A   B   C   D   E   F   G   H   I   K   L   M   N   O   P   R   S   T   W   Y

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 lower-case 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 three-dimensional starmaze. Bastions are shaped like ten-story 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 nine-dimensional 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 nine-dimensional 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 1-4-6-4-1 always sum to the value of two raised to the dimension of the chart (1+4+6+4+1 = 16 = 2⁴). These numbers correspond to the coefficients of a binomial expansion like (x+y)⁴, which equals 1x⁴ + 4x³y + 6x²y² + 4xy³ +1y⁴, 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 4-D binary trickle chart with the dominant room at the apex. Rooms within each house in the three-dimensional starmaze are arranged on levels and are given position numbers according to the trickle chart. And the entire starmaze can be mapped to a 9-D 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: 1-9-36-84-126-126-84-36-9-1. 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 N-dimensional 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 5-cubes, 672 6-cubes, 144 7-cubes, and 18 8-cubes (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 three-by-three 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.

Center-Pipes

The pipes in the starmap which represent center-cell passages. All the center-cell passages from one microcosm lead to corresponding rooms in another microcosm. These two microcosms are connected by a single center-pipe. Unlike macropipes, center-pipes 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 center-pipe and travel to the corresponding room in the microcosm at the other end of the center-pipe. So each center-pipe contains 16 center-cell passages, half of one color, half of the other.

Chirality

Left- or right-handedness, 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 color-based 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 three-dimensional 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 n-dimensional hypercube is equal to 2ⁿ so, for example, a 9-dimensional 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 three-dimensional starmaze which lead to a different house. Crossings are routed through multi-story 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 non-center 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 four-step 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 2-ring room. There are a total of 512 distinct cycles in the maze. A cycle is present whenever an open non-center cell is adjacent to a closed non-center 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 four-step 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 non-involved non-center cells. For each of these 6-bit 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, counter-clockwise 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 three-dimensional 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 three-dimensional 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 rough-hewn tunnels. Its ivory outer walls reflect the light of the oculus onto the other deeps, but little of this light penetrates the dimly-lit 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 higher-dimensional hypercubes. The 9-D hypercube which holds the starmaze can be decomposed into an inner 8-cube and an outer 8-cube. In addition to the nine binary dimensions corresponding to the cells, I sometimes refer to the starmap as a two-dimensional maze because it can be drawn on a sheet of paper (although technically it is non-planer, that is, it has lines which cross over and under each other, so is not actually two-dimensional). The three-dimensional version of the maze has the same number of rooms connected in the same way, but with passages that twist and curve in normal three-dimensional space (instead of extending all at right angles as they would in nine-space). I have also explored generalized versions of the original puzzle which have more or less than nine dimensions. I have, for example, mapped the 5-D form of the starmaze puzzle and stepped through the solution of the 25-dimensional 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 one-way 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 three-dimensional 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 nine-dimensional 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 one-way 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 zero-ring 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 eleven-step path from start to end represents a journey from a spiritual death to a spiritual rebirth.

Female House

Any house in the three-dimensional 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 one-way 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 three-dimensional 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 nine-dimensional 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 nine-dimensional 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 non-center edge of the starmaze exactly once. It does this by following the four possible disjoint hamiltonian circuits for the inner 8-cube, traversing the center dimension from room 16 to room 325, then continuing with four more hamiltonians in the outer 8-cube. 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 non-center cell in a particular order. Because all 2048 non-center 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 non-center 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 n-dimensional hypercube by touring the (n-1) dimensional "lower" hypercube, moving up along the nth dimension, and then touring the (n-1) 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. one-way), 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 three-dimensional starmaze. The four deeps rise from the waters of the grotto, which remains flooded even at low tide. The bottom-most public rooms of the deep are below water level and have thick glass windows which allow views of sea-creatures. 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 edge-disjoint 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 8-dimensional 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 8-cube and one for the outer 8-cube 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 2⁶ 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 non-moving. 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 non-moving 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 non-center 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 three-dimensional starmaze what a microcosm is to the two-dimensional 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 center-pipes and reside in separate micorocosms. But in the three-dimensional 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 center-pipe. 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 inward-facing 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 four-dimensional cube is also known as a tesseract. A hypercube can also be defined as a graph in n-dimensional 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 self-correcting 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 nine-dimensional hypercube, or 9-cube.

