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# Towards a new Paradigm of Board Games

"There is no reason that forces the game industry to restrict itself on a paradigm of games that is repetitive, addictive, regressive and is denying the right of the user to develop his/hers intellectual capabilities of creativity. It is a strategic decision of the game industry to stupidify its costumers." ThinkArt Lab To introduce the paradigm of morphic (board) games, I start with two simple questions. Why are classical games producing regressive addiction and boredom? What are the differences between classical and morphic games? My first answers to the questions might be summarized as follows: Classical games are based on the perceptive acts of identification and separation of the elements of the game by the rules of the game. Morphic Games are involved into differentiations and structurations of interactive and reflectional patterns (morphograms) in complex constellations. For a more conceptual answer of the two questions I connect the proposed new kind of games to the theory and practice of morphogrammatics. What do I understand by morphogrammatics? Morphogrammaitics is a pre-semiotic theory of inscription. It is studying and formalizing the 'deep-structure' of semiotics. Mathematically, morphograms, as the fundamential patterns of morphogrammatics, are representations of Stirling numbers of the second kind. Formal semiotics consists of an sign repertoire and rules of maipulating its signs. This is established by a strict difference of operators and oprands (signs). In contrast, morphograms are playing a double role: they are involved in a chiastic interplay of patterns (operands) and rules (operators). This is in decisive conterast to identity-based semiotic systems that are based on atomic signs. Strings of signs are based on a set of signs with cardinality m and its potentiation (n): m^n. Hence for m=4 and n=4, there are exactly 4^4 = 256 different semiotic strings possible. But on a morphogrammatical level there are just exactly Sn(4,4) = 1+6+7+1=15 morphograms for m=n=4 possible. In this sense, those 15 morphograms are presenting the 'deep-structure' of the set of semiotic strings of length 4.

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Towards a new Paradigm of Board Games

Dr. phil Rudolf Kaehr

ThinkArt Lab Glasgow

ISSN 2041-4358

( work in progress, vs. 0.3.5, May/April 2014 )

MorphoBoard Games

Motivations for MorphoBoard Games

MorphoGames are well motivated by themselves. But there is no reason not to accept another motivation

too. MoprphoGames offer ideal approaches for a general understanding of morphogrammatics

and polycontextural logics that had been developed on a much more abstract level in the past.

MorphoGames offer a playful environment to learn and enjoy the essential structurations and transformation

rules of morphogrammatics.

Morphogrammatics is a theory and a system of pre-semiotic patterns and their transformations that

is fundamental for a study of the morphosphere.

Semiotic games, like Leibniz games are ‘localized’ on the level of inscription of the semiosphere. The

basic axiom of the semiosphere is based on perception: What you see is what it is (WYSIWYGapproach).

MorphoGames of Morphogramatics are ‘localized’ on the level of the morphosphere where the main

experience is based on cognition: What you see is not how it acts.

Polycontextural game theory had been developed in the context of an explication and formalization of

the concepts and stratagems of ‘interactionality’ and ‘reflectionality’, especially on the level of formal

systems (languages) and MAS (Multi-Agent Systems).

MorphoBoard Games are not to be confused with Morpho Games from other sources.

Mathematical Theories of Games

http : // www.wisdom.weizmann.ac.il/~fraenkel/Papers/GamesHandbk.pdf

MorphoBoard Game Definition

Board program

The parameters of the Game are easily changed.

The parameters are : Width, Heigth, Patterns (colors) and Randomness.

The initialization of the Board depends on the chosen values for the parameters.

Further changes of the rules are obvious.

This presentation of some very first results of the paradigm of MorphoGames are following closely

Yves Papegay’s contribution to classical Board Games programmed in Mathematica.

Yves Papegay, Exploring Board Game Strategies, A Recreational Application of GUIKit

The Mathematica Journal, Volume 10, Issue 2, 2006

http : // www.mathematica - journal.com/2006/09/exploring - board - game - strategies/

Aim of the paper

The aim of this paper is three-fold:

1. Reconstruction of the classical concepts, definitions and programs for Board Games as presented

by Papegay's Mathematica program.

2. Elaboration of some dynamics of the parameters of the program to learn the scope of the internal

possibility of the concept and possible extensions beyond the concept of classical Board Games.

3. Sketch of the paradigm of Morpho Games and first steps of developing procedures for it in

Mathematica.

The aim of this paper is three-fold:

2 MorphoBoardGames.nb

1. Reconstruction of the classical concepts, definitions and programs for Board Games as presented

by Papegay's Mathematica program.

2. Elaboration of some dynamics of the parameters of the program to learn the scope of the internal

possibility of the concept and possible extensions beyond the concept of classical Board Games.

3. Sketch of the paradigm of Morpho Games and first steps of developing procedures for it in

Mathematica.

Game

Game definition

"As quoted, the most important point when designing a board game implementation is to have a

proper idea of the board itself, which represents the complete status of the game at each play and

the rules that determine whether or not a play is legal." (Papegay)

“The game denotes a complete set of interactions between the program—or the physical support of

the game, whatever it is—and the player. Hence, it is a succession of three phases:

† an initialization phase when the game starts or restarts

† a playing phase (i.e., when the user is playing). This is the most common behavior of the game

and consists in a sequence of successive plays.

† a termination phase at the end of the game.”

Board

“The board represents not only the physical board but, by extension, the complete status of the

game at a given time. By definition, in a board game, this status is well defined by a mapping

between a two-dimensional set of locations and additional information (usually qualitative or

discrete) for each location.”

Board: Designing, Initializing, Interpretation

The design of the board follows decisions about its topology. Out of the multitude of possibilities,

classical Board Games are deciding for a ‘Euclidean’ topology with its Height and Width.

Interpretation

The category of interpretation for Board Games is reduced for classical Games to an Interpretation in

the modus of identity. Because identity is ubiquitous for the classical approach it is not necessary to

be specially mentioned. It is obvious that the elements of a Game, i.e. the value occupations on the

Board, are subsumed under the law of identity.

Definition of a Board

Here, the parameters of the board are set as follows:

Width=22;

Height=15;

Patterns= {...}.

In a further development, a menu will allow the user to set the values.

MorphoBoardGames.nb 3

Width = 22;

Height = 15;

Patterns =

8Item@v, Background Ø Green, Frame Ø TrueD,

Item@e, Background Ø Red, Frame Ø TrueD, Item@u, Background Ø Blue, Frame Ø TrueD,

Item@w, Background Ø Yellow, Frame Ø TrueD, Item@s, Background Ø Pink,

Frame Ø TrueD, Item@z, Background Ø Gray, Frame Ø TrueD Patterns@@xDDD

s s u s z w z u s u s v v w s s e z w v z s

w w w z z u e v v u u w z v w z z w z e v v

w e v e v s u e z e v z v z z u s s e e u w

w w w u v z u e s z u w u w z e w v u u z w

w s e w v s w v e z u w w e s z z u u s z s

z z v v u e v v z u v s s s z w w e v e u u

w e u z s u w w z z z v w e w s v z w z s z

w w e e u s u w v w e e z s v e v s s z e s

u z z s u w v u e z v e v v e w v z e u z u

z v s e z e w s e z z e s e e s v w e s v w

z v s e w w z u v w w z e u e e e s w w v s

w s z w z e z e v s w u v v s u e s v w s v

s z e v w z e z e z w s s s e w w e e w s v

u e z v w z u w u s z s u s z w s s e w e e

v u v v w e e v u w v u z w s v s e e v v u

Numeric labeling of the colored board

4 MorphoBoardGames.nb

Width = 22;

Height = 15;

