Hexagonal Interacting States>

Hexagonal Interacting States

A new DLL which operates with MCell has been developed. This DLL implements a new family of rules called 'Hexagonal Interacting States'. Specific rules in the family are selected by setting parameters in an input dialog window shown below.

Each cell can be in one of 9 possible states. On each iteration, the states of a cell's 6 neighbors are counted and the cell changes state according to the resulting counts. If a cell is in state 0, it changes to a state in the range 1 thru 8 if one or more of the conditions specified in the 'Birth Control' is satisfied. Similarly, if a cell is not in state 0, it remains in its current state or returns to state 0 according to the conditions specified in the 'Survival Control'. In the example dialog shown above the settings in row 1 of the 'Birth Control' state:

If a cell is in state 0, it will change to state 1 if it has 2 OR MORE neighbors in state 5 OR if it has 3 OR MORE neighbors in state 7.

And the settings in row 6 of the 'Survival Control' state:

If a cell is in state 6, it will remain in state 6 if it has 2 neighbors in state 3 AND 1 neighbor in state 4. Otherwise it will change to state 0.

The 'Birth Control' grid has 8 rows. Each row contains settings which specify the conditions that must be satisfied in order for a cell to change from state 0 to state n where n is the row number. The conditions in the rows are tested in numerical order and the first condition that is satisfied determines the next state of the cell. If no condition is satisfied, the cell remains in state 0.

The 'Survival Control' grid also has 8 rows. Each row contains settings which specify the conditions that must be satisfied for a cell to remain in state n where n is the row number. Again, if no condition is satisfied, the cell returns to state 0.

Grid values are changed by simply left clicking them. Each click rotates the value thru the range 0..8, incrementing with normal clicks and decrementing if the shift key is held down.

Notice the two buttons labeled 'OR >=' and 'AND ='. These are toggle buttons which rotate thru the values 'OR =', 'OR >=', 'AND >=' and 'AND ='. To see the relevance of these buttons carefully study the example described above.

For convenience, buttons are provided to reset each grid and to randomly select candidate conditions. Each of the random select buttons has an associated control which limits the generated counts. When clicked, these random select buttons generate two random numbers in the range 0 thru 'limit' for each row and stores them in two random positions in that row. The limit values are changed in a manner similar to the grid values.

A process for finding interesting rules that works quite well is:

Reset both rule grids.
Set the random limit value in the birth control to 3.
Click the random button in the birth control.
Test the generated rule using random seeding.
Reset the birth control grid and repeat steps 3 thru 4 until an interesting rule is found.
Set the random limit value in the survival control to 1, 2 or 3.
Reset the survival grid and click its random button.
Test the rule variant.
Repeat steps 6 and 8.
Increment or decrement grid values one at a time and examine the effect.
Repeat step 10.

To illustrate the potential of this new family of rules, a set of example patterns has been assembled. Several interesting rules have been selected and a few examples are given for each.

The convention used to name the example patterns is:

Name Pattern

Cxxx Chos

DFxxx Diagonal Flotilla

DGxxx Diagonal Gun

DPxxx Diagonal Puffer

DSxxx Diagonal Ship

DSTxxx Diagonal Stretcher

EPxxx Expanding Pattern

Fxxx Fuse

Gxxx Gun

Ixxx Interaction

IGxxx Irregular Gun

KGxxx Knight Gun

KPxxx Knight Puffer

KSxxx Knight Ship

Mxxx Maximum Tiling

MGxxx Moving Gun

MRxxx Moving Replicator

Oxxx Oscillator

Pxxx Puffer

Rxxx Replicator

Sxxx Ship

SCxxx Stretching Chaos

SDxxx Speed Comparison

SDPxxx Stretching Diagonal Puffer

SDSxxx Stretching Diagonal Ship

SExxx Symmetrical Expansion

SLxxx Still Life

SPxxx Stretching Puffer

SSxxx Stretching Ship

STxxx Stretcher

USxxx Unusual Ship

Name	Pattern
Cxxx	Chos
DFxxx	Diagonal Flotilla
DGxxx	Diagonal Gun
DPxxx	Diagonal Puffer
DSxxx	Diagonal Ship
DSTxxx	Diagonal Stretcher
EPxxx	Expanding Pattern
Fxxx	Fuse
Gxxx	Gun
Ixxx	Interaction
IGxxx	Irregular Gun
KGxxx	Knight Gun
KPxxx	Knight Puffer
KSxxx	Knight Ship
Mxxx	Maximum Tiling
MGxxx	Moving Gun
MRxxx	Moving Replicator
Oxxx	Oscillator
Pxxx	Puffer
Rxxx	Replicator
Sxxx	Ship
SCxxx	Stretching Chaos
SDxxx	Speed Comparison
SDPxxx	Stretching Diagonal Puffer
SDSxxx	Stretching Diagonal Ship
SExxx	Symmetrical Expansion
SLxxx	Still Life
SPxxx	Stretching Puffer
SSxxx	Stretching Ship
STxxx	Stretcher
USxxx	Unusual Ship

