuits consist of an even number of steps.
 Coincidental Room
 A room which has the same pattern for its open and closed cells as it does for its
binary coordinates. In these rooms all the squares are filled and circles empty.
There are only eight such patterns:
room 0,
room 56,
room 146,
room 170,
room 325,
room 381,
room 471, and
room 495. If you take the inverse of a coincidental pattern and then flip the center cell,
the result is another coincidental pattern.
 Coloring
 Coloring is used in the starmap to indicate the direction of each passage without
the need to draw arrows. Instead, movement through the map follows one simple rule: if you
enter a room through a passage (pipe) of one color, you must leave through one of the other
color. There are only two colors  black and white in some diagrams, blue and red in the actual
starmap. Because each room diamond in the map can be entered from either color direction, the
room diamond actually represent two distinct patterns, each the inverse of the other.
See coloring diagram.
 Color Maze
 A maze in which movement is restricted based on colors, popularized in the 1980s by Adrian Fisher.
In a color maze, the passages between decision points are colored and wanderers must either follow a repeating
sequence of colors or follow a colorbased rule, usually that you can't leave by a passage of the same color
as the one you entered by. The starmap is a color maze. See coloring.
 Constellations
 Coherent, repeating shapes which often appear in distance maps.
These shapes are based on recurring sets of relationships. Examining them in detail often
yields insights about the innermost workings of the starmaze.
 Courtyard
 One of the four house types in a threedimensional starmaze.
Courtyards are built as gardens in five descending triangular terraces which face inward and converge
on the oculus in the center of the maze.
There are four courtyards, one for each of the four temples.
The north courtyard, comprising the House of Rumor, gathers water trickling
from the Tower of Rain and channels it through a series of winding streams, ponds,
and waterfalls.
The east courtyard, comprising the House of Desire, is riddled with pleasant bowers,
gazebos, and discrete benches and nooks hidden beneath a canopy of trees.
The south courtyard, comprising the House of Laughter, is an oasis of palm trees and trickling fountains.
The west courtyard, comprising the House of Innocence, is a hedge maze dotted with statues and topiary.
 Coverage
 The percentage of the maximum possible permutations of filled and unfilled cells occurring
in the patterns of a particular hypercube puzzle. The maximum permutations
for any ndimensional hypercube is equal to 2^{n} so,
for example, a 9dimensional hypercube puzzle in which 128 of the possible 512 permutations occurred
would have a coverage of 25%. The starmaze has a coverage of 100%, which means that every possible
permutation of open and closed cells occurs once and only once. The minimum possible amount of coverage
is 2 patterns. In a minimum coverage hypercube puzzle, one pattern occurs in all the
sun rooms and the other pattern occurs in all the moon rooms.
Minimum coverage puzzles include a set of maximum cycle puzzles. For more information on
coverage, see Puzzle Keys.
 Crossing
 Passages in the threedimensional starmaze
which lead to a different house.
Crossings are routed through multistory bridges between towers,
inside rampart walls connecting bastions,
over and under the garden walls separating courtyards,
through tunnels between the subterranean deeps,
and even through a system of automatic cockleshell boats in the grotto.
Half of all crossings result in a change of level.
Like all passages, crossings are open when connecting public chambers
and enclosed when connecting private chambers.
In the starmap, they correspond to macropipes connecting one
microcosm to another.
Crossings always leave from yin cells. There are 1024 crossings in the starmaze.
 Crossroads
 The grand passage tour is marked by hexagrams assigned
to each noncenter cell of every pattern. Because the base hexagram of each
arrival cell matches the base hexagram of the departing cell for the next step in the tour,
the hexagrams form a continuous chain. Most patterns contain four different pairs of base hexagrams;
each step of a pilgrim's tour is accomplished by departing through the only cell with the hexagram which matches the cell he
entered by. Crossroads are the 50 rooms in which one base hexagram occurs more than two times.
In 49 of these 50 rooms, the base hexagram occurs four times, which means that a pilgrim entering the crossroad through
that one base hexagram will be confronted with three matching hexagrams of which only one is the correct departure cell
to stay on the tour. In one room (room
11),
the base hexagram occurs six times, which provides the pilgrim with five choices instead of the usual three.
This choice is encountered twice for each crossroad, and three times in room 11  a total of 101 times.
 Cycle
 A fourstep circuit which leads from a room back to itself.
Geometrically, each cycle is a square within the hypercube involving
four 90 degree turns across two neighboring dimensions.
The shortest path from a room back to itself is always a cycle for every room with only four exceptions:
the pit, the source, the starting pattern, and
the ending pattern. Some rooms have two cycles, some have four, some have six, and
some have eight, as shown in the room's cycle diagram.
A room with two cycles is called a 2ring room.
There are a total of 512 distinct cycles in the maze. A cycle is present whenever
an open noncenter cell is adjacent to a closed noncenter cell. You can follow a
cycle by choosing this open cell, then its neighboring cell (which was closed but becomes open in the subsequent
pattern), then the first cell again, and then the second again to return to the original pattern.
Every cycle in the starmaze contains four transitions:
one ebbing, one flowing,
one waxing, and one waning.
The four patterns which comprise a cycle are assigned seasons: Spring, Summer, Fall, and Winter.
A cycle is considered to be forward if the patterns proceed in that order, backward if they proceed in the reverse
order. A spring pattern can lead to summer patterns through forward cycles, or to winter patterns through
backward cycles, but never to an autumn pattern. Movements between spring and summer or between fall and winter
involve a change of houses (yin cell); other movements change rooms within the same house.
The 512 fourstep cycles in the starmaze fall into 8 groups, starting with the 64 cycles involving cells
9 and 2 and proceeding clockwise. Within a group, a cycle exists for each of the 64 patterns with
binary coordinates defined by the 6 bit number formed by the other 6 noninvolved noncenter cells.
For each of these 6bit numbers, a valid cycle will exist for a coordinate with the center cell's bit
set to either 0 or 1, but not both.
For more information about cycles, see Cycles and Seasons.
 Cycle Diagram
 A diagram used to convey the number and nature of the cycles emanating from
each pattern. The room at right (room 9) has four cycles:
two forward, and two backward. The cycles are represented by circular arrows positioned
between the two cells generating that cycle. The arrows are drawn clockwise if the
seasonal progression is forward, counterclockwise if it is backward.
In a cycle diagram, open cells are indicated by filled diamonds.
The state of the center cell is not shown, since the center cell is not involved in any cycles.
Instead the center displays a sun if the pattern is a sun room,
or a moon if the pattern is a moon room.
For more examples of cycle diagrams, see Cycles and Seasons.
 Dais
 One of four rooms reached by a crossing from a throne
in the threedimensional starmaze.
Each room is called "a dais" of its throne; together they form the second level
of rooms extending from that throne. All dais rooms occur in male houses
and are associated with the element of water.
 Day
 Any journey in the starmaze passes through an alternating series of sun rooms
and moon rooms. In this context, each sun room is referred to as a day. That is,
a five step path might be described as a journey of three days and two nights.
 Deep
 One of the four house types in a threedimensional starmaze.
Deeps are formed from the bases of the four towers
and are located in a vast grotto beneath the maze.
They rise up from the waters of the grotto;
the lowest rooms in three of the deeps are beneath the waterline and contain thick glass
windows which function as aquariums.
Rooms on the fourth level contain piers which provide crossings
to adjacent deeps via a system of automated cockleshell boats.
There are four deeps, one for each of the four temples.
The north deep, comprising the House of Judgement, contains a series of courtrooms and dungeons.
The south deep, comprising the House of Lamentation, provides a secluded retreat for those
recovering from great loss who are not yet ready to face the world again.