Patterns =

8Item@1, Background Ø Green, Frame Ø TrueD, Item@2, Background Ø Red,

Frame Ø TrueD, Item@3, Background Ø Blue, Frame Ø TrueD,

Item@4, Background Ø Yellow, Frame Ø TrueD, Item@5, Background Ø Pink,

Frame Ø TrueD, Item@6, Background Ø Gray, Frame Ø TrueD Patterns@@xDDD

6 3 4 4 6 3 4 5 2 3 1 1 4 3 5 4 1 6 4 1 3 2

5 1 4 2 6 5 6 2 5 4 2 2 4 1 3 6 1 3 1 4 1 5

5 1 4 1 5 2 3 3 4 2 6 5 3 3 5 6 6 5 6 4 6 2

2 6 1 4 5 2 4 6 1 4 1 6 5 1 1 2 4 3 3 1 6 3

6 3 6 3 5 3 1 1 3 5 5 6 2 3 3 3 3 4 6 4 6 6

1 3 2 1 6 1 5 2 5 6 6 3 4 6 3 3 4 2 6 5 5 4

4 5 3 1 2 1 2 3 1 1 6 1 5 5 3 5 2 1 5 4 6 2

2 4 6 3 3 4 4 4 6 4 6 2 5 3 2 3 4 4 5 6 1 2

2 6 5 1 3 6 1 2 4 1 1 5 3 2 5 6 1 3 2 3 1 1

4 2 4 1 2 4 4 6 6 1 2 1 4 1 3 2 3 1 2 6 6 3

3 1 2 2 4 4 1 5 5 4 4 2 6 6 4 6 3 1 3 6 5 4

2 2 5 1 6 6 2 6 2 6 1 4 5 1 1 1 6 4 2 4 5 1

5 5 6 1 4 6 1 3 1 3 3 4 1 5 2 6 3 1 4 3 3 2

1 3 5 5 1 3 5 3 5 5 5 4 2 3 5 4 6 3 4 3 6 4

4 1 2 3 2 6 2 4 1 2 2 5 6 4 3 5 5 1 5 1 2 6

Board example of the presentation of MorphoGames

Needs@"GraphUtilities`"D

Boardinit@D :=

Patterns =

8Item@v, Background Ø Green, Frame Ø TrueD,

Item@e, Background Ø Red, Frame Ø TrueD, Item@u, Background Ø Blue, Frame Ø TrueD,

Item@w, Background Ø Yellow, Frame Ø TrueD, Item@s, Background Ø Pink,

Frame Ø TrueD, Item@z, Background Ø Gray, Frame Ø TrueD,

Item@l, Background Ø Cyan, Frame Ø TrueD, Item@m, Background Ø LightBlue,

Frame Ø TrueD, Item@n, Background Ø LightRed, Frame Ø TrueD

MorphoBoardGames.nb 5

w e s u s e l z v l l

s w z m m l w z w m n

e s w s s l l n m u n

n m m l e n s l s s z

v u e z v l l u w n e

v l w z v w m n l u e

l n l v v s n m z m s

v w z m m e u u v v e

s n l v w u l l m n w

Interpretation of a Board

The category of interpretation of a board is not necessarily a specific topic of a classical definition of a

game. Classical games are conceived as ruled by the identity during a play of its board (width,

height), modality (randomness, usw), elements (patterns) and rules.

This is in concordance with the definition of an elementary formal system (EFS) in the sence of Melvin

Fitting and Raymond Smullyan.

But there are other approaches to an interpretations of a board and its use available.

This proposal is distinguishing, at first, between Leibniz, Brownian, Mersennian and Stirling games.

Classical games are understood as Leibniz games.

Play

“By definition, a play is one step of the playing phase. At each play, the player has to select which

action, among the legal (valid) ones, to perform. “

For poly-Games, this decision function of selection is complemented with the election function that

decides what kind game (Leibniz, Stirling, etc.) shall hold for the next steps.

Rules

“The rules are the set of constraints that determine what can be played and how the status of the

game should be modified by a play.”

“How to Undo Remember we said our board representation needed to handle undo operations.There

are two possible methods :

(1) Keep a stack in which each stack item holds a whole board representation; to make a move push

it on the stack and to undo a move pop the stack.Probably this is too slow ...

(2) Keep a stack storing only the move itself together with enough extra information to undo the

move and restore all the information in the board position. E.g. in chess you would need to store the

identity of a captured piece (if any) and enough information to restore castling and en passant capturing

privileges.”

http://www.ics.uci.edu/~eppstein/180a/970408.html

MorphoGame interpretation of a Board

For a Stirling approach to Board Games, the fact that the concept of patterns,where the ordered

strings or morphograms of identity-free elements, are crucial, leads to the following elementary rules.

MorphoGame rules

Rules in colors

Rule1. Ê = ‡

Rule2. Ê Ê = ‡ ‡

Rule3. Ê ‡ = ‡ Ê

Rule4. Ê Ê Ê ¹≠ Ê Ê ‡ ¹≠ Ê ‡ Ê ¹≠ Ê ‡ ‡ ¹≠ Ê ‡ Á

Rule5. Ê Ê ¹≠ Ê.

Rule5 is resolved in metamorphic MorphoGames.

6 MorphoBoardGames.nb

Rule1. Ê = ‡

Rule2. Ê Ê = ‡ ‡

Rule3. Ê ‡ = ‡ Ê

Rule4. Ê Ê Ê ¹≠ Ê Ê ‡ ¹≠ Ê ‡ Ê ¹≠ Ê ‡ ‡ ¹≠ Ê ‡ Á

Rule5. Ê Ê ¹≠ Ê.

Rule5 is resolved in metamorphic MorphoGames.

Identification Rules

In contrast, the rules for identity-based, i.e. classical games, like Leibniz games, are given by the

following postulates :

Pos1. Ê ¹≠ ‡,

with the natural consequences of

Pos2. Ê Ê ¹≠ ‡ ‡

Pos3. Ê ‡ ¹≠ ‡ Ê, and

Pos4: Ê Ê ¹≠ Ê .

Wordings of morphogrammatic constellations

For a Stirlingian game with 3 elements, some typical situations occur.

1. Ê Ê Ê ª ‡ ‡ ‡ ª Á Á Á,

Ê Ê ‡ ª ‡ ‡ Ê ª Á Á Ê

etcetera

2. Ê Ê ‡ ª rev(Ê ‡ ‡) : reversion

3. Ê ‡ Ê ª rev( Ê ‡ Ê) : self-symmetry

Ê Ê Ê ª rev(Ê Ê Ê)|

Ê ‡ Á ª rev( Ê ‡ Á)

4. Rules for blanks within the MorphoBoard Game.

Elimination of blanks: blank|element1|blank|element2 ï element1|element2.

The rules for morphic patterns are defining the rule-set of the MorphoBoard Games.

Patterns of Rule4 for Sn (4, 4)

Ê Ê Ê Ê

Ê Ê Ê ‡

Ê Ê ‡ Ê

Ê Ê ‡ ‡

Ê ‡ Ê Ê

Ê ‡ Ê ‡

Ê ‡ ‡ Ê

Ê ‡ ‡ ‡

Ê ‡ ‡ Á

Ê ‡ Ê Á

Ê ‡ ‡ Á

Ê ‡ Á Ê

Ê ‡ Á ‡

Ê ‡ Á Á

Ê ‡ Á Ï

Meta-Rule for morphograms

Two arbitrary morphograms @mgD i and @mgD j , i, j œ Sn2, of the same complication (length) are morphogrammatically

equivalent if they don’t belong to the class of morphograms defined by the generalized

Rule4 with Sn(n,n).

Range of morphic board constellations

What ever happens on a MorphoBoard boils down to a system or structuration of morphograms ruled

numerically by the Stirling numbers of the second kind and their summations by the Bell numbers.

Therefore, the range of possible constellations is never infinite but restricted by the definition of the

morphic structurations, counted by the Stirling numbers of the second kind.

MorphoBoardGames.nb 7

What ever happens on a MorphoBoard boils down to a system or structuration of morphograms ruled

numerically by the Stirling numbers of the second kind and their summations by the Bell numbers.