The following tables show the number of examples by type for each rule:

Rule 1

Type Number

DS 2

O 1

S 2

Rule 2

Type Number

DS 1

S 2

Rule 3

Type Number

DP 1

DS 1

O 1

P 2

S 1

Rule 4

Type Number

DST 1

Rule 5

Type Number

DST 2

Rule 6

Type Number

R 1

Rule 7

Type Number

DP 3

DS 2

O 2

R 2

Rule 8

Type Number

DST 1

Rule 9

Type Number

R 1

Rule 10

Type Number

DS 2

R 1

Rule 11

Type Number

DS 3

Rule 12

Type Number

DP 1

Rule 13

Type Number

O 1

Rule 14

Type Number

DS 1

Rule 15

Type Number

DS 1

MR 3

Rule 16

Type Number

O 1

Rule 17

Type Number

DS 2

R 3

Rule 18

Type Number

DG 7

DP 3

DS 1

I 1

O 5

Rule 19

Type Number

DG 6

Rule 20

Type Number

DG 14

DP 12

DS 8

O 8

Rule 21

Type Number

DP 8

DS 2

Rule 22

Type Number

DS 1

S 2

Rule 23

Type Number

DS 5

S 2

SD 1

The following is a cross reference showing which rules support which types:

DG
[18, 19, 20]

DP
[3, 7, 12, 18, 20, 21]

DS
[1, 2, 3, 7, 10, 11, 14, 15, 17, 18, 20, 21, 22, 23]

DST
[4, 5, 8]

I
[18]

MR
[15]

O
[1, 3, 7, 13, 16, 18, 20]

P
[3]

R
[6, 7, 9, 10, 17]

S
[1, 2, 3, 22, 23]

SD
[23]

The formal definition of each rule follows:

Rule 001
HexagonalInteractingStates
,TB1,TS0,B1;00300002,B2;00000100,B3;01003000,B4;00100000,B5;00000030,B6;00000000,B7;30100000,B8;
00020010,S1;00100100,S2;00000000,S3;00001000,S4;00000001,S5;00000001,S6;00100000,S7;00000000,S8;10100000

Rule 002
HexagonalInteractingStates
,TB1,TS0,B1;00300002,B2;00000100,B3;01003000,B4;00100000,B5;00000030,B6;00000000,B7;30100000,B8;
00020010,S1;00000100,S2;10000000,S3;00000001,S4;10000100,S5;00100000,S6;00000000,S7;01000001,S8;00001000

Rule 003
HexagonalInteractingStates
,TB1,TS0,B1;00300002,B2;00000100,B3;01003000,B4;00100000,B5;00000030,B6;00000000,B7;30100000,B8;
00020010,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000001,S8;00000000

Rule 004
HexagonalInteractingStates
,TB1,TS0,B1;00010000,B2;03000300,B3;03000100,B4;00200000,B5;30000000,B6;00000120,B7;00020020,B8;
00200003,S1;00001010,S2;00001000,S3;00000000,S4;00001000,S5;10000000,S6;00000000,S7;00000010,S8;00010001

Rule 005
HexagonalInteractingStates
,TB1,TS0,B1;00010000,B2;03000300,B3;03000100,B4;00200000,B5;30000000,B6;00000120,B7;00020020,B8;
00200003,S1;00000000,S2;00001000,S3;00000000,S4;00001000,S5;10000000,S6;00000000,S7;00000010,S8;00010001

Rule 006
HexagonalInteractingStates
,TB1,TS0,B1;00302000,B2;00330000,B3;00000300,B4;00010000,B5;00000000,B6;00003000,B7;00200002,B8;
20000000,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

Rule 007
HexagonalInteractingStates
,TB1,TS0,B1;00200000,B2;00100000,B3;10000000,B4;00000000,B5;00100300,B6;30001000,B7;00300000,B8;
00002000,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

Rule 008
HexagonalInteractingStates
,TB1,TS0,B1;00200000,B2;00100000,B3;10000000,B4;00000000,B5;00100300,B6;30001000,B7;00300000,B8;
00002000,S1;00000010,S2;00000100,S3;01010000,S4;01000000,S5;10010000,S6;10000000,S7;00010010,S8;01000000