The west deep, comprising the House of Whispers, includes a series of vents in the rock
through which sigh the winds of the nearby ocean and is the seat of a mysterious oracle.
The east deep, comprising the House of Darkness, is partially encased in the surrounding rock
with interior rooms connected by roughhewn tunnels. Its ivory outer walls reflect the light
of the oculus onto the other deeps, but little of this light penetrates
the dimlylit rooms within.
 Dimension
 Dimension refers to the number of coordinates necessary to describe every point in a space.
Because the starmaze is best described as filling a binary space of nine dimensions,
each cell in a starmaze pattern can be thought of as representing one of those nine dimensions.
The dimensions are numbered in the same way the cells are, according to the Lo Shu, so
"neighboring dimensions" are dimensions with consecutive Lo Shu numbers.
Each dimension in the starmaze has acquired a symbolic meaning, so
every movement through the maze takes on meaning according to the dimension traversed. Surprisingly,
each starmaze dimension can also be assigned a graphic symbol derived from it's key.
Geometric concepts from our normal Euclidean space, such as squares and cubes, can be generalized to higherdimensional
hypercubes. The 9D hypercube which holds the starmaze can be decomposed into an
inner 8cube and an outer 8cube.
In addition to the nine binary dimensions corresponding to the cells, I sometimes refer to the
starmap as a twodimensional maze
because it can be drawn on a sheet of paper (although technically it is nonplaner, that is, it has lines which cross over
and under each other, so is not actually twodimensional). The threedimensional version of the maze has the same
number of rooms connected in the same way, but with passages that twist and curve in normal
threedimensional space (instead of
extending all at right angles as they would in ninespace). I have also explored generalized versions of the original puzzle which
have more or less than nine dimensions. I have, for example, mapped the 5D form of the starmaze puzzle and
stepped through the solution of the 25dimensional form.
For more information, see Other Dimensions.
 Distance
 The shortest number of steps required to travel from one room in the starmaze to another, also known as
the normal starmaze distance or legal distance. Because the maze is comprised
of oneway passages, the distance from room A to room B will not necessarily equal the distance from room B to room A.
The distance from any room back to itself is four (with four exceptions). The distance from the starting pattern
to the ending pattern is eleven. No room in the maze is more than thirteen steps distant from another.
For every room in the maze, there exists another room that is at least 9 steps distant (to or from).
The distance from a room to its inverse is always an odd number.
The average (mean) distance from one room to another within the maze is 5.775951.
The geometric distance, in contrast, is the number of steps from one room to another if the unidirectional
restrictions of the puzzle are ignored. The geometric distance from room A to room B does equal the distance from
room B to room A, and the geometric distance from a room to itself is zero.
 Distance Maps
 A distance map for distance d is a matrix of 512 rows and 512 columns which displays a pixel at position (r,c)
if and only if the distance between room r and room c is exactly d. Varying the distance d
creates a striking series of diagrams that helps illuminate the relationship of rooms within the starmaze.
Another even more revealing series of distance maps can be created by changing the normal order of rooms to
grand tour order. For examples and more information, see Distance Maps.
 Divination
 The practice of gaining insights into a situation by randomly forming and meditating upon
a starmaze pattern. The pattern can be formed by flipping coins or casting stones
or rolling dice to determine an open or closed state for each of the nine cells.
Additional layers of meaning can then be derived by considering the season,
element, form and house associated with that pattern,
as well the hexagrams associated with its incoming and outgoing passages.
Practitioners of the I Ching can cast yarrow sticks or toss coins in the usual way
and then use the Hexagram Concordance to associate that hexagram
with a starmaze pattern and a particular date and time. See astrology.
 Dominant Rooms
 In the color scheme of the starmap, the room in a
microcosm which has four black exit pipes. This corresponds to
a pattern with all four yang cells open (filled). In the
threedimensional starmaze, the dominant room is placed at the highest level of each house.
The dominant room can be thought of as residing at the apex of a binary trickle chart.
 Ebbing
 A transition which originates in a sun room
and which involves a binary coordinate change from 1 to 0
(a square cell to a circle).
 Edge
 The connection between two corners of a hypercube. Since the rooms of the starmaze correspond to the corners
of a ninedimensional hypercube, an edge is equivalent to the passage between two adjacent rooms. 'Edge' and 'passage'
are used interchangably, but 'edge' is favored when discussing geometric aspects of the maze. A directed edge is a
oneway passage. All the passages in the starmaze are directed edges. Each edge travels in one of nine dimensions,
numbered according to the Lo Shu. Each cell in a starmaze pattern corresponds to an edge. Inbound edges are represented
by closed (empty) cells, outbound edges by open (filled) cells.
 Elements
 A property assigned to each starmaze pattern based on the how quickly
the pattern can change into its inverse.
Element names are based on the five classic Chinese elements arranged in order of speed and fluidity.
A fire pattern can reach its inverse in 5 steps, a water pattern in 7 steps,
a wood pattern in 9 steps, an earth pattern in 11 steps, and a metal pattern in 13 steps.
Pattern 0 (the pit) is assigned the element of emptiness.
There are a total of 32 fire patterns, 192 water patterns, 238 wood patterns, 45 earth patterns,
and 4 metal patterns (gold, silver, bronze, and iron).
A pattern's element is largely determined by its number of open yin cells.
Every movement from room to room within the starmaze can represent a transmutation of elements
but there are limitations on how the elements can change.
For more information, see Elements.
 Ending Pattern
 A certain pattern in the starmaze which occurs in
room 495. This pattern consists of a closed center cell surrounded by eight open cells. The original
starmaze puzzle consists of finding the shortest path from the starting pattern to its
inverse, the ending pattern. The ending pattern is one of the four zeroring rooms
in the maze that requires more than four steps to return to itself (it requires six steps). It is one of only eight coincidental rooms. Symbolically, the ending pattern is related to the unreachable source,
and represents the closest one can reach to perfection. Thus, the elevenstep path from start to end represents a journey
from a spiritual death to a spiritual rebirth.
 Female House
 Any house in the threedimensional starmaze in which
movement is allowed from private to public chambers,
but not the reverse. Each private chamber in a female house contains a
hidden spiral staircase leading up through a oneway door through the monolith
in the public chamber above. Eight of the sixteen houses are female and each female house is surrounded
by four adjacent male houses. Movement from any public chamber in a female
house to any private chamber in that house requires travel through one of the adjacent male houses.
The pattern for every room in a female house contains an even number of yin cells
in the season of either Winter or Spring.
Each female house contains two thrones and two rooms which are the terminus
of thrones in the opposite house; all remaining rooms are steps of other thrones.
The eight female houses are the houses of Darkness, Bones, Laughter, Sand, Whispers, Dreams, Rumor, and Numbers.
Each of these houses is cared for by a female keeper who lives in the sixteen private chambers.
The role of keeper is normally passed from mother to daughter.
 Flip
 To reverse the value of a cell in a pattern or a bit in a
binary coordinate. Flipping an open (filled) cell makes it closed (empty) and also causes
some adjoining cells to flip; flipping a closed cell makes it open. Similarly, in a binary address, flipping
a 1 bit turns it into a 0 and flipping a 0 turns it into a 1. Each movement in the starmaze involves flipping
a single bit in the binary address of the starting room.
 Flowing
 A transition which originates in a sun room
and which involves a binary coordinate change from 0 to 1
(a circle cell to a square).
 Forms
 Each of the 512 possible patterns come in one of 20 basic forms.
Each form consists of all the unique rotations of a pattern with a closed center cell,
the same rotations with the center cell open, and the inverse of both sets.
