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TopoGo
Topological Go, or TopoGo, is a game that dates from pre-Singularity Earth. The inventor of the game was reputedly a baseline interested in modelling wormhole networks. It is a variation of the classical game of 2-Go, with several levels of play. The deeper levels add increasing complexity to the game.

TopoGo is typically played by youth to develop spatial intuition and facility with accounting, and by strategists to develop game theory algorithms. It is rumored that playing higher levels of the game can provide a fairly accurate model of historical resource conflict among the Archai. The popularity of the game has waxed and waned throughout the centuries, with reports of particularly talented players being tutored and/or subsumed by various avatars. Interestingly, the rating
system, scoring, and topological variants on gameplay have been the most well documented among the Metasoft Version Tree.

Indeed, the classic Metasoft algorithms for TopoGo are considered canonical by serious contestants, and superb play can bring substantial economic incentives.

At various points in time, there have been contests in TopoGo from such normally dissociated empires such as Metasoft and the Zoefic Biopolity. In 8300 A.T., a thought-coral reef was reputed to have
defeated a MoonBrain and 5,000 other KnownNet participants in a Version Tree 23.5 Game. However, there have been no recorded contests between players of greater than S3....

Non-Metasoft players typically decide upon a Version Tree protocol to fix the rules of the game, which tends to completely describe the parameter space for the Board. Although a highly mathematical game, sophonts with good spatial modelling/visualization can usually grasp the basic rules.

However, published higher level contests often have enigmatic results. There are numerous examples of games that have a declared winner different from what a standard algorithmic assessment of the rules would indicate. There is some speculation that deep study of these published contests lends insight into higher toposophic game theory, and some ascensions have been reputedly linked to that line of study.




The TopoGo Board:
1. TopoGo is played on a continuous, rather than discrete, board. 3+1 TopoGo, otherwise known as standard TopoGo, typically using a 3D display. The defined boardspace can be any size desired, but beginning boards start out as a cube 500 units on a side, marked identically to a 3-Go game cube for ease of use. Adjacent is defined to be any Distance (as defined by the boardspace metric) < 1.

2. Any number of sides > 1 may contest the board; beginning games start with two sides. Each side has a fixed Starting Point from which to begin their moves. This Starting Point provides a set number of Resources per unit of Time, and may be adjusted to determine the Handicap.

3. The boardspace has a number of defined resource Nodes scattered throughout its volume; each Node provides Resource equal to its value per unit of Time. Distribution of Resources across the board also determines the Handicap.

Actions and Interrupts:
4. Interrupts occur whenever an Action by a side is possible. An Action consists of Creating Pieces, Movement, Acquiring Resource, Weighting, Destroying Pieces, or in advanced versions of the game,Capture, Transport, and Declaring/Breaking Alliances. For basic games, an Interrupt happens by default for every integer unit of Time.

5. Any number of Actions may be performed by a side at any time as long as the total of their Resources > 0.

Accounting:
6. Accounting of Time, Distance, Weight, and Resource is kept to whatever degree of precision desired. Basic games proceed in integer units of Time; full games proceed by Interrupt.

Creating Pieces:
7. A side may create at its Starting Point one zero Weight Pair of Home/Travel stones in ten units of Time for each unit of Resource.

Legal Moves:
8. The basic piece in the game consists of a Pair of stones, called Home and Travel, linked by a Red Edge (or line) between them. A side may move any number of pieces in its possession, subject to Resource constraints (see Rule 5).

9. Except for the one Red Edge, all stones in the game are linked by Black Edges to every other stone. The value of the Black Edge between each and every stone is their Euclidean distance (henceforth Distance).

10. The Home stone remains fixed in location; the Travel stone moves forward to occupy territory. Each unit of Distance traversed requires one unit of Time.

11. The value of a particular Red Edge is a fixed constant times the Distance between the Pair of stones. In basic games, this constant is 1/3.

12. It costs one unit of Resource to move a Travel stone 10 units of Distance. This cost must be prepaid in full before the Travel stone can be moved. Only Weight 0 Travel stones may be moved. The value of the particular Red Edge and all Black Edges for all Travel stones is updated during each Interrupt.

13. It costs one unit of Resource to give the Home/Travel Pair a Weight of one unit. There is no way to reduce the Weight of a Pair of stones.

Acquiring Resource:
14. A non-zero Weight Home/Travel Pair adds the Resource value of any node it is Adjacent to to the sides Resource.

Destroying Pairs:
15. Unlike standard 2-Go, Pairs in TopoGo are destroyed, rather than captured.

16. A Pair in TopoGo is destroyed if it forms a cycle on the board such that the value of the Red Edges in that cycle exceeds the value of the Black Edges in that cycle.

Aim of the Game:
17. The purpose of the game is to obtain more Resource than all your opponents. The game is finished when all Resource nodes are occupied, when no more moves can be made, or when all sides have agreed upon a winner.

Handicaps: Resources can be adjusted to give weaker players an advantage over a stronger ones.




These basic rules can be layered with increasingly more complicated algorithms. Some examples include:

Level 2:
Creating Pieces:

7a. Creation of Pairs requires 10 Resource-Time Units.That is, one Unit of Resource for each of 10 Time Units, two Units of Resource for each of 5 Time Units, etc. If the expenditure of Resource falls below one unit per unit Time, the Pair is destroyed.

7b.Creation of Pairs requires a new type of stone called a Forge, which can be created using 50 Resource-Time Units. Each Starting Point begins with one Forge. A Forge can be Created at any Node which has the appropriate amount of Resource per unit Time. If the expenditure of Resource falls below one unit per unit Time, the Forge is destroyed.