Rule 009
HexagonalInteractingStates
,TB1,TS1,B1;01010020,B2;00001002,B3;00020002,B4;00000030,B5;00003010,B6;03000000,B7;00200001,B8;
00010200,S1;01000000,S2;00000101,S3;00000001,S4;10000100,S5;10010000,S6;00000000,S7;01000000,S8;00000000

Rule 010
HexagonalInteractingStates
,TB1,TS1,B1;01010020,B2;00001002,B3;00020002,B4;00100030,B5;00003010,B6;03000000,B7;00200001,B8;
00010200,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

Rule 011
HexagonalInteractingStates
,TB1,TS1,B1;01010020,B2;00001002,B3;00020002,B4;00100030,B5;00003010,B6;03000000,B7;00200001,B8;
00010200,S1;00000010,S2;00000010,S3;00000000,S4;00000001,S5;00010010,S6;00010010,S7;00001000,S8;00001000

Rule 012
HexagonalInteractingStates
,TB1,TS1,B1;02000010,B2;00020100,B3;20000010,B4;00001300,B5;00003030,B6;00303000,B7;00010002,B8;
02000030,S1;10000000,S2;00000000,S3;01000001,S4;00001000,S5;00000000,S6;00000010,S7;00000010,S8;10000000

Rule 013
HexagonalInteractingStates
,TB1,TS1,B1;02000010,B2;00020100,B3;20000010,B4;00001300,B5;00003030,B6;00303000,B7;00010002,B8;
02000030,S1;00000000,S2;00000000,S3;00010000,S4;00000000,S5;00000011,S6;00000000,S7;01000000,S8;00100001

Rule 014
HexagonalInteractingStates
,TB1,TS1,B1;02000010,B2;00020100,B3;20000010,B4;00001300,B5;00003030,B6;00303000,B7;00010002,B8;
02000030,S1;00001010,S2;00000000,S3;00000000,S4;00000000,S5;00100000,S6;00000000,S7;00000000,S8;01000000

Rule 015
HexagonalInteractingStates
,TB1,TS1,B1;00030000,B2;00001000,B3;03010000,B4;00001300,B5;00300010,B6;00023000,B7;30000001,B8;
03200000,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

Rule 016
HexagonalInteractingStates
,TB1,TS1,B1;00030000,B2;00001000,B3;03010000,B4;00001300,B5;00300010,B6;00023000,B7;30000001,B8;
03200000,S1;00110000,S2;01000100,S3;01100000,S4;00001000,S5;00000010,S6;00010001,S7;00100000,S8;10100000

Rule 017
HexagonalInteractingStates
,TB1,TS1,B1;02000000,B2;00200000,B3;00000001,B4;00001002,B5;02000000,B6;00000002,B7;00030000,B8;
00010003,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

Rule 018
HexagonalInteractingStates
,TB1,TS2,B1;02000000,B2;00200000,B3;00000001,B4;00001002,B5;02000000,B6;00000002,B7;00030000,B8;
00010003,S1;00000000,S2;00000000,S3;00010000,S4;00010000,S5;00000100,S6;00000000,S7;00010000,S8;00000000

Rule 019
HexagonalInteractingStates
,TB1,TS2,B1;02000000,B2;00200000,B3;00000001,B4;00001002,B5;02000000,B6;00000002,B7;00030000,B8;
00010003,S1;00000000,S2;00000000,S3;00010000,S4;00300000,S5;00000100,S6;00000000,S7;00010000,S8;00000000

Rule 020
HexagonalInteractingStates
,TB1,TS2,B1;02000000,B2;00200000,B3;00000001,B4;00001002,B5;02000000,B6;00000002,B7;00030000,B8;
00010003,S1;20000000,S2;00000000,S3;00010000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

Rule 021
HexagonalInteractingStates
,TB1,TS3,B1;02000000,B2;00200000,B3;00000001,B4;00001002,B5;02000000,B6;00000002,B7;00030000,B8;
00010003,S1;20000000,S2;00000000,S3;00010000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

Rule 022
HexagonalInteractingStates
,TB1,TS0,B1;00000201,B2;02001000,B3;00000000,B4;10200000,B5;00001000,B6;00020000,B7;00002100,B8;
00000000,S1;00000000,S2;00001010,S3;01000100,S4;00000000,S5;00000000,S6;10000000,S7;00000001,S8;00000000

Rule 023
HexagonalInteractingStates
,TB1,TS0,B1;00100000,B2;00200000,B3;00020200,B4;00020010,B5;00300000,B6;01000010,B7;00000002,B8;
00000010,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