Most forms have 8 unique rotations, and so include a total of 32 patterns.
But three of the forms have only 4 unique rotations, one form has only 2 rotations, and the final two forms have only 1 rotation.
The forms are arranged in order of the number of stars (open cells) in the initial pattern.
The first form has one star, forms 2 to 5 have two stars, forms 6 to 12 have three stars,
forms 13 to 19 have four stars, and the final form has no stars.
Each form is given a name reminiscent of a yoga position, e.g. "Clouds", "Facing the Lion", "Cranes and Ravens", etc.
Every movement within the maze causes a transformation of one form into another.
Each form can change into 2 to 9 other forms. Only 4 of the 20 forms can change into themselves.
All of the patterns in a given form emanate the same number of cycles.
For more information, see Forms.
 Foundation
 One of four rooms reached by making three crossings
from a throne
in the threedimensional starmaze.
Each room is called "a foundation" of its throne; together they form the fourth level
of rooms extending from that throne.
 Geometric Distance
 The number of steps from one room to another if the unidirectional
restrictions of the puzzle are ignored. The geometric distance between two rooms is equal to the number of differing
bits in their binary coordinates, a concept also known as the Hamming Distance.
This number is never greater than nine. Two rooms with a
geometric distance of nine are on opposite corners of the ninedimensional hypercube
and are said to be geometric inverses.
The normal starmaze distance between two rooms is always greater than or equal to their
geometric distance. For example, the geometric distance from the starting pattern
to the ending pattern is five, but the normal starmaze distance between these two rooms is eleven.
 Geometric Inverse
 A pattern with inverse binary coordinates (zeros changed to ones and vice versa).
Geometric inverse rooms occupy opposite corners of the ninedimensional hypercube.
This means that the geometric distance between geometric inverses is always 9.
Curiously, the legal or normal starmaze distance between geometric
inverses is also always 9, even though normal starmaze distances are often longer than geometric distances.
The term inverse, as opposed to geometric inverse, refers to a pattern's
open and closed cells, not the bits of its underlying binary coordinate.
 Grand Passage Tour
 A path through the starmaze that starts with the starting pattern and traverses every noncenter
edge of the starmaze exactly once.
It does this by following the four possible disjoint hamiltonian circuits for the inner 8cube,
traversing the center dimension from
room 16 to
room 325,
then continuing with four more hamiltonians in the outer 8cube. In the process, it visits each room exactly four times.
During this tour the normal rules restricting movement between rooms are lifted; passages may be traversed even if their cells
are closed. The steps of the grand passage tour are marked by placing hexagrams in every noncenter cell in a particular order.
Because all 2048 noncenter edges are traversed exactly once, and since the hexagrams at either end of each passage transform according
to their moving lines (that is, each edge represents both hexagram A moving to hexagram B and B moving back to A),
all 4096 possible hexagram transitions are encountered by a pilgrim following the tour. The grand passage tour is the means
by which the starmaze is linked to the hexagrams of the I Ching and is the basis for
starmaze astrology. For more information about the tour, see crossroads.
 Grand Tour
 A path through the starmaze that moves through the entire starmaze from the pit
to the source and back again, visiting each room exactly once (as opposed to the
grand passage tour which visits every noncenter edge).
It does this by suspending the normal
rules restricting movement only through open cells and following the
remarkably simple recursive algortithm shown at right. The algorithm tours an ndimensional hypercube by touring the
(n1) dimensional "lower" hypercube, moving up along the nth dimension, and then touring the (n1) dimensional "upper" hypercube.
At the lowest level it simply moves one unit in the first dimension. The sequence of movements as the value of n increases is
1, 121, 1213121, etc. The result is a list of rooms in "grand tour order." If the binary coordinates
of the rooms are listed in grand tour order, the result is a "gray code" which has many interesting applications.
It is sometimes quite revealing to use this order in place of the normal order when plotting
distance maps and in other studies. As an added bonus, grand tour ordering can be used to solve
the nine disk Tower of Hanoi problem.
 Graph
 A diagram consisting of one or more nodes and a set of edges connecting those nodes.
A hypercube is a type of graph. If the edges of the graph are directed (i.e. oneway),
the graph is called a directed graph or digraph. Graph theory is the study of such diagrams.
 Grotto
 A feature in the subterranean part of the threedimensional starmaze.
The four deeps rise from the waters of the grotto, which remains flooded
even at low tide. The bottommost public rooms of the deep are below water level and have thick
glass windows which allow views of seacreatures. A system of automated cockleshell boats
provide crossings between piers located
in four public rooms of the deep's fourth level.
The grotto is lit during the day by the oculus.
 Hamiltonian
 A circuit that visits each node exactly once. Disjoint or edgedisjoint
hamiltonians are multiple hamiltonian circuits within the same graph that have no edges in common.
The figure at right shows two disjoint hamiltonians in a tesseract.
In an 8dimensional hypercube
it is possible to find four disjoint hamiltonians; each hamiltonian uses two of the eight
edges belonging to each node in the cube. And since each hamiltonian begins and ends in the same node, all four hamiltonians
can be joined together to create a larger circuit which visits each node four times and each edge exactly once. Two
such circuits, one for the inner 8cube and one for the outer 8cube can then
be joined to form the grand passage tour of the starmaze.
 Hexagram
 One of 4096 symbols which form the basis of the I Ching.
Each hexagram consists of six lines, numbered from bottom to top. Each line may be either solid or broken.
Since there are 2^{6} ways of arranging six solid or broken lines, this yields a total of 64 base hexagrams,
each associated with an image and a judgement. In addition, each line of a base hexagram can be either moving or nonmoving.
Moving lines are in the process of changing into their opposite; a moving solid line will become broken,
a moving broken line will become solid. For example, the figure at right shows base hexagram 59, "Dispersion",
which consists of 3 broken lines and 3 solid lines. Lines 2 and 4 (counting from the bottom) are moving.
This means that base hexagram 59 is in the process of changing into base hexagram 12, "Stagnation" (and vice versa).
Each of the 64 base hexagrams can, depending on its moving lines, change into any other base hexagram.
Or to put it another way, there are 64 possible arrangements of moving and nonmoving lines for each of the
64 base hexagrams. This yields a grand total of 4096 hexagrams.
4096 also happens to be the combined number of noncenter cells (exits) from every room in the starmaze.
So every passage in the starmaze has a hexagram carved at either end and each hexagram
is assigned a transition number from 1 to 4096.
The two transition numbers always differ by 1, and the two hexagrams share the same moving lines.
For example, the transtion from Dispersion to Stagnation shown in the diagram occurs in the passage leading through
cell 2 from room 433 to room 227. In room 433, this passage is labeled with the first hexagram and corresponds to transition number 285;
the other end of the passage, in room 227, is labeled with the second hexagram and corresponds to transition number 286.
The overall effect of this scheme is that every possible hexagram occurs once, and only once, within the starmaze.
And the hexagram transitions are numbered so that following them in order leads pilgrims along the
grand passage tour. Pilgrims relying solely on the base hexagrams carved over each passage, however,
will encounter fifty crossroads, rooms with more than one base hexagram to choose from.
They must make a total of 101 correct decisions to stay on the true path. For more information on starmaze hexagrams,
see the Hexagram Concordance.
 House
 A house is to the threedimensional starmaze what a microcosm is to the
twodimensional starmap. There are 16 houses, each containing 32 chambers
(16 rooms with 2 chambers each). In the starmap, inner rooms and outer rooms
are connected by centerpipes and reside in separate micorocosms.