Therefore, the range of possible constellations is never infinite but restricted by the definition of the

morphic structurations, counted by the Stirling numbers of the second kind.

This fact of finiteness of the morphic constellations enables interesting classifications and reduction

rules of the range of constellations.

Hence given a situation with 4 positions, there are by Sn2(4,4), just 15 morphogrammatic constellations

possible. Therefore, there are just 15x15 = 225 morphic confrontations between two morphograms

of length=4 possible.

For the phenotypical realization of a board and its constellations, the range is counted by

Example

For m=3, k=2

m!

Hm-kL! .

For the case of just 2 elements involved in a constellation of 3 positions, the abstract morphogram,

[Êıı], has a representation of 6 concrete realizations, i.e. the set {[abb], [acc], [baa], [bcc], [caa],

[cbb]}, all representing the morphogram [Êıı].

Hence, the number of confrontations between the phenotypical representations of [Êıı] and [Êıı]

on a MorphoBoard is 6x6=36.

Equivalence classes of games

The distinctions of different phenotypical representations of genotypical constellations, morphograms,

enable to define a theory of equivalence classes of MorphoGames.

MorphoGames that appear phenotypically as different may still be morphogrammatically equivalent.

This not to confuse with the trivial statement that what we can play in red we can also play in green.

MorphoGame strategies

At first, there are two simple strategies to consider:

1. Strategy: Elimination

Morphic sameness is eliminating the morphograms.

a.) vertical and horizontal

u w m n

z s v z

ï [ ]

b.) horizontal

n l e w

v e m w ï [ ]

c.) vertical

n l e

v e m ï [ ]

2. Strategy: Reduction

Morphic sameness is reducing the morphograms up to one morphogram.

Rules have to specify which morphic representation of the reduction owith the second strategy

survives.

a.) vertical and horizontal

u w m n

z s v z

ï

m

v

n

z

b.) horizontal

n l e w

v e m w ï v e m w

c.) vertical

n l e

v e m ï e m

8 MorphoBoardGames.nb

n l e

v e m ï e m

Explanation of the elinination rules

Rule1: Ê = ‡

If there are just two elements on the board as neighbors available, then they get eliminated independently

of being the same or different. Also the blanks between the elements are eliminated.

Ñ Ñ Ñ Ñ Ñ Ñ

Ñ v Ñ Ñ l Ñ

ï É

v Ñ Ñ l

Ñ Ñ Ñ Ñ

ï v l ï [ ] : horizontal

v

l

ï@D : vertical (for final steps)

Rule2: Ê Ê = ‡ ‡

The same as for Rule1 with two or more elements.

v Ñ l

v Ñ l

ï@D : vertical

v Ñ v

l Ñ l

ï@D : horizontal

Rule3: Ê ‡ = ‡ Ê

m

n

n

m

ï [ ] : horizontal+vertical

Iteration of Rule3

l

m

n

u

n

m

ï [ ] : horizontal+vertical

n

w

n

l

z

l

ï [ ] : horizontal+vertical

Rule4: Ê Ê ‡ ¹≠ Ê ‡ Ê ¹≠ Ê ‡ ‡ ¹≠ Ê ‡ Á

This rule marks the difference between patterns of elements. It hold in both directions: vertically and

horizontally.

Following Rule4: Ê ‡ Á ¹≠ Ê ‡ Ê , the two patterns

therefore not eliminable.

MorphoPalindromes

v

w

e

and

l

s

l

are morphogrammatically different and

A higher level of sophistication is achieved with the strategy to detect not just morphic equivalences

but morphic palindromes.

MorphoBoardGames.nb 9

Morphic palindromes are a subclass of morphograms. The detection of Morphic palindromes is demanding

for a new abstraction of detection and separation.

Therefore, palindromic games are of a higher gaming level than the basic morphogrammatic games.

Two arbitrary morphograms @mgD i and @mgD j , i, j œ Sn2, of the same complication (length) are morphogrammatically

equivalent if they are morpho-palindromic.

There are three main categories to distingush for a detection of morphic palindromes.

Case one: Head and body are morphogrammaticall equivalent

The MorphoGame related strategy to detect and eliminat patterns is based on the presumption that

the head and the body, defined by repetition, reversion and accretion, are of the same length.

Odd palindromes, i.e. morphogrames of length 2n+1, entail a medium pattern between the two

parts, head and body. For even palindromes, the medium part is empty.

Even palindromes

Therefore, the head and the body of the palindrome shall be ‘parallelized’ horizontally in the game.

If two patterns are building together a morphic palindrome then they shall be eliminated.

1. Repetition

2. Reversion

3. Accretion

Iterability scheme for @1, 2, 2, 3D ; 14

@1, 2, 2, 3D

á ¯ ä

reversion accretion repetition

@2, 3, 3, 1D

@3, 2, 2, 1D

@3, 4, 4, 1D

@4, 2, 2, 1D

@4, 5, 5, 1D

@2, 3, 3, 4D

@3, 1, 1, 2D

@3, 2, 2, 4D

@3, 4, 4, 5D

@4, 1, 1, 2D

@4, 2, 2, 5D

@4, 5, 5, 6D

@1, 2, 2, 3D

@1, 4, 4, 3D

- [1,2,2,3] = tnf[3,4,4,1];

val it = true : bool

- [1,2,2,3,1,4,4,3] = [1,2,2,3,2,3,3,1];

val it = false : bool

Example

The two neighbor patterns [ Ê ‡ Ê Á ‡ Á ] and [ Á ‡ Á Ê ‡ Ê ] are morphic palindromes, hence they

are eliminable.

In this case, the two patterns are also just morphogrammatically equivalent.

[1,2,1,3,2,3] = mg [3,2,3,1,2,1]

[1,2,2,3,3,4,4,1] = mg [1,4,4,3,3,2,2,1]

-ispalindrome@1, 2, 2, 3, 3, 4, 4, 1D;

val it = true : bool

Odd palindromes

Serial case

10 MorphoBoardGames.nb

n

n

z

e

e

,

l

l

n

w

w

,

l

s

l

w

l

s

l

: vertical

e s u s e : horizontal

If @mgD is a vertical or horizontal palindrome it is eliminable :

@mgD œ PAl : @mgD ï @D

Counter example

m

z

z

z

l

l

v

w

ï not-[]

Examples

-ispalindrome[1, 2, 1, 3, 1, 2, 1];

val it = true : bool

-ispalindrome[1, 2, 3, 2, 1];

val it = true : bool

-ispalindrome[1, 1, 2, 3, 3];

val it = true : bool

Case two: Head and body are morphogrammatically different

A more intriguing case is given if the body and the head of two even palindromes are morphogrammatically

different.

Trivially, the morphogram [1,1,1,1] is a palindrome. The same holds for the morphogram [1,2,1,2].

But both are morphogrammatically different, [1,1,1,1] mg ¹≠ [1,2,1,2].

Hence, the new abstraction is ruled by the property of being a palindrome.

Both morphograms of the example are palindromes of the same length, hence they can be eliminated.

In contrast, the morphogram [1,1,1,2] is not a palindrome. Hence, the comparison of [1,1,1,2] and

[1,1,1,1] is not eliminative.