The formal definitions of some potentially interesting rules which have not been fully explored follows:

T001.mcl
HexagonalInteractingStates
,TB1,TS1,B1;00102000,B2;00001000,B3;00010000,B4;00001300,B5;00300010,B6;00023000,B7;30000001,B8;
03200000,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

T002.mcl
HexagonalInteractingStates
,TB1,TS1,B1;00101000,B2;00001000,B3;00010000,B4;00001300,B5;00300010,B6;00023000,B7;30000001,B8;
03200000,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

T003.mcl
HexagonalInteractingStates
,TB1,TS1,B1;00102000,B2;00001000,B3;00010000,B4;00001300,B5;00300010,B6;00023000,B7;30000001,B8;
03200000,S1;00000000,S2;00000001,S3;00000000,S4;01000000,S5;00000000,S6;00100001,S7;00000000,S8;00011000

T004.mcl
HexagonalInteractingStates
,TB1,TS1,B1;00102000,B2;00001000,B3;00010000,B4;00001300,B5;00300010,B6;00023000,B7;00000001,B8;
03200000,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

T005.mcl
HexagonalInteractingStates
,TB1,TS1,B1;20000000,B2;00301000,B3;00000020,B4;01000000,B5;00001000,B6;20000000,B7;00010300,B8;
02000000,S1;00000100,S2;00010000,S3;01000000,S4;10000000,S5;10001000,S6;00010000,S7;10000100,S8;00000001

T006.mcl
HexagonalInteractingStates
,TB1,TS1,B1;20000000,B2;00301000,B3;00000020,B4;01000000,B5;00001000,B6;20000000,B7;00010300,B8;
02000000,S1;02100000,S2;01000000,S3;00000000,S4;00010001,S5;00100000,S6;00000000,S7;00000000,S8;00002002

T007.mcl
HexagonalInteractingStates
,TB1,TS1,B1;02000000,B2;00200000,B3;00000001,B4;00001002,B5;02000000,B6;00000002,B7;00030000,B8;
00010003,S1;00000000,S2;00000100,S3;00000010,S4;10000000,S5;00000000,S6;00000000,S7;00000011,S8;00000000

T008.mcl
HexagonalInteractingStates
,TB1,TS1,B1;02000000,B2;00200000,B3;00000001,B4;00001002,B5;02000000,B6;00000002,B7;00030000,B8;
00010003,S1;10000010,S2;01000000,S3;00000000,S4;00001000,S5;00001000,S6;00000001,S7;00001000,S8;00000000

T009.mcl
HexagonalInteractingStates
,TB1,TS1,B1;02000000,B2;00200000,B3;00000001,B4;00001002,B5;02000000,B6;00000002,B7;00030000,B8;
00010003,S1;10000100,S2;00000000,S3;00000000,S4;00001001,S5;10010000,S6;01000001,S7;00001100,S8;00010000

T010.mcl
HexagonalInteractingStates
,TB1,TS2,B1;02000000,B2;00200000,B3;00000001,B4;00001002,B5;02000000,B6;00000002,B7;00030000,B8;
00010003,S1;00000000,S2;00000000,S3;00010000,S4;00030000,S5;00000100,S6;00000000,S7;00010000,S8;00000000

T011.mcl
HexagonalInteractingStates
,TB1,TS1,B1;02000000,B2;00200000,B3;00000001,B4;00001002,B5;02000000,B6;00000002,B7;00030000,B8;
00010003,S1;00000000,S2;00000000,S3;10010000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

T012.mcl
HexagonalInteractingStates
,TB1,TS3,B1;00033000,B2;00020000,B3;00000020,B4;00300003,B5;00000001,B6;02300000,B7;00030030,B8;
00001000,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

T013.mcl
HexagonalInteractingStates
,TB1,TS3,B1;00010001,B2;00000000,B3;00002000,B4;00000000,B5;02000002,B6;00002000,B7;00000300,B8;
00001000,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

T014.mcl
HexagonalInteractingStates
,TB1,TS0,B1;00000201,B2;02001000,B3;00000000,B4;10200000,B5;00001000,B6;00020000,B7;00002100,B8;
00000000,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000

T015.mcl
HexagonalInteractingStates
,TB1,TS0,B1;00100000,B2;20000000,B3;03300000,B4;00000102,B5;00310000,B6;30010000,B7;00000000,B8;
00000000,S1;00000000,S2;00000000,S3;00000000,S4;00000000,S5;00000000,S6;00000000,S7;00000000,S8;00000000