But in the threedimensional starmaze, each inner room corresponds to a private chamber
directly below the public chamber corresponding to its outer room, so both
reside in the same house. This means that half of the rooms (the outer rooms) in any microcosm are located
in the house at the position of the microcosm at the other end of the centerpipe.
The 16 houses have colorful names ('the House of Bones', 'the House of Birds', etc.) and are arranged
in a ring of four temples as shown in this diagram.
Each temple consists of four houses:
an underground deep which leads up to an outward facing bastion and
an inwardfacing courtyard which in turn lead up to a tower.
There is a temple for each of the four cardinal directions.
Each tower also connects to its nearest two towers, each deep to its nearest two deeps, etc.
The result is that the 16 houses form a torus. Each house, then, may be described either by its name
or by its position (south bastion, west courtyard, etc.)
The houses are connected by a series of over a thousand crossings.
Four separate crossings are required to journey from a house to its opposite.
The rooms within a house or organized into five levels.
Houses are considered either male or female
and are assigned male or female keepers who live in the private chambers of the house.
For floorplans of each house, see House Charts.
 Hypercube
 A cube extended to four or more dimensions. A fourdimensional cube is also known
as a tesseract. A hypercube can also be defined as a graph in ndimensional
binary space where two nodes are connected if and only if
their binary coordinates
differ by a single bit. Because of this property, hypercubes are often studied in connection with selfcorrecting
codes. The number of components (edges, squares, cubes, etc.) within a hypercube can be determined by
consulting Cartan's Triangle.
The hypercube is a very important concept for the starmaze because the rooms and
passages of the maze can be mapped directly to the 512 corners and 2304 edges
of a ninedimensional hypercube, or 9cube.
 Hypercube Puzzle
 A set of interconnected patterns based on a hypercube with a specific assignment of directed edges.
Patterns are defined for each node of the hypercube as an arrangement of circles and squares to represent
the 0s and 1s of that node's binary coordinates. Each circle or square is filled if its associated
edge is outbound, unfilled if inbound. The origin and behavior of a given hypercube puzzle will vary according
to its assignment of directed edges, as defined by its unique key. The starmaze
is an example of a hypercube puzzle. For more information about hypercube puzzles, see
Puzzle Keys.
 I Ching
 The Book of Changes, perhaps the oldest surviving book in the world. Annotated over the years with many
layers of poetry and philosophy, it consists at its heart of 64 images, each associated with a hexagram,
and a system which defines how these images can change one to another. Like the starmaze, it is rooted in a
binary space of yin and yang. Because of this, many patterns and ideas in the I Ching
resonate strongly with features of the starmaze. In a sense, the starmaze can be seen as a ninedimensional extension of the I Ching.
Those interested in using the I Ching for divination can use the
Hexagram Concordance to further enhance their interpretation of any hexagram.
 Inner Rooms
 Rooms in the maze having binary coordinates
with the fifth (centerdimension) bit set to 1.
Half of the rooms in the maze are inner rooms, half are outer. The set of all inner rooms forms
an eightdimensional hypercube. The starmaze can be seen as an inner 8cube, an outer 8cube,
and the set of centercell edges connecting each inner room to its corresponding outer room.
In the threedimensional starmaze, inner rooms are represented by private chambers
which are dark and completely enclosed while outer rooms are open to the elements. Each house
consists of 16 inner chambers and 16 outer chambers. Symbolically, inner rooms represent introspection and the inner life.
 Inverse
 A pattern in which the open cells are replaced by closed cells and vice versa.
The starting pattern is the inverse of the ending pattern.
In the starmap, each position is shared by inverse rooms.
The geometric inverse, in contrast, is based on binary coordinates,
so involves circles changing to squares instead of filled changing to empty. The inverse of a pattern is
never the geometric inverse of that pattern because inversion only flips the yin and center coordinates.
But because flipping these five bits always switches the parity of a pattern, the
path from a room to its inverse always consists of an odd number of steps. Or, to put it another way,
inversion flips all nine cells, but only five bits;
the distance between inverse rooms is an odd number between 5 and 13, but the
geometric distance is always 5. The distance between a room and its inverse
determines the element assigned to that room.
In the threedimensional starmaze, the inverse of a room is always located
in the opposite chamber (public vs. private)
on the same level of the opposite house.
 Keeper
 A person charged with caring for a house in the threedimensional starmaze.
Male houses are kept by male keepers and female houses by female keepers.
The role of keeper is normally passed from father to son or from mother to daughter.
The keeper dwells within the sixteen private chambers of his or her house.
 Key

A unique assignment of directed edges that defines the behavior of a specific
hypercube puzzle. Keys are represented either as a string of 0s and 1s or
a block of white and black squares where each digit or square defines the direction of a specific edge.
The number of digits in a key is equal to the number of edges in the ndimensional hypercube
the puzzle is based on, n*2^{n1} (2304 in the case of a 9cube).
For more information about keys, see Puzzle Keys.
 Legal
 Respecting the unidirectional restriction of the puzzle. A legal circuit or tour only
traverses edges in the allowed direction (through open cells only).
A circuit like the grand tour is illegal because it includes movement through closed cells.
 Level
 A property of all rooms in the threedimensional starmaze.
The rooms of each house are arranged in five levels, numbered from top to bottom.
And since each room consists of an upper public chamber
and a lower private chamber, every level is two stories high.
Each house has one level 1 room, four level 2 rooms, six level 3 rooms, four level 4 rooms, and one level 5 room.
The level of a room can be deduced from the yang cells of its pattern;
a public level 1 room has 4 open yang cells, level 2 rooms have 3 open yang cells, etc.
(The reverse is true for patterns corresponding to private chambers.)
Within a house, the public rooms flow downwards in a binary trickle
with the dominant room on level 1;
level 1 has 4 internal exits, level 2 rooms have 3 exits, etc.
The private rooms flow in the opposite direction, from level 5 up to level 1.
The 1024 crossings between houses may or may not cause level changes
as shown in the diagram at right.
Crossings to or from a level 3 room will vary depending on whether
its two open yang cells share a common yin cell (3a) or are in opposition (3b).
A room and its inverse always share the same level.
For example, the solution path leads from
a private level 1 room in the East Deep
to a public level 1 room in the West Tower.
From a symbolic perspective, levels convey the arc of the story told by the public chambers of each house.
Level 1 introduces the topic,
level 2 focuses on each aspect,
level 3 compares and contrasts aspect pairs,
level 4 adds a third aspect,
and level 5 brings all four aspects together.
The rooms extending from a throne are also arranged in five levels.
In this way the hard to grasp movement through crossings is made analogous to
the more familiar shifts within a house.
 Lo Shu
 An ancient Chinese symbol which is used to number the cells of a
starmaze pattern. The magic square derived from the Lo Shu is shown at right.
Each row, column, and diagonal sum to 15. The Lo Shu figures prominently in the I Ching,
plays a key role in the art of Feng Shui, and was used to structure many ancient Chinese temples.
Applied to the nine cells of a starmaze pattern, it defines the sequence of dimensions.
The four corner yin cells are numbered 2, 4, 6, and 8; the yang cells
are numbered 1, 3, 7, and 9; and the center cell is dimension number 5.
In addition to defining cell numbers, the Lo Shu is also used to define pattern or room numbers.
This is done by replacing each value n with 2^{(n1)}. The result is the multiplicative magic square
shown at right. Each row, column, and diagonal multiply to 4096, which also happens to be the number of
noncenter passages in the starmaze and the number of unique hexagrams (counting moving lines)
in the I Ching. The pattern number is calculated by summing the values in all the open (filled) cells. For example,
pattern 400 has cells 5, 8, and 9 open, and 16 + 128 + 256 = 400.
The system yields room numbers ranging from 0 to 511.