@mgD mg 1 ¹≠ @mgD 2 Ô @mgD 1 ,@mgD 2 œ PAl : @mgD 1 || @mgD 2 ï []

Example

horizontal palindromes:

l n l

v w z

s n l

ï l n l œ Pal, v w z

s n l œ Pal,

l n l ¹≠ mg s n l , v w z ï []

Program to detect morphic palindromes

Bisymmetric e/v-version

BiSymTest[x] =

TabView[

Grid /@

{x ,

"Palindrome" -> SameQ[

Grid[

Reverse[First[Grid[Reverse /@

x]]]] ,

Grid[

Reverse /@ First[Grid[Reverse[

x]]]],

Grid[Transpose[x]]],

Clear[x]

BiSymTest[x] =

TabView[

Grid /@

{x ,

"Palindrome" -> SameQ[

Grid[

Reverse[First[Grid[Reverse /@

x]]]] ,

Grid[

Reverse /@ First[Grid[Reverse[

x]]]],

Grid[Transpose[x]]],

Clear[x]

}]

MorphoBoardGames.nb 11

Wee MorphoPalindrome Checker Palin (4, 2)

« Å▸ ¡

v

e v

e v e

List Version

PalindromeQ[i_String] := StringReverse[i] == i

MorphoPalindromeQ[i_String] := ReLabel[StringReverse[i]] == i

Example for lists

@1,2,3,1D

@1, 1, 2, 3, 1, 1D rule1 @1, 1, 2, 3, 1, 1D

@4, 1, 2, 3, 1, 4D rule3 @1, 2, 3, 4, 2, 1D

@4, 1, 2, 3, 1, 5D rule4 @1, 2, 3, 4, 2, 5D

@2, 1, 2, 3, 1, 3D rule1 @1, 2, 1, 3, 2, 3D

@3, 1, 2, 3, 1, 2D rule2 @1, 2, 3, 1, 2, 3D

The MorphoBoard configuration

The MorphoBoard configuration is defined by the board measures, the number of elements and a

random function that maps the elements onto the Board.

The MorphoTransition rules

The Transition mappings are considering different modi of parallelism of the reduction rules.

Termination and Evaluation of the MorphoPlay

The scores of the play are determined by the numbers of steps and the number of not resolved

patterns of the reduction game.

Examples for a manual play

The examples show the strategy how to achieve the results of:

First play: 18 steps, with 2 patterns left.

Second play: 15 steps, with 3 patterns left.

12 MorphoBoardGames.nb

Classical games

Obviously, classical board games are ruled by classical two-valued logic. This is abbreviated by the

identity rules of the classical game: Pos1: Ê ¹≠ ‡. An element of a game has a unique value: it is or it

is not on the board. Logically speaking, it is true (false) that an element occupies a place on the

board.

For the game, what counts are sequences of the identical and unique elements. Therefore, the classical

two-valued logic is ruling the rationality of the game. A blank on a classical board has no logical

significance. It just marks the elimination of a value.

Hence, the rules of a classical game are concerning the status of the sequences of connected identical

elements on the board.

Classical sequences of the same elements might be ordered horizontally and vertically, but there are

also possibilities to connect the elements diagonally.

u

v

A diagonal connection is illustrated by the example

m

s

u

u

. The diagonal rule reduces the blue diago-

n

e

nal sequence

u

Ñ

Ñ

v

u

u

to the figure

Ò

m

s

n

v

Ò

e

.

All three modi of connection, horizontal, vertical and diagonal, are gap-free.

This reflection on the logic of games shouldn’t be confused with the genre of Logic Games.

MorphoGames

Things are different for MorphoGames. What counts for the game are not the elements and their

values as such but the differences between the elements that are localized on the board. Because of

this differential definition of the states of a MorphoGame, not only the involved and localized elements

are of importance but the gaps, represented by the blanks, too.

To put it in a abbreviation, the logic of MorphoGames is not two-valued but a combination of at least

two 2-valued logics and the logics between the two 2-valued logics covering the logical status of the

gaps inscribed as blanks.

Hence, the rules of MorphoGames are concerning the status of the sequences of horizontally and

vertically connected complex differences and their gaps on the board.

For example:

w Ñ blanks Ñ m

w Ñ blanks Ñ m

ï

w w

m

m

ï Ñ Ñ

: vertical + horizontal move and

elimination.

Desiderata

The MorphoBoard Games are not yet programmed.

This note gives just the general concept with its rules and the program for the Board constellations.

The rules have to be applied by a human player. A programmed version will hopefully follow soon.

Board configuration one

n z l v l s z s l z n

m e z m w e n m e z w

n l n u u l u l s s l

MorphoBoardGames.nb 13

n u z l n l n v n w v

v w u w m n m z z z m

m w z s v z m n s e e

w l e l u e l u v z m

w s v l u v z m u w l

m n w n v z n s u v z

n l e w l v m n e w v

v e m w s l l w v l m

Decompositions of board one

One possible solution of the Board is given by the following 18 steps of the Game with 2 patterns left

(omitting any moves of the squares).

This demonstration of the MorphoGame with ‘board one’ is quite static and relies only on the existing

configurations of the board. There are also no ‘diagonal’ interpretations involved.

Therefore, the action of moving elements (squares) on the board to produce new constellations is not

yet included in this description.

The rules are horizontally and vertically applied to the patterns.

1 Ñ Ñ Ñ Ñ Ñ Ñ Ñ 1 6 6

1 Ñ Ñ Ñ Ñ Ñ Ñ Ñ 1 6 6

17 Ñ Ñ Ñ Ñ 17 5 5 13 13 l

17 Ñ Ñ Ñ Ñ 17 5 5 13 13 v

12 12 2 Ñ Ñ 2 18 8 Ñ 8 m

12 Ñ 2 Ñ Ñ 2 18 8 Ñ 8 e

18 10 10 11 11 15 Ñ 15 v 9 9

18 10 10 11 11 15 Ñ 15 u Ñ Ñ

4 Ñ Ñ 4 14 Ñ 14 s u 9 9

Ñ Ñ Ñ Ñ Ñ Ñ Ñ 3 Ñ Ñ 3

4 Ñ Ñ 4 14 Ñ 14 3 Ñ Ñ 3

The number at the corners of a deleted pattern (configuration) indicates the number of the steps of

the play and its empty frame.

Vertical application of Rule3

1 = n z l v l s z s l

m e z m w e n m e

ï Ñ Ñ ,

with vertical application of Rule3 : Ê ‡ = ‡ Ê

2 = u w m n

z s v z

ï Ñ Ñ

, with Rule3 : Ê ‡ = ‡ Ê,

3 = n e w w

w v l l

:

n

w = e v ï w l

= w l

ï Ñ Ñ : vertical,

with Rule3 : Ê ‡ = ‡ Ê for

n

w = e v ,

and Rule3 as Identity Rule for

w

l

= w l

,

14 MorphoBoardGames.nb

w

l

w

l

ï Ñ Ñ

: vertical + horizontal.

4 =

m n w n

n l e w

v e m w

: horizontal is not valid becaus of double "n",

and not valid for vertical because of double "w" :

m n w n ¹≠ n l e w

v e m w : vertical,

Hence, what holds is :

m n w

n l e

v e m

vertically + horizontally and n l e w

v e m w horizontally.

5 = u l

n v

ï@D : vertical + horizontal.

6 = z n

z w

ï@D : horizontal.

7 = l n

u z

ï@D : vertical + horizontal.

8 = z z z

n s e : vertical

z

m

9 =

w

l

: vertical + horizontal

v

z

10 = l e

s v

: vertical + horizontal

11 = l u

l u

: vertical + horizontal

12 = v w

m w : horizontal

13 = s s

n w : vertical

14 =

v z n

l v m

s l l

ï@D : vertical

15 = e l u

v z m

ï@D : vertical + horizontal

16 = u u

l n : vertical

17 =

n 7 7 16 16 l

n 7 7 16 16 l

: horizontal move, neglecting blanks

MorphoBoardGames.nb 15

n 7 7 16 16 l

n 7 7 16 16 l

ï n l

n l

ï @D : horizontal move, removing blanks

18 =

12 12 2 Ñ Ñ 2 m

12 Ñ 2 Ñ Ñ 2 m

w 10 10 11 15 Ñ Ñ

w 10 10 11 15 Ñ Ñ

4 Ñ Ñ 4 14 Ñ 14

Ñ Ñ Ñ Ñ Ñ Ñ Ñ

4 Ñ Ñ 4 14 Ñ 14

ï w Ñ Ñ Ñ Ñ Ñ m

w Ñ Ñ 4 14 Ñ m ï w w

m

m

ï @D : vertical + horizontal move

According to the rules, this final constellation cannot be further resolved .