Legal Moves:

10a. The is no distinction between Home and Travel stones in a Pair. Either stone may be moved.

11a. The value of the Red Edge depends upon the relative Velocity (Distance per unit Time) between the two stones.

12a. (The Grazer rule part I) A non-zero Weight stone can be moved, at a cost of (Weight)^2 units of Resource per unit of Distance.

13a. (The Grazer rule part II) A Home/Travel pair can be given a variable Weight at one unit of Weight per (Weight)^2 units of Resource.

13b. (The Grazer rule part III) The total Weight of a Pair can be distributed separately between stones.

Acquiring/Transporting Resource:

14a. Each Node accumulates Resource per unit of Time, unless moved. Transport of Resource along Red Edges in a Pair takes a small constant amount of Time. Transport of Resources along Black Edges requires the same cost as a legal move. Resources are considered to have a Weight of 0 for purposes of Capture and travel cost, and may be escorted by Fleets (see Rule 18).

Destroying Pairs:

16a. A Pair in TopoGo is destroyed if:

1) It forms a cycle on the board such that the value of the Red Edges in that cycle exceeds the value of the Black Edges in that cycle

2) The Weight of that Pair is less than the weight of the other Pairs in that cycle.

Capturing Pairs:

18. A new type of stone called a Fleet can be constructed at any Node for a cost of 10 Resource-Time Units * (Weight)^2. Fleets are not harmed if expenditure of Resource halts during their construction; the Weight of the Fleet is given by Sqrt(Resources/10).

19. Fleets can be broken down separately or combined, with Weights given by the Resource formula in 18. Breaking down or combining requires a small constant unit of Time.

20. A Fleet can traverse an Adjacent Pair in a small constant of Time as long as the Weight of the Pair on both stones exceeds the Weight of the Fleet.

21. It costs one unit of Resource per unit of Weight to move a Fleet 10 units of Distance. This cost must be prepaid in full before the Fleet can be moved.

22. A Fleet captures any stone when the Weight of the Fleet exceeds the Weight of the Adjacent stone. For purposes of Capture, the Weight of a Forge is equal to Sqrt(Resources/50).

Capitols and Alliances:

23. Each side must have a stone called a Capitol, which is by default at the Starting Point. A Capitol may be moved to any Node along any sequence of Red Edges.

24. Capture of a sides Capitol by an opposing Fleet transfers control of that sides Nodes and Pairs to the opposing side. Any Fleets in control of the side with the captured Capitol remain under their control, however.

25. For a game with n>2 sides, a side may declare or break an Alliance
with another side. Declaration of an Alliance begins at the point of
contact between the two sides, and spreads along all Red Edges in the Alliance in a small constant unit of Time. Along Black Edges, the declaration spreads at one unit of Distance per unit of Time. An Alliance cannot be declared between two sides not in contact.

26. Alliances are non-transitive. (If A is allied to B, and B is allied to C, A is not allied to C.)

27. Alliances may Acquire/Transport Resource as given in Rule 14a, and combine Fleets as given in Rule 19.

28. An Alliance may be broken by any side at any time. Breaking of an Alliance begins at the Capitol, and propagates as given in Rule 24. Any Combined stones (Fleets, Pairs, Capitols, Forges) immediately revert to control of the side with the greatest individual Weight (see rule 22).

Level 3:
The TopoGo Board:

1a. The Boardspace is defined by the metric of its topological space.

Legal Moves:

9a. The Boardspace metric replaces the Euclidean distance in determining Distance and the values of the Black and Red Edges.

10a. Each unit of Distance travelled requires a variable amount of Time depending upon the local metric of the Travel piece.

11a. The value of the Red Edge depends upon the local metric of the Travel piece.

12a. It costs a variable amount of Resource depending on the local metric to move the Travel piece ten units of Distance.

Level 4:
Information and Signal Propagation (the Affine rules):

29. All stones (Pairs, Nodes, Forges, Capitols, Fleets, and Resources) have an initial Information equal to their Weight.

30. A Signal can be broadcast from any stone at any time. The Information value of the Signal is equal to the cube root of the Information of the broadcasting stone. A Declaration is considered to be a broadcast Signal of Information value 0 originating from the Capitol.

31. A Signal can be narrowcast from an origin stone to a destination stone along the Black Edge between the two stones. The Information value of a narrowcast Signal is equal to Sqrt(Information of narrowcast stone).

32. All Signals start at their declared origin, and propagate through Black Edges at 1 unit of Distance per unit of time.

33. All Signals may propagate through Red Edges, if allowed by the controlling side, for a small constant unit of time per Red Edge.

34.The Information of any stone can be increased at a Resource cost equal to the square of its Fibonacci sequence of its new Information (e.g. Info 4 costs (1+2+3+4)^2 = 100)

35. A Signal, upon intersecting a stone, will capture the stone if its Information value exceeds either the Information or the Weight of that stone.

36. If a Signal fails to capture a stone, that stone may broadcast a counter-Signal of Information 0 which will grant immunity for any receiving stones against that particular Signal. (Note: to be effective, the counter-Signal must traverse at least one Red Edge in order to arrive before the Signal).

Level 5:
Reputedly played using Resource/Distance/Metrics obtained from detailed galactic maps.

Level 6:
Reputedly played using brane-theoretic landscapes.

Level 7:
Reputedly the real thing ....
 
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Development Notes
Text by Adam Getchell

Initially published on 29 December 2006.

 
 
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