 Macropipes
 The pipes in the starmap which represent yin cell passages.
Macropipes connect microcosms.
Each macropipe represents or contains the 16 pipes leading from the rooms in one microcosm
to their corresponding rooms in one of four neighboring microcosms. Each macropipe is drawn as a bundle
of four large colored pipes, each of which branches out to one of seven points directly
north, south, east, or west of a row or column of rooms in a microcosm. To follow a macropipe on the starmap,
start at one of the four corners of a diamondshaped room and move your finger in a straight line until it
reaches one of the branched openings of a macropipe. If the color of that pipe allows you to move, follow
that particular large pipe in the bundle to a neighboring microcosm and, when the pipe branches out again,
take the branch with leads to a room in the same row and column as the room you left from.
By consolidating 16 pipes into a single bundle, macropipes greatly simplify and clarify the structure of the map.
In the threedimensional starmaze, macropipes appear as crossings.
 Male House
 Any house in the threedimensional starmaze in which
movement is allowed from public to private chambers,
but not the reverse. Each public chamber in a male house contains a fireman's pole
hidden in the monolith leading to the private chamber beneath.
Eight of the sixteen houses are male and each male house is surrounded
by four adjacent female houses. Movement from any private chamber in a male
house to any public chamber in that house requires travel through one of the adjacent female houses.
The pattern for every room in a male house contains an odd number of yin cells
in the season of either Summer or Fall.
Every room in a male house is either a dais or a foundation.
The eight male houses are the houses of Desire, Judgement, Lamentation, Birds, Innocence, Time, Words, and Rain.
Each of these houses is cared for by a male keeper who lives in the sixteen private chambers.
The role of keeper is normally passed from father to son.
 Mandala
 A geometric design used as an aid in meditation. Both the starmap and
the threedimensional starmaze can be used as mandalas.
 Memory Palace
 An ancient rhetorical technique of remembering long speeches by visualizing each topic
as a room in an imaginary palace. The threedimensional starmaze has become
a personal memory palace which I use not to memorize a speech but to practice moving through
the starmaze. See also mandala.
 Microcosm
 One of 16 clusters of rooms in the starmap.
Each microcosm consists of 16 diamondshaped rooms arranged in 7 rows and 7 columns to form a larger diamond shape.
The individual rooms of a microcosm are connected by micropipes,
and the microcosms themselves are connected by macropipes and centerpipes.
The connections within a microcosm form a tesseract
and can be mapped to a binary trickle chart.
As you move within a microcosm, the center and yin cells of the patterns remain constant;
only the yang cells vary. The 16 small diamond shapes comprising a microcosm are often referred to as
rooms, but in fact each small diamond actually represents both a room and its inverse, depending on
what color of pipe you entered it through.
So there are really 32 rooms or patterns within a microcosm.
In the atlas, each microcosm is given a label from A to P.
Each microcosm has a different orientation and a different dominant room.
A microcosm corresponds to a house in the threedimensional starmaze.
 Micropipes
 The pipes in the starmap which represent yang cell passages.
Micropipes connect rooms within each microcosm.
Unlike macropipes or centerpipes, each micropipe corresponds to a single edge.
Each micropipe is colored to indicate direction.
Micropipes are often referred to simply as pipes.
 Monolith
 One of 256 structures in the threedimensional starmaze which
serve to mark the location of each room and serve as a secret passage between
public and private chambers.
Every public room in the starmaze contains a stone monolith, five feet square and eight feet tall
which displays the starmaze pattern and other markings. If the center cell of that pattern is open,
pushing it will open a narrow, weighted door which will close upon entering the monolith.
In male houses, the monolith conceals a fireman's pole
leading from public chambers down to the private chambers beneath.
In female houses, the monolith conceals a narrow, winding staircase
which allows movement in the reverse direction.
 Moon Rooms
 A room with a binary coordinate which contains an odd number of 0s
(and an even number of 1s).
Because of maze's parity, a moon room always leads to a sun room and vice versa.
Half of the rooms in the maze are moon rooms. When discussing a journey through the maze, moon rooms are referred
to as nights.
 Moving Lines
 The lines in a hexagram that are in the process of changing from broken to solid or vice versa.
In the I Ching, an additional meaning is associated with each moving line.
 Neighbor
 Two rooms or patterns are considered to be neighbors
if one can be reached from the other in a single move.
Neighbors have a distance of one. Only .8% of all possible room pairs are neighbors.
 Neutral Key
 A puzzle key consisting of all zeros or all ones in each dimension. In a neutral key
hypercube puzzle there is a onetoone correspondence
between the open and closed cells of every pattern
and its underlying binary coordinates.
Neutral key puzzles have one source, one sink,
no cycles, and 100% coverage.
All their passages flow one way from source to sink in a binary trickle.
For examples of both 3D and 9D neutral keys, see Puzzle Keys.
 Night
 Any journey in the starmaze passes through an alternating series of sun rooms
and moon rooms. In this context, each moon room is referred to as a night.
 Node
 A corner of a cube or, more generally, a point or vertex in a graph.
Every room in the starmaze is a node.
 Oculus
 A round opening in the center of the threedimensional starmaze
where the bottommost rooms of the four courtyards meet.
The oculus is lined with stone and penetrates through the roof of the grotto;
a coin thrown into the oculus will fall for more than a hundred feet before splashing into the waters below.
For these reasons, the oculus is often referred to as the well when viewed from above.
But from the rooms and piers of the deeps the oculus is the primary source of light
within the groto, showing as a brilliant blue circle during the day and admitting starlight by night.
 Opposite Houses
 A pair of houses the threedimensional starmaze
which are as far apart as they can be, in opposite types
(bastions are the opposite of courtyards,
towers the opposite of deeps)
of opposite temples (north is the opposite of south, east the opposite of west).
The House of Darkness, in the east deep, is opposite the House of Dreams, in the west tower.
The opposite of a male house is also male;
the opposite of a female house is also female.
It requires four crossings to move between opposite houses.
In the starmap, opposite houses correspond to microcosms
on opposite ends of a common centerpipe.
Inverse rooms are always located in opposite houses
(on the same level but in opposite chambers).
Crossings can be seen as sets of binary trickles leading from one
house to its opposite in a way directly analogous to the descent of levels
within a house. For this reason, the movement between opposite houses is an
important aspect of navigation within the starmaze. The following poem is helpful
in remembering the pairs of opposite houses:
Rumor to Sand,
Innocence to Words,
Laughter to Numbers,
Judgement to Birds.
Time turns to Desire,
Lamentation to Rain,
Bones become Whispers,
Again and again.
But Dreams go to Darkness
And Darkness to Dreams
And nothing between them
Is at all what it seems.
 Orientation
 Each room in the starmap
is connected by four macropipes and four micropipes
which correspond, after a 45 degree clockwise twist, to the eight directions
of noncenter cells in a pattern: north, northeast, east, etc.
However, a north pipe does not always correspond to movement through the north cell.
In order to reduce the complexity of pipes in the starmap,
each room is assigned one of 16 different orientations. In some orientations,
the north/south poles are switched. In others, both the northwest/southeast poles
and east/west poles are switched. Thus, a separate orientation diagram is required
to fully understand the starmap.
 Origin
 The corner of a cube with a binary coordinant of zero (or 000000000 in the case of the 9dimensional starmaze).
The pit is located at the origin of the starmaze. The pattern displayed at the
origin is a defining feature of other hypercube puzzles.
 Outer Rooms
 Rooms in the maze having binary coordinates
with the fifth (centerdimension) bit set to 0.