Hence the play ends after 18 steps with 2 patterns unsolved. The score is (18, 2).

5 13 13 l

5 13 l

5 13 13 v

8 Ñ 8 m

8 Ñ 8 e

15 v 9 9

15 u Ñ Ñ

ï

5 13 v

8 Ñ m

8 Ñ e

15 v Ñ

15 u Ñ

ï

Ñ

Ñ

s

v

u

u

l

v

m

e

.

s u 9 9

s u Ñ

Test of (4.) with ReLabel

m n w n

n l e w

v e m w

88m, n, w, n

16 MorphoBoardGames.nb

Ñ

Ñ

s

v

u

u

l

v

m

e

ï

Ñ

Ñ

s

v

Ñ

Ñ

l

v

m

e

. Following this path, further reductions are possible:

Ñ

Ñ

s

v

Ñ

Ñ

l

v

m

e

ï Ò, with the figure

Ñ

Ñ

s

v

Ñ

Ñ

plus the reduction of the blanks and finally the equivalence

of s and v by Rule1 we get the terminal state Ò.

l

Ñ

And by a self-application of Rule1 onto the figure

v

m

it reduces it via

Ñ

Ñ

to the terminal state Ò.

e

Ñ

As a result we have the situation that by applying the morpho-rules consequently what includes its

self-application, all MorphoGames are re-deducible to a terminal state Ò or ‡. That is, all MorphoGames

terminate in the final state Ò or ‡.

But this approach makes sense only for the final steps of the difference-theoretic concept of the

MorphoGame.

A second run

A second approach to the previous configuration of the Board for the run one is given by the following

15 steps with 3 patterns left as shown by the resulting constellation of the board by run two.

n z l v l s z s l z n

m e z m w e n m e z w

n l n u u l u l s s l

n u z l n l n v n w v

v w u w m n m z z z m

m w z s v z m n s e e

w l e l u e l u v z m

w s v l u v z m u w l

m n w n v z n s u v z

n l e w l v m n e w v

v e m w s l l w v l m

15 7 Ñ Ñ Ñ Ñ Ñ Ñ 7 14 15

15 7 Ñ Ñ Ñ Ñ Ñ Ñ 7 14 15

9 Ñ Ñ Ñ Ñ 9 u 11 11 6 6

9 Ñ Ñ Ñ Ñ 9 n Ñ Ñ Ñ Ñ

13 12 10 Ñ Ñ 10 12 11 11 Ñ Ñ

13 12 10 Ñ Ñ 10 12 n s Ñ Ñ

4 Ñ Ñ Ñ 4 5 Ñ 5 v Ñ Ñ

4 Ñ Ñ Ñ 4 Ñ Ñ Ñ 14 Ñ Ñ

13 3 3 13 v 5 Ñ 5 14 6 6

1 Ñ 1 Ñ 2 Ñ Ñ Ñ Ñ Ñ 2

1 Ñ 1 13 2 Ñ Ñ Ñ Ñ Ñ 2

1. = n l e

v e m

ï @D : vertical

2.

l v m n e w v

s l l w v l m

ï @D : vertical

In contrast, the horizontal interpretation of H2.L doesn' t hold.

88l, v, m, n, e, w, v

18 MorphoBoardGames.nb

v

m

n

v

n

13. =

4

4

w

w

ï

m

m

w

w

ï @D

m

14. = z z

u

u

ï @D

15. = n m

n

w

ï @D

This final constellation cannot be resolved according the rules.

Hence, the score is (15,3).

Ñ 9 u 11 11

Ñ 9 n Ñ Ñ

Ñ 10 12 11 11

Ñ 10 12 n s

4 5 Ñ 5 v

4 Ñ Ñ Ñ 14

ï

v

u

n

n

Ñ

s

v

.

v 5 Ñ 5 14

2 Ñ Ñ Ñ Ñ

2 Ñ Ñ Ñ Ñ

Self-application of rules

A further reduction is possible with the idea of a self-application of the difference rules. Again, this

step makes sense only after the difference-oriented run is exhausted. Applied from the beginning

would ruin the game.

v

u

n

n

Ñ

s

v

ï

v

n

Ñ

s

v

ï v n ï

Strategies for morphograms

The two examples show clearly the strategy how to detect and separate morphogrmmatically similar

patterns and how to eliminate them.

First, a pattern has a clear frame which is defined by its environment.

Hence two or more patterns that are similar must have the same horizontal and vertical frame. The

frames are defined by their local Width and Height.

A frame is separated, horizontally and vertically, from its environment by different other not overlapping

patterns.

board

1.

z z z m

n l e

v e m

w

w

ï

z z z m

n l e

v e m

w

w

: part of the board

MorphoBoardGames.nb 19

2.

z z z m

n l e

v e m

w

w

horizontal z z z m ëvertical

w

w

environments of the pattern

n l e

v e m .

3.

n l e

v e m :

horizontally separated pattern by environments w w and z z z m .

4.

n l e

v e m ï n v = l e = e m ¹≠ w w

: vertically, reduction by Rule3, with

environments w w and z z z m .

n

v = l e = e m ï @D.

5. n l e w = v e m w : horizontal reduction by Rule3, including w .

6. Horizontal strategy with additional separation criteria

board

w s v l u v z m u w l

m n w n v z n s u v z

n l e w l v m n e w v

v e m w s l l w v l m

board

n l e w l v m n e w v

v e m w s l l w v l m

n l e w

v e m w

l v m n e w v

s l l w v l m

n l e w

v e m w ; l

s ; v m n e w v

l l w v l m

horizontal separation of

n l e w

v e m w by l s .

Program-assisted recognition of patterns with ReLabel

Program-assisted recognition of patterns

The cognitive training necessary to play MorphoGames might be supported by some simple but

helpful programs. They might be implemented as tools into the game.

20 MorphoBoardGames.nb

ReLabel@L_ListD := L ê.

Map@Ò@@1DD Ø Ò@@2DD &, Transpose@8DeleteDuplicates@LD, Range@Length@Union@LDDD

MorphoBoardGames.nb 21

n l e w = v e m w ï @D

ReLabel[{n, l, e, w, l, v, m}] ¹≠ ReLabel[{v, e, m, w, s, l, l}]

{1, 2, 3, 4, 2, 5, 6} ¹≠ {1, 2, 3, 4, 5, 6, 6},

Therefore these patterns cannot be eliminated by the existing rules.

n l e w l v m ¹≠

v e m w s l l

A further separation beyond n l e w = v e m w implying

n l e w l v m ¹≠ v e m w s l l

or more elements, stops with the double occurrence of the element "l".

Also the identity of the elements doesn't count, their order is of relevance.

Numeric presentation of a MorphoBoard

2 5 4 2 1 5 3 6 5 3 3

2 4 6 3 1 5 2 1 2 5 5

4 1 2 6 2 5 5 2 6 6 6

1 3 6 5 3 4 6 6 6 3 1

2 4 6 2 3 4 4 2 3 6 6

2 2 5 4 6 1 5 4 4 6 6

4 1 6 5 2 5 5 6 5 5 3

2 4 5 2 6 6 2 1 5 1 1

6 2 1 2 3 2 5 3 5 2 2

1 6 3 4 4 4 6 6 4 1 6

5 3 6 1 5 3 1 2 1 3 2

1 1

1

2 2 is not accepted because it has a prolongation of 1 in 2,

1 6

1

1

hence is not separated or the same as the neighbor 2

6

.