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 nine-dimensional 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 (center-dimension) 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 eight-dimensional hypercube. The starmaze can be seen as an inner 8-cube, an outer 8-cube, and the set of center-cell edges connecting each inner room to its corresponding outer room. In the three-dimensional 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 three-dimensional 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 three-dimensional 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 n-dimensional hypercube the puzzle is based on, n*2^n-1 (2304 in the case of a 9-cube). 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 three-dimensional 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^(n-1). 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 non-center 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 diamond-shaped 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 three-dimensional starmaze, macropipes appear as crossings.

Male House

Any house in the three-dimensional 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 three-dimensional 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 three-dimensional 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 diamond-shaped 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 center-pipes. 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 three-dimensional starmaze.

Micropipes

The pipes in the starmap which represent yang cell passages. Micropipes connect rooms within each microcosm. Unlike macropipes or center-pipes, 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 three-dimensional 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 one-to-one 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 3-D and 9-D 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 three-dimensional starmaze where the bottom-most 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 three-dimensional 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 center-pipe. 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 non-center 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 9-dimensional 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 (center-dimension) 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 eight-dimensional hypercube. The starmaze can be seen as an outer 8-cube, an inner 8-cube, and the set of center-cell edges connecting each outer room to its corresponding inner room. In the three-dimensional 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 1x⁴ + 4x³y + 6x²y² + 4xy³ +1y⁴. 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 N-dimensional hypercube, is a variant of Pascal's Triangle.

Passage

One of 2304 one-way connections between rooms in the three-dimensional starmaze. A passage corresponds to an edge in the starmaze hypercube or a pipe in the starmap. In the three-dimensional 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 one-way 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. Non-center 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 center-pipes. 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 nine-dimensional 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 zero-ring 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 three-dimensional 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 (17-n). The precise assignments within a level use the 4-bit 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 three-dimensional 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 upward-leading stairs with one-way 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 n-dimensional starmaze pattern, there is 1 center cell, (n-1)/2 yang cells, and (n-1)/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 (n-1) 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 three-dimensional 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 one-way 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 2-ring room. There are 4 0-ring rooms (which have no cycles at all), 112 2-ring rooms, 280 4-ring rooms, 112 6-ring rooms, and 4 8-ring rooms. The inverse of an n-Ring room is an n-Ring room. That is, the inverse of a 2-ring room is also a 2-ring room, the inverse of a 4-ring room is a 4-ring room, etc. The direction of each ring reverses in the inverted room. Movement through the center cell from an n-ring room leads to an (8-n) ring room. That is, the center cell in a 2-ring room leads to a 6-ring room, the center cell in a 4-ring room leads to another 4-ring room, and the center cells of the four 8-ring rooms (170, 186, 325, and 341) lead to a 0-ring 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 three-dimensional 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 counter-clockwise 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 three-dimensional 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 three-dimensional 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 N-dimensional starmaze. So for the classic 9-D starmaze, 11 steps are required, for a total solution path of 12 patterns including both start and end. There are 1920 different 11-step 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 seven-phase 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 9-D starmaze requires eleven steps, its solution path includes twelve pattern and so is said to have a length of twelve. In the three-dimensional 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 zero-ring 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 (v-neck, turtle-neck, 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 tic-tac-toe grid. That's why I called my version of the puzzle the "starmaze". See Origins.

Starmap

A two-dimensional 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 zero-ring 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 three-dimensional 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 three-dimensional 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 inward-facing 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 three-dimensional 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 commonly-used name for a four-dimensional 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 4-D binary trickle chart.

Three-Dimensional Maze

A full-sized, three-dimensional representation of the maze which may someday be built as a virtual reality. The 3-D 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 inward-facing 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 three-dimensional 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 three-dimensional 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 columbarium-like 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, 1-2-1-3-1-2-1 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 nine-disk Tower of Hanoi puzzle. In the same way, each room in the starmaze can be associated with a particular arrangement of the nine-disk 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 three-dimensional starmaze or the hexagram concordance are assigned transition numbers. Non-center 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. Non-center 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.