Half of the rooms in the maze are outer rooms, half are inner. The set of all outer rooms forms
an eightdimensional hypercube. The starmaze can be seen as an outer 8cube, an inner 8cube,
and the set of centercell edges connecting each outer room to its corresponding inner room.
In the threedimensional starmaze, outer rooms are represented by public chambers
which are open to the elements or have ample windows, while inner chambers are enclosed. Each house
consists of 16 inner chambers and 16 outer chambers. Symbolically, outer rooms represent relations between the self and the
world around it.
 Parity
 The starmaze can be divided into two types of rooms, sun rooms and moon rooms.
Every sun room leads only to moon rooms, and every moon room leads only to sun rooms. Thus any journey through the
maze consists of an alternating rhythm of sun to moon to sun to moon, or day to night.
This property of the maze is called parity. Parity occurs because each room in the maze has a
binary coordinate with an even or odd number of zeros and every movement flips a single bit
in that coordinate, increasing or decreasing the number of zeros by one. For example, the pit
is a moon room because its binary address, 000000000, has an odd number of zeros (9 zeros). Each of the nine rooms
adjacent to the pit has a coordinate with one 1 and eight 0s, so each of these are sun rooms. Because of parity,
we know that any path leading from a room to a room with the opposite parity (such as the path from a room to its
inverse) must have an odd number of steps. And any path leading from a room to a room with
the same parity (such as any circuit) must have an even number of steps.
 Pascal's Triangle
 A triangular array of numbers known in antiquity but named after the mathematician Blaise Pascal.
Each number in the triangle is the sum of the two numbers immediately above it.
For example, the number 6 in row four (starting from row zero) is the sum of 3 and 3 in row three.
Pascal's triangle has many interesting properties. Each row sums to a power of two.
The values of row N provide the coefficients of the binomial expansion of (x+y)^{N}.
For example, (x+y)^{4} equals
1x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} +1y^{4}.
These rows also appear in binary trickle charts and other patterns which often arise in the starmaze.
For example, item K (starting from 0) of row 9 gives the number of starmaze patterns
with K open (filled) cells.
Cartan's Triangle, which can be used to find the number of components in an Ndimensional
hypercube, is a variant of Pascal's Triangle.
 Passage
 One of 2304 oneway connections between rooms in the
threedimensional starmaze. A passage corresponds
to an edge in the starmaze hypercube or a pipe
in the starmap. In the threedimensional maze,
each public passage includes a steep stone slide
which establishes its direction. Outgoing passages (which correspond to open cells) have
downward slides. This same slide prevents motion back through incoming passages.
A passage which leads to a different house is called a crossing;
a passage which leads to a different room of the same house is called a shift.
Secret passages concealed behind a panel in the central pillar of each room lead from upper public chambers
to lower private chambers, or vice versa.
In male houses, which only allow movement from public to private chambers,
the pillar conceals a fireman's pole.
Female houses use a spiral staircase and oneway doors which only allow movement
from private to public.
Passages from one private room to another are enclosed; passages between public rooms are open.
Each passage is labelled with a hexagram and transition number
at either end. Noncenter passages are numbered according to the grand passage tour.
Center passages are numbered according to the grand tour.
 Pattern
 One of the 512 possible positions in the starmaze puzzle, represented as a three by three array
of cells. Various schemes have been used to represent a starmaze pattern. The simplest
is a tic tac toe grid with Xs or stars indicating open cells.
For many years the scheme at right was used.
Here, three different shapes are used for the yin (corner), yang,
and center cells. The shapes incorporate arrows which suggest the forces implicit in the rules governing
movement within the maze. The yin cell shapes point to adjacent yang cells and to the center because these cells
will flip whenever a yin cell is selected. Similarly, the yang cell shapes point
to adjacent yin cells and the center shape points to all four yang cells. The
scheme now in use does not distinguish between yin and yang cells,
but instead uses circles and squares to convey the binary coordinate
associated with every pattern. Open cells are filled, closed cells are empty, but the shape used is a circle
if the underlying address bit is a zero, a square if it's a one.
In some contexts, pattern refers only to the arrangement of open and closed cells in the original puzzle;
in others, pattern refers to this scheme which incorporates both cells and bits at once. Also, since
each puzzle position corresponds to a room in spatial maze, the terms 'pattern' and 'room'
are often used interchangably. In general, 'pattern' is preferred when referring to abstract properties of starmaze
or when referring specifically to the graphic representation of the nine cells. 'Room' is preferred when
referring to movement within a spatial realization of the maze or to a position on the starmap.
Pattern numbers are calculated by summing the powers of two for each of the open cells using the Lo Shu.
Each pattern is assigned a season, form, element,
and time: sun for day patterns or moon for night patterns.
 Pipes
 The representation of a passage from one room to another in the starmap.
There are three types of pipe:
macropipes, micropipes, and centerpipes.
Pipes are colored to indicate direction.
 Pit
 A certain pattern in the starmaze which occurs in
room 0. This pattern consists of nine closed cells.
Because all its cells are closed, there is no escape from this room.
The pit has the binary coordinate 000000000, and is thus occupies the origin
of the hypercube in ninedimensional binary space.
The starmaze is actually a network with one sink (the pit)
and one source, which is the pit's inverse.
Because there is no escape from the pit, it is one of the four
zeroring rooms in the maze that requires more than four steps to return to itself.
It is also one of eight coincidental rooms.
 Position
 A number from one to sixteen indicating a room's location within a house
of the threedimensional starmaze.
Room positions are numbered according to the binary trickle chart, with position 1 on top,
positions 2 to 5 on the second level, etc.
The same position number applies to both public and private chambers.
The first 8 positions are considered to be in the upper half of the house, the final 8 in the lower half.
Within a house, the opposite of position n is position (17n).
The precise assignments within a level use the 4bit numbers from the trickle chart
with cell 9 of the public chamber as the first (highest) bit, then cells 7, 3, and 1.
For example, room 134, which has cell 3 open and cells 9, 7, and 1 closed (0010),
is the public chamber of the thirteenth room in the House of Sand
because 0010 occurs in the thirtenth position of the trickle chart counting top to bottom, left to right.
Rooms corresponding to private chambers are positioned using the inverse of their yang cells.
 Private Chamber
 The lower enclosed half of a room in the threedimensional starmaze.
Private chambers correspond to inner rooms.
The sixteen private chambers of a house comprise the residence of the house's keeper.
These chambers are arranged in five levels and connect through a series
of upwardleading stairs with oneway access panels hidden beneath the public slides.
In male houses, each private chamber is accessible from the public chamber
above it via a fireman's pole hidden in the monolith,
but there is no direct access from private back to public.
Instead the keeper must make a journey of 5 steps involving a crossing
to an adjacent female house. (The highest private male room is even harder,
requiring 7 steps and an additional house.) The reverse is true for private chambers
in female houses, any of which climb directly to a public chamber via a secret spiral staircase,
but which cannot be reached from above without crossing to an adjacent male house.
 Proper Starmaze
 A term used in the study of the generalized starmaze.
In an ndimensional starmaze pattern, there is 1 center cell, (n1)/2 yang cells, and (n1)/2 yin cells.
A starmaze is considered "proper" if there are an even number of yin cells and an even number of yang cells,
that is, if (n1) is a multiple of 4. So puzzles of dimension 5, 9, 13, 17, 21, 25, etc. are proper starmazes.
For more information, see Other Dimensions.
 Public Chamber
 The upper open half of a room in the threedimensional starmaze.
Public chambers correspond to outer rooms.
The sixteen public chambers of a house are arranged in five levels
and connect through a series of oneway stone slides in a binary trickle.
See private chamber.
 Ring Rooms
 The path from a room back to itself is called a cycle or ring.