ReLabel@82, 5, 4, 2, 1, 5, 3, 6, 5, 3, 3

22 MorphoBoardGames.nb

Definition of the environments of patterns

Classical situation

Positions of colored tiles are given by a succession of calls to the pseudorandom number generator.

(Papegay )

Visualizing the Board

We are now able to initialize the game given the vlues for its Heights, Width and Patterns (colors).

Choosing some colors, we can define the function View for a nicer display of the board. (ibd)

Transition function

“To deal with corner and boundary situations, we define the BoardValue function to access the values

of the board. It returns -1 if the arguments for location are outside the bounds of the board. This

allows us to ignore the boundaries of the board when considering the neighbors of a location.” (ibd)

Full Mathematica Program for the Board Game MHaki by Yves Papegay

Needs@"GraphUtilities`"D

Needs@"GUIKit`"D

NewGame@D ê; Not@NeedRandomnessD := HBoard = InitBoard@D; InitPlay@D;L

NewGame@s_: SeedD := HSeed = s; SeedRandom@sD; Board = InitBoard@D; InitPlay@D;L

NewGame@s_: SeedD := HSeed = 7; SeedRandom@7D; Board = InitBoard@D; InitPlay@D;L

NewGame@"new"D := HSeedRandom@D; Seed = Random@Integer, 31 991D;

SeedRandom@SeedD; Board = InitBoard@D; InitPlay@D;L

InitBoard@D := Table@InitPosition@i, jD, 8i, Height

MorphoBoardGames.nb 23

Function@x, BoardValue@xD ã BoardValue@First@yDDDD &, yDDDD, 8p

24 MorphoBoardGames.nb

NBPlayGame@"new"D

With a Parameter Interface (to do)

H* Frame for Board *L

Width = 22;

Height = 11;

Patterns = Range@6D;

NeedRandomness = True

H* Colors *L

HMakiColors = 8Green, Blue, Red, Yellow, White, Pink<

H* HMakiColors@@x+vDD *L

View@D := Show@Graphics@Raster@Transpose@Map@Reverse, Transpose@BoardDDDD ê.

x_Integer :> HMakiColors@@x + 2DDDD

H* FaceNeighbours@p_D *L

FaceNeighbours@p_D :=

Map@Plus@p, ÒD &, 880, 1

MorphoBoardGames.nb 25

Width = 20; Height = 15; Patterns = Range@5D; NeedRandomness = True; PixelSize = 12;

Clear@ScoreD

InitScore@D := Score = 80, 0, Width Height<

UpdateScore@D := Module@8n = Width Height - Length@Select@Flatten@BoardD, Ò == 0 &DD

26 MorphoBoardGames.nb

In the simplest case of a MorphoGame it starts with a parallelism by the tuples of similar neighboring

values.

BoardValue@posD = 888i, j

MorphoBoardGames.nb 27

That is:

a = mg

Hence, in ML:

b iff ReLabel(a) == ReLabel(b).

fun teq a b = (tnf a = tnf b);

And more in the sense of differences of E=equal and N=nonequal:

fun teq a b = (ENstructure a) = (ENstructure b);

Difference ε/n-notation of morphograms

The fact that the presentation of the morphograms by specific elements is arbitrary has to be considered

as crucial. Therefore, not the elements are determining the morphic patterns but the differences

between the elements.

This is well depicted for the example [Ê ‡ Ê].

Ê ‡ Ê : morphogram

\ê \ê

1. n n 3. : ε ê n - structure

2. ε

A useful notation is given with the matrix of the ε/n-structures.

Ê Ê Ê Ê Ê ‡ Ê ‡ Ê Ê ‡ ‡ Ê ‡ Á

ε -

ε

ε

ε -

n

n

n -

ε

n

n -

n

ε

n -

n

n

Example for the ε/n-structures of

z

w

v

m

l

z

z

w

v

= n -

n n , m

l

z

= n -

z

n n , hence w

v

mg

=

m l

z

.

With ReLabel:

ReLabel@L_ListD := L ê.

Map@Ò@@1DD Ø Ò@@2DD &, Transpose@8DeleteDuplicates@LD, Range@Length@Union@LDDD

28 MorphoBoardGames.nb

iê j 1 2 3 4 5 6 7

1 e z m w e n m

2 l n u u l u l

3 u z 8i, j< 8i, j + 1< l n v

4 w u 8i + 1, j< 8i + 1, j + 1< n m z

5 w z s v z m n

6 l e l u e l u

7 s v l u v z m

iê j 1 2 3 4 5 6

1 e z m blank w m

2 l n u Ñ u l

3 u z 8i, j< blank 8i, j + n< v

4 w u 8i + 1, j< blank 8i + 1, j + n< z

5 w z s Ñ v n

6 l e l Ñ u u

7 s v l blank u m

BoardValueMorpho@8l1_, c1_, l2_, c2_

MorphoBoardGames.nb 29

Bval@BoardValueMorpho@Hi, j - 1L, Hi + 1, j - 1LDD

mg

=

val@BoardValueMorpho@Hi, j L, Hi + 1, j LDD,

mg

=

val@BoardValueMorpho@Hi, j + 1L, Hi + 1, j + 1LDDF

mg

¹≠

val@BoardValueMorpho@Hi, j + 2L, Hi + 1, j + 2LDD

short :

8i, j + 1<

8i + 1, j + 1<

8i, j + 2<

8i + 1, j + 2<

val@BoardValueMorpho@Hi, j + 1L, Hi + 1, j + 1LDD

mg

¹≠

val@BoardValueMorpho@Hi, j + 2L, Hi + 1, j + 2LDD

Horizontal environment, upper

8i - 1, j - 1< 8i - 1, j< 8i - 1, j + 1<

8i, j -1< 8i, j< 8i, j + 1<

val@BoardValueMorpho@Hi, j - 1L, Hi , jL, Hi, j + 1LDD

¹≠ mg

val@BoardValueMorpho@Hi - 1, j - 1L, Hi - 1, jL, Hi - 1, j + 1LDD

MorphoBoard Games in the Framework of Graphematics

The distinction between classical and morphogrammatic board interpretations and rules motivates to

involve additionally to the Stirling games of MorphoGames a group of other graphematic systems

(structurations).

An elementary group of not mixed approaches or paradigms is listed as: Stirling, Mersenne, Brown

and Leibniz systems.

Stirling structurations are the domain of morphogrammatics and therefore of MorphoGames.

Leibniz structurations are the domain of identity based systems of abstract logical, arithmetic and

semiotic calculi, therefore of classical board games.

Brownian and Mersennian structurations are two non-orthodox systems that are not genuinely morphogrammatic.

The additional structurations are becoming relevant for game theory and games if they are set into a

interactional context that involves parallelism.

Brownian games are commutative, while Mersennian games are iteration invariant.

Leibniz games are special cases of such a parallel setting: they collapse with their neighbor systems.

30 MorphoBoardGames.nb

To define a reasonable board game in the framework of Brownian, Mersennian and Stirling structurations,

a simple parallelism of the path (steps) of the game is of necessity.

Tabularity vs. linearity

With this, the emphasis on the tabularity of the game, with its board and its planar rules for the

distinction of patterns, environments and successions, a new approach, compared to the previous

studies of graphematic calculi, is promoted.

First, common to all games is that they need a workspace (board) on which parts are selected for

manipulation by the rules typical for the chosen rationality (kind) of the game. The kind of the game

defines the characteristics of the elements (operands) for the applied rules (operators).

Second, a reasonable game has a beginning and an end. Hence it is ruled by the initial and final

conditions of the game.

The mentioned categories are stable. There is no interplay between the categories. Between board

and part, elements and rules their is no interplay. A board is a board and a part of a board is not the

board.

Metamorphic games where the basic categories are involved in complex interplays are possible only

as a multitude of discontextural games. The shall be called poly-Games.