Some rooms have two cycles, some have four, some have six, and some have eight  as shown in
the room's cycle diagram.
A room with two cycles is called a 2ring room. There are 4 0ring rooms (which have no cycles at all),
112 2ring rooms, 280 4ring rooms, 112 6ring rooms, and 4 8ring rooms. The inverse of an nRing
room is an nRing room. That is, the inverse of a 2ring room is also a 2ring room, the inverse
of a 4ring room is a 4ring room, etc. The direction of each ring reverses in the inverted room.
Movement through the center cell from an nring room leads to an (8n) ring room.
That is, the center cell in a 2ring room leads to a 6ring room, the center cell in a 4ring room
leads to another 4ring room, and the center cells of the four 8ring rooms
(170,
186,
325, and
341)
lead to a 0ring room.
For more information about ring rooms, see Cycles and Seasons.
 Room
 One of the 512 locations in the starmaze. Each room corresponds to a pattern. These
terms may be used interchangably, but room is preferred whenever
referring to movement within a spatial realization of the maze or to a position on the starmap.
In the threedimensional starmaze, a "room" actually consists of two rooms,
an upper public chamber and a lower private chamber.
Both chambers connect to four passages internal to the house,
four crossings to other houses, and an additional secret passage hidden behind a panel
in a central pillar leading between the two.
The public chambers, which correspond to outer rooms,
are either open to the elements or have ample windows;
the private chambers, which correspond to inner rooms, are dark and completely enclosed.
The rooms of each house are assigned a position from first to sixteenth.
In the starmap, a room and its inverse both occupy a diamond shape
connected to other rooms with various pipes,
and public and private chambers appear as corresponding rooms in different microcosms
connected by a center pipe.
See sun rooms, moon rooms,
ring rooms, crossroads, and coincidental rooms.
 Seasons
 An assignment given to each room in the starmaze so that every cycle
will have a Spring, Summer, Fall, and Winter room. Assignments are based on the number of "square"
yin and yang cells (cells with an underlying
binary coordinate of 1). Spring patterns have an even number of square yin and yang cells,
Summer an odd number of square yin cells, Winter an odd number of square yang cells, and Fall an odd number of
both square yin and yang cells. The four seasons present in every cycle always occur in either the
correct order ("forward time") or the exact reverse order ("backward time"). The order of the seasonal
progression appears in cycle diagrams respectively as clockwise or counterclockwise
circular arrows. Seasons can be assigned to hypercubes of any dimension.
For more information, see Cycles and Seasons.
 Seat
 A room containing a throne
in the threedimensional starmaze.
The room is called "the seat" of its throne and is the highest of five levels
of rooms extending from that throne.
Although the actual throne is located in the public chamber of the seat,
the private chamber is also considered part of the seat.
 Shift
 The act of making an internal transition, from one room to another,
within the same house of the threedimensional starmaze.
There are three types of transition: a switch from one chamber to the other,
a crossing to another house, or a shift to another level.
A shift corresponds to choosing a yang cell in a starmaze pattern.
 Sink
 In a directed graph, a node which you can enter but never leave from.
A starmaze of any size has only one sink, known as the pit,
and only one source.
 Solution
 The fewest number of steps needed to move from the starting pattern
to the ending pattern. N + 2 steps are required to solve an Ndimensional starmaze.
So for the classic 9D starmaze, 11 steps are required, for a total solution path
of 12 patterns including both start and end. There are 1920 different 11step solutions to the starmaze.
Only 96 of the 512 possible patterns are part of a solution. All 1920 solutions require traveling through
the center cell three times, through all four yin cells,
and twice through half of the four yang cells. As a result, the sum total of
Lo Shu cell numbers of any solution is always 55.
For other forms of the starmaze, a solution can always be found by following a sevenphase algorithm
described in Other Dimensions. For more details about the classic
starmaze solution, see Reading The Solution.
 Solution Path
 An ordered set of patterns comprising a solution to the starmaze puzzle.
Also, the visualization of that sequence plotted on the starmap.
A solution path includes both starting and ending patterns. So although the solution to the classic 9D starmaze
requires eleven steps, its solution path includes twelve pattern and so is said to have a length of twelve.
In the threedimensional starmaze, solution paths lead from the the Room of Awakening,
a private chamber in the first room of the House of Darkness in the east deep,
to the Room of Transcendence, a public chamber atop the west tower in the House of Dreams.
 Source
 In a directed graph, a node which you can leave from but never return to.
In the starmaze, the source refers to a certain pattern which occurs in
room 511. This pattern consists of nine open cells. Because all its cells are open, once you leave
this room, there is no way to ever return to it.
The starmaze is actually a network with this one source
and only one sink, the pit, which is the source's inverse.
Because there is no way to return to the source, it is one of the four
zeroring rooms in the maze that requires more than four steps to return to itself.
 Space
 When most people hear the term space, they think of the kind we live in, which requires three coordinates
(length, breadth, and height), or four if you count time. But "space" can be used to describe any collection of data
and so can have any number of dimensions. A space of sweaters
might have five dimensions: one for color (bone, aqua, rust, etc.), one for size (small, medium, large, or extra large), one for style
(vneck, turtleneck, cardigan, etc.), one for fiber (wool, cotton, acrylic, etc.), and one for manufacture (hand or machine).
A binary space has only a zero or a one in each dimension, but as many dimensions as needed.
 Star
 An open cell in a pattern. The computer game that inspired this strange obsession
was drawn as a pattern of stars (asterisks) in a tictactoe grid.
That's why I called my version of the puzzle the "starmaze". See Origins.
 Starmap
 A twodimensional representation of the starmaze consisting of sixteen microcosms
connected by a system of colored pipes. The original starmap was
devised in 1987. In 1998 the starmap was revised to eliminate an arbitrary twisting of pipes
in the original design. See Atlas and Revised Atlas.
For information about the history and design of the starmap, see Mapping the Maze.
For details on how to read the starmap, see Reading the Solution.
 Starmaze Explorer
 A HyperCard program I wrote to investigate the starmaze. The Starmaze Explorer can record any sequence
of patterns, plot that sequence against a simplified representation of the starmap,
display information about individual patterns, find a shortest path between any two patterns, and do
many other useful things. It also contains a log book of various experiments and findings. For more
information see Starmaze Explorer.
 Starting Pattern
 A certain pattern in the starmaze which occurs in
room 16. This pattern consists of a open center cell surrounded by eight closed cells. The original
starmaze puzzle consists of finding the shortest path from this pattern to its
inverse, the ending pattern. The starting pattern is one of the four
zeroring rooms in the maze that requires more than four steps to return to itself (it requires six steps).
Symbolically, the starting pattern is related to the pit,
and represents a kind of spiritual crisis. Although technically, the starmaze is a closed network with no
exits or entrances, this room is considered to be the one and only entrance to the maze.
 Step
 One of six rooms reached by making two crossings
from a throne
in the threedimensional starmaze.
Each room is called "a step" of its throne; together they form the third level
of rooms extending from that throne.
 Subway
 A starmaze subway is a legal circuit that visits
all 16 houses without traversing the center dimension.
Shown at right is the currently designated route. Because it takes 14 days and
14 nights to come full circle it is called the fortnight express.
It's not yet clear whether a shorter subway is possible.
We do know that no subway could take less than 24 steps. Moving to a new house requires a yin movement.
But after 4 such moves, all yin cells are exhausted and require 2 yang moves to replenish. So the best
possible sequence of yin and yang moves would be 4+2+4+2+4+2+4+1+1=24. We also know there has to be an
even number of steps. So unless a 24 or 26 step circuit can be found,
a fortnight circuit like this one is the shortest possible subway.
 Sun Rooms
 A room with a binary coordinate which contains an even number of 0s
(and an odd number of 1s).
Because of maze's parity, a sun room always leads to a moon room and vice versa.
Half of the rooms in the maze are sun rooms. When discussing a journey through the maze, sun rooms are referred
to as days.
 Temple
 A collection of rooms in the threedimensional starmaze. There is a temple for each of the four
cardinal directions. Each temple consists of four houses: an underground
deep which leads up to an outward facing bastion and
an inwardfacing courtyard which in turn lead up to a tower.
The East Temple includes the houses of Words, Bones, Desire, and Darkness and overlooks a forest.
The South Temple includes the houses of Sand, Birds, Laughter, and Lamentation and overlooks a desert.
The West Temple includes the houses of Time, Dreams, Innocence, and Whispers and overlooks an ocean.
The North Temple includes the houses of Numbers, Rain, Rumor, and Judgement and overlooks a mountain.
 Terminus
 The room opposite a throne
in the threedimensional starmaze.
A journey from the seat of a throne through all available crossings
(all open yin cells) in any order
will lead to the terminus of that throne; at that point the traveller cannot escape
the house without first moving to a different level.
 Tesseract
 A commonlyused name for a fourdimensional hypercube.
A tesseract has 16 corners, 32 edges, 24 squares, and 8 cubes. Each corner has 4 mutually perpendicular edges projecting from it.
(In the diagram at right, the fourth dimensional edges are shown in red.) Diagrams such as this are merely
projections of a true tesseract, like shadows cast on a wall. Tesseracts occur frequently within the starmaze.
Each microcosm in the starmap is a tesseract, and the entire map
is itself a tesseract with each microcosm as a corner and macropipes as edges.
A tesseract can also be mapped directly to a 4D binary trickle chart.
 ThreeDimensional Maze
 A fullsized, threedimensional representation of the maze which may someday be built as a virtual reality.
The 3D starmaze consists of four large temples arranged around
the points of a compass. Each temple consists of four structures called houses:
a subterranean house called a deep, an outward facing house called a bastion,
an inwardfacing terraced house called a courtyard, and an elevated house called a tower.
The houses are connected to each other by a system of over a thousand crossings.
Each house consists of sixteen public chambers and sixteen enclosed private chambers.
An oculus in the center of the maze opens into a grotto beneath.
The sketch shown here is a simplified mockup without the grotto, bridges, and other details.
For more information, see Starmaze 3D.
 Throne
 A room in the threedimensional starmaze
which contains four open crossings.
The throne itself is located in the public chamber of the throne room,
but the private chamber is also considered to be part of the throne room.
The four open crossings from a particular throne room descend to 15 other rooms in other houses
all sharing allegiance to that throne; these 16 rooms (each with a public and private chamber)
form a kind of virtual house arranged in levels which trickle
just as rooms do in an actual house.
The room containing the throne is called the seat of the throne.
Following crossings in any order leads to one of four rooms, each called a dais,
then to one of six rooms, each called a step,
then to one of four more rooms, each called a foundation,
and finally to the terminus of the throne,
always located in the opposite house from the seat.
You can always travel from a throne room to its inverse room in five steps
by simply taking all four crossings in any order, switching chambers once along the way.
For this reason each of the 32 public and private throne rooms is associated
with the element of fire.
There are two thrones in opposite positions
within each of the eight female houses, for a total of 16 thrones in all.
 Tower
 One of the four house types in a threedimensional starmaze.
Towers are rooted in the subterranean deeps and rise up
between bastions and courtyards on either side.
There are four towers, one for each of the four temples.
The north tower, comprising the House of Rain, has a fountain on top which drips water
down the inner face of the tower and into the courtyard below.
The south tower, comprising the House of Birds, is topped by columbariumlike mews for desert falcons and is
often circled by falcons, owls, and other birds of prey.
The east tower, comprising the House of Bones, is built entirely of ivory and is sometimes referred to as
the "ivory tower"; the striking arch at its summit is known as the Gate of Ivory, a reference to Virgil's
assertion that true dreams exit through the Gate of Horn, false dreams through the Gate of Ivory.
The west tower, comprising the true House of Dreams, contains the corresponding (and much more narrow) Gate of Horn;
its magnificent summit with sweeping views of the ocean can be
reached only by a spiral staircase from the private chamber below and has on display
an antique flying machine which could in theory provide the only possible escape from the maze.
 Tower of Hanoi
 A classic puzzle involving different sized disks resting on three spindles. The object is to move a stack
of disks from one spindle to another without ever placing a larger disk on top of a smaller disk. If the smallest
disk is number 1, the next largest number 2, etc., the solution for three disks is to move disk 1 to spindle 2,
disk 2 to spindle 3, disk 1 to spindle 3 (atop disk 2), disk 3 to spindle 2, disk 1 to spindle 1, disk 2 to spindle 2
(atop disk 3), and finally disk 1 to spindle 2. This sequence of disks, 1213121 is the
grand tour order for a cube. Similarly, the grand tour of the starmaze produces a sequence of movements
which could be used to solve the ninedisk Tower of Hanoi puzzle. In the same way, each room in the starmaze
can be associated with a particular arrangement of the ninedisk Tower of Hanoi.
 Trace List
 A list of room numbers compiled and displayed in the Starmaze Explorer
computer program. The program allows a starmaze researcher to run any number of experiments to calculate which rooms
in the maze have a given property; these rooms are then placed in the trace list. The researcher can then step through this
list, displaying the patterns one at a time, or plot the entire list on the starmap.
 Transition
 The change from one pattern to another
along a particular dimension.
Transitions are represented by edges,
passages,
and pipes;
the term "transition" is preferred when speaking about patterns and the meanings associated with them.
Transitions within the threedimensional starmaze
or the hexagram concordance
are assigned transition numbers.
Noncenter transitions originating in sun rooms are said to
either ebb or flow;
those originating in moon rooms are said to
either wax or wane.
Every cycle in the starmaze contains
one ebbing, one flowing, one waxing, and one waning transition.
Transitions through the center are said to either
"turn inward" or "turn outward".
Yin transitions are called crossings;
yang transitions are called shifts.
 Transition Number
 Numbers carved at either end of every passage in the starmaze, from 1 to 4608.
Noncenter transitions are associated with the 4096 hexagrams of the
I Ching and, if followed in order, trace the grand passage tour.
Center transitions are numbered according the grand tour.
In both cases, the numbers on either end of a passage always differ by 1.
 Transmutation
 The changing of elements by moving from one starmaze pattern
to another. For the limitations on how the five elements can transmute, see Elements.
 Waning
 A transition which originates in a moon room
and which involves a binary coordinate change from 1 to 0
(a square cell to a circle).
 Waxing
 A transition which originates in a moon room
and which involves a binary coordinate change from 0 to 1
(a circle cell to a square).
 Well
 A common name for the oculus when seen from above.
 Yang Cells
 The four cells in a starmaze pattern located north, south, east, and west of the center cell,
sometimes referred to as the "compass" cells. In Chinese philosophy, Yin and Yang are the
two opposing forces responsible for all change in the universe. The yang principle is associated
with light, the masculine, and many other things including odd numbers. The compass cells
are considered yang because those cells are odd numbers in the Lo Shu.
 Yin Cells
 The four corner cells in a starmaze pattern. In Chinese philosophy, Yin and Yang are the
two opposing forces responsible for all change in the universe. The yin principle is associated
with darkness, the feminine, and many other things including even numbers. The corner cells
are considered yin because those cells are even numbers in the Lo Shu.