General frameworks

Systems

Leibniz

a a b b

a b a b

Stirling turn

Mersenne

Stirling

á ä

a a b

a b a

ä á

a a

a b

a a b

a b b

Brown

Pascal

á ä

Brown ¬ Stirling Ø Mersenne

ä ¯ á

Leibniz

types\values aa ab ba bb combinatorics

Leibniz aa ab ba bb m n

Mersenne aa ab ba - 2 n - 1

Brown aa ab - bb J n + m- 1

N

Stirling aa ab - - ⁄ M

k=1

S Hn, kL

n

http://memristors.memristics.com/Handouts/Kindergarten%20and%20Differences-Handouts.html

http://memristors.memristics.com/Kindergarten%20and%20Differences/Kindergarten%20and%20Diff

erences.html

Leibniz Games (Identification)

Graaphematic identity rules

‡ ‡ ¹≠ ‡

‡ ¹≠ Á

Wording

Two elements are not equal one element.

Different elements are different and not equal.

MorphoBoardGames.nb 31

Board frames

" i, j œ Board HHeigth i, Width jL :

val@BoardValue@Hi, jLDD id

=

H* vertical *L

val@BoardValue@Hi, jLDD = id

val@BoardValue@Hi + 1, jLDD,

valBBoardValue@Hi, j + 1LD, H* horizontal *L

val@BoardValue@Hi, jLDD = id

valBBoardValue@Hi, j + 1L, Hi + 1, jL, Hi + 1, j + 1LD,

with = id œ Pos1

Leibnizian game rules

r1:

n

n

n

fl [] : vertical

r2: n n n fl [] : horizontal

r1.2:

Ñ n Ñ

n n n

Ñ n Ñ

fl [] : mixed

Grid@

88Item@" ", Background -> White, Frame -> TrueD,

Item@n, Background -> Blue, Frame -> TrueD,

Item@" ", Background -> White, Frame -> TrueD Blue, Frame -> TrueD,

Item@n, Background -> Blue, Frame -> TrueD,

Item@n, Background -> Blue, Frame -> TrueD White, Frame -> TrueD,

Item@n, Background -> Blue, Frame -> TrueD,

Item@" ", Background -> White, Frame -> TrueD

32 MorphoBoardGames.nb

n l n u u l u l s s l

n u z l n l n v n w v

v w u w m n m z z z m

m w z s v z m n s e e

w l e l u e l u v z m

w s v l u v z m u w l

m n w n v z n s u v z

n l e w l v m n e w v

v e m w s l l w v l m

Mersenne Games (Differentiation)

The basic graphematic rules for the Mersenne differentiation calculus

Á ‡ ¹≠ ‡ Á

Á Á = ‡ ‡

The basic rules of the calculus of differentiations

Rule 1. () () = Ø

Rule 2. (()) = ()

3. Substitution rules

Wording

Rule1: A differentiation between 2 differentiations is an absence of a differentiation.

Rule2: A differentiation of a differentiation is a differentiation.

In colors

Rule1. ‡ ‡ = Ø

Rule2. ‡ = ‡

Board frames

" i, j œ Board HHeigth i, Width jL :

val@BoardValueMersenne@HHi, jL, Hi + 1, jLLDD

Mers

=

val@BoardValueMersenne@HHi, j + 1L, Hi + 1, j + 1LLDD,

with Mers = œ Rule1, Rule2

Mersennian game rules

Rule1 : Vertical HserialL

z

z

ï @D : Rule1,

Applications of Rule1

l

u

l u fl l l

fl @D

Rules2: Horizontal (parallel)

z z ï z : Rule2

Applications of Rule2

z z z ï z : Rule2

MorphoBoardGames.nb 33

e z w

e z w

s s l

ï

Ñ s l

: Rule2, environment

n w v

n w v

First steps of a run for a Mersennian Game

n z l v l s z s l z n

m e z m w e n m e z w

n l n u u l u l s s l

n u z l n l n v n w v

v w u w m n m z z z m

m w z s v z m n s e e

w l e l u e l u v z m

w s v l u v z m u w l

m n w n v z n s u v z

n l e w l v m n e w v

v e m w s l l w v l m

n z l v l s z s l Ñ n

m e z m w e n m e Ñ w

Ñ l n u u Ñ u l s s l

Ñ u z l n Ñ n v n w v

v Ñ u w m n Ñ z z z m

m Ñ z s v z Ñ n s e Ñ

Ñ l e Ñ Ñ e l u v z m

Ñ s v Ñ Ñ v z m Ñ w l

m n w n v z n s Ñ v z

n l e Ñ l v m n e w v

v e m Ñ s l l w v l m

Vertical

Rule1 :

l

l

,

u n

u , n

fl @D

Rule1 :

m

m ,

w z

w , z

fl @D

Horizontal, Rule2

e e fl e ,

s s , l l , u u ,

z z z

Brownian Games (Distinction)

The basic graphematic rules for the Brownian distinction calculus

Á ‡ = ‡ Á

Á Á ¹≠ ‡ ‡

Basic rules for the Brownian distinction calculus based on the graphematic rules.

Rule 1. () () = ()

Rule 2. (()) = Ø

3. Substitution rules

34 MorphoBoardGames.nb

Rule 1. () () = ()

Rule 2. (()) = Ø

3. Substitution rules

Wording

Rule1: A distinction of 2 distinctions is a distinction.

Rule2: A distinction of a distinction is no distinction.

In colors

Rule1. ‡ ‡ = ‡

Rule2.

‡ = Ø

Board frames

" i, j œ Board HHeigth i, Width jL :

val@BoardValueBrown@HHi, jL, Hi + 1, jLLDD

Brown

=

val@BoardValueBrown@HHi, j + 1L, Hi + 1, j + 1LLDD,

Brown

with = œ Rule1, Rule2

Examples

The genuine Brownian rules might be translated into the two rules:

Brownian rules :

Rule1 :

w

w

ï w : vertical, serial

Rule2 : w w ï @D : horizontal, parallel

Graphematic rules :

w

w

w

v

w

w

v

w

ï w w ï@D : vertical, horizontal

ï @D : vertical

l

u

l u ï u u

l

l

: vertical

l

u

l u

ï @D : horizontal

With horizontal chain and environment

e z w

s s l

n w v

ï

e z w

Ñ Ñ l

n w v

Direct Brownian game rules

Rule1

Vertical (linear) setting. Rule1 is sufficient to deal with uni-linear events selected from the board.

z

z

ï

z

Rule2

Has no corespondence in a uni - linear setting. HOverlapping is excludedL

Horizontal (parallel) setting.

Rule2 demands for a planar (tabular) definition of the workspace. This is realized by a parallel setting

of the event chains.

MorphoBoardGames.nb 35

z z z ï z

s s l ï l

First steps of a run for a Brownian Game

n z l v l s z s l z n

m e z m w e n m e z w

n l n u u l u l s s l

n u z l n l n v n w v

v w u w m n m z z z m

m w z s v z m n s e e

w l e l u e l u v z m

w s v l u v z m u w l

m n w n v z n s u v z

n l e w l v m n e w v

v e m w s l l w v l m

n z l v l s z s l Ñ n

m e z m w e n m e z w

Ñ l n u u Ñ u l s s l

n u z l n l n v n w v

v Ñ u w m n Ñ z z z m

m w z s v z m n s e e

Ñ l e Ñ Ñ e l u v z m

w s v l u v z m Ñ w l

m n w n v z n s u v z

n l e Ñ l v m n e w v

v e m w s l l w v l m

Comparison of Brownian and Mersennian games

Games under Mersenne and Brown rules are complementary. There are also dual games for each

type of games.

Stirling Games (Difference)

Stirling games have a first representation by MorphoGames.

Board frames

" i, j œ Board HHeigth i, Width jL :

val@BoardValueMorpho@HHi, jL, Hi + 1, jLLDD

mg

=

val@BoardValueMorpho@HHi, j + 1L, Hi + 1, j + 1LLDD,

with = mg œ Rule1 - Rule4

Elementary rules for morphoGames: Frame rules

Very first elementary rules for morphoGames are given by the rules for frames only. This approach is

abstracting from the specific rules (or logic) of the constellations (patterns, morphograms), and is

just taking the abstract frames, independent of their internal structure or coloring, defined by their

width and heights only and separated by their (empty) environment into account.

Frame rule1

36 MorphoBoardGames.nb

Frames of the same size which are separated, vertically or horizontally or both, by an empty environment

can be eliminated. This applies for single patterns too.

This might happen automatically by a first run. And then by pattern construction by the frame rule2.

Frame rule2

The composition of constellations (patterns) can be manipulated in all directions: the horizontal,

vertical and the diagonal.

Frame rule3

The minimum of adjacent (adjacency) of the squares have to be chosen to define the game.

The minimum adjaceny of the neigborhod of frames is obviously 1.

Example

First steps for a morphFrame game

Example of the first step of a morphoFrame Game with an adjency of 1.

ï

First steps of a run for a Stirling Game

n z l v l s z s l z n

m e z m w e n m e z w

MorphoBoardGames.nb 37

n l n u u l u l s s l

n u z l n l n v n w v

v w u w m n m z z z m

m w z s v z m n s e e

w l e l u e l u v z m

w s v l u v z m u w l

m n w n v z n s u v z

n l e w l v m n e w v

v e m w s l l w v l m

vertical, first steps

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ z n

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ z w

n Ñ Ñ Ñ Ñ l Ñ Ñ Ñ Ñ Ñ

n Ñ Ñ Ñ Ñ l Ñ Ñ Ñ Ñ Ñ

v w Ñ Ñ Ñ Ñ m Ñ Ñ Ñ Ñ

m w Ñ Ñ Ñ Ñ m Ñ Ñ Ñ Ñ

w Ñ Ñ Ñ Ñ Ñ Ñ Ñ v Ñ Ñ

w Ñ Ñ Ñ Ñ Ñ Ñ Ñ u Ñ Ñ

Ñ Ñ Ñ n Ñ Ñ Ñ Ñ u Ñ Ñ

Ñ Ñ Ñ w Ñ Ñ Ñ Ñ e Ñ Ñ

v e m w Ñ Ñ Ñ Ñ v Ñ Ñ

vertical, final first steps

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ n

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ w

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ

v Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ

m Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ v Ñ Ñ

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ

Ñ Ñ Ñ n Ñ Ñ Ñ Ñ Ñ Ñ Ñ

Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ e Ñ Ñ

v e m Ñ Ñ Ñ Ñ Ñ v Ñ Ñ

Horizontal, first steps

n z l v l s z s l z n

m e z m w e n m e z w

n l n u u l u l s s l

n u z l n l n v n w v

v w u w m n m z z z m

m w z s v z m n s e e

w l e l u e l u v z m

w s v l u v z m u w l

m n w n v z n s u v z

n l e w l v m n e w v

v e m w s l l w v l m

Test with ReLabel

EqualBReLabelB: n z l v l s z s l z n >F,

ReLabelB: m e z m w e n m e z w >FF

ReLabel@8n, z, l, v, l, s, z, s, l, z, n

38 MorphoBoardGames.nb

ReLabel@8n, z, l

MorphoBoardGames.nb 39

6 3 4 4 6 3 4 5 2 3 1 1 4 3 5 4 1 6 4 1 3 2

5 1 4 2 6 5 6 2 5 4 2 2 4 1 3 6 1 3 1 4 1 5

5 1 4 1 5 2 3 3 4 2 6 5 3 3 5 6 6 5 6 4 6 2

2 6 1 4 5 2 4 6 1 4 1 6 5 1 1 2 4 3 3 1 6 3

6 3 6 3 5 3 1 1 3 5 5 6 2 3 3 3 3 4 6 4 6 6

1 3 2 1 6 1 5 2 5 6 6 3 4 6 3 3 4 2 6 5 5 4

4 5 3 1 2 1 2 3 1 1 6 1 5 5 3 5 2 1 5 4 6 2

2 4 6 3 3 4 4 4 6 4 6 2 5 3 2 3 4 4 5 6 1 2

2 6 5 1 3 6 1 2 4 1 1 5 3 2 5 6 1 3 2 3 1 1

4 2 4 1 2 4 4 6 6 1 2 1 4 1 3 2 3 1 2 6 6 3

3 1 2 2 4 4 1 5 5 4 4 2 6 6 4 6 3 1 3 6 5 4

2 2 5 1 6 6 2 6 2 6 1 4 5 1 1 1 6 4 2 4 5 1

5 5 6 1 4 6 1 3 1 3 3 4 1 5 2 6 3 1 4 3 3 2

1 3 5 5 1 3 5 3 5 5 5 4 2 3 5 4 6 3 4 3 6 4

4 1 2 3 2 6 2 4 1 2 2 5 6 4 3 5 5 1 5 1 2 6

Vertical

3

6

6

5

5

2

2

3

= palin

3

1

1

3

1

1

2

1

1

5

¹≠ palin,

1

1

3

1

1

2

1

1

= palin,

1

6

2

3

6

5

,

3 1

3 1

6 4

3 1

¹≠ palin, but morpho - equivalent

Horizontal

4 4 4

6 1 2

= palin1 »» palin2, palin1 ¹≠ mg palin2

3 1 1 4

4 2 2 4

= palin »» palin

4 4 1 5 5 4 4 2 6 6 = palin

1 5 5 4 4 2 = palin

2 4 4 6 6 1 = palin

Palindrome tests

40 MorphoBoardGames.nb

-ispalindrome Htnf@3, 6, 6, 5, 5, 4, 4, 3DL;

val it = true : bool

-ispalindrome Htnf@3, 1, 1, 3, 1, 1, 2, 1, 1, 5DL;

val it = false : bool

-ispalindrome Htnf@1, 1, 3, 1, 1, 2, 1, 1DL;

val it = true : bool

-ispalindrome@1, 2, 3, 1D;

val it = true : bool

-ispalindrome Htnf@1, 6, 2, 3, 6, 5DL;

val it = true : bool

-ispalindrome Htnf@3, 1, 1, 4DL;

val it = true : bool

-ispalindrome Htnf@4, 2, 2, 4DL;

val it = true : bool

-ispalindrome Htnf@6, 6, 1, 5, 5, 4, 4, 2, 6, 6DL;

val it = true : bool

- ispalindrome Htnf@1, 5, 5, 4, 4, 2DL;

val it = true : bool

- ispalindrome Htnf@2, 4, 4, 6, 6, 1DL;

val it = true : bool

Augmenting games with new elements

Up to now, the scope of the patterns, i.e. the number of elements on the board had been, once set,

constant.

From a classical, identity-oriented point of view, an addition of elements is straight forwards and

simple.

Morphogrammatics is, again, not based on identical elements, but on patterns, i.e. morphograms.

Hence, an extension of the scope of the patterns has to take the specific laws (properties, characteristics)

of morphogrammatics into account.

This leads to a morphogrammatic concept of addition (coalition). And in more complex situation to a

morphogrammatic approach to multiplication (cooperation).

Coalition

Coalition in morphogrammatics is a retro-grade recursive and super - additive operation.

Hence, a morphogrammatically adequate extension of the range of elements is not abstract but retrograde

defined by the ‘elements’ already in the game.

Steps towards a program for MorphoBoard Games

FaceNeighboursMorpho@p_, q_D :=

Map@Plus@p, q, ÒD &,

880, 1

SpotMorpho@p_, q_D := FixedPoint@

Function@y1, Union@y1, Apply@Union, Map@Select@FaceNeighboursMorpho@ÒD, Function@

x1, BoardValueMorpho@x1D ã BoardValueMorpho@First@y1DDDD &, y1DDDD, 8p