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Clifford Algebra VoDou Physics of Wolfram's 256 CellularAutomata Rules

 

The 256 Cellular Automata Rules ofWolfram correspond tothe 256 basis elements of the 256-dimensional CliffordAlgebra of 16x16 Real Matrices, Cl(8) and Cl(1,7), and itsdiscrete counterpart, as well as to the 256 Odu of IFA.

To visualize the correspondence, write the rule numbers in binarynotation, put them with the pictures of the 30-level actions of eachof the 256 rules as shown on pages 55 and 56 of Wolfram'sbook A New Kind ofScience, and organize them with respect to the

1 + 8 + 28 + 56 + (35+35) + 56 + 28 + 8 + 1 = 256 = 16x16 =2^8

graded structure of Cl(8). Hereis some more about Cellular Automata.

Graded structure details include:


Here are 78 of the 256, including:

The vertical line is a symmetry line of the symmetry of exchanging0s and 1s in the binary numbers.

Note that:


 

Here are all 28 rules for each of grades 2 and 6.

 

Note that:

 


 

Here are all 56 rules for each of grades 3 and 5.

Note that:

 


Here are all 35+35 = 70 rules for grade 4.

 

Note that:

 


 

Here is a numerical (no pictures) list of all 256, with gradeindicated:

Grade: 0         1         2         3         4         5         6         7         8000 00000000001           00000001002           00000010003                     00000011004           00000100005                     00000101006                     00000110007                               00000111008           00001000009                     00001001010                               00001010011                               00001011012                     00001100013                               00001101014                               00001110015                                         00001111016           00010000      017                     00010001018                     00010010019                               00010011020                     00010100021                               00010101022                               00010110023                                         00010111024                     00011000025                               00011001026                               00011010027                                         00011011028                               00011100029                                         00011101030                                         00011110031                                                   00011111032           00100000033                     00100001034                     00100010035                               00100011036                     00100100037                               00100101038                               00100110039                                         00100111040                     00101000041                               00101001042                               00101010043                                         00101011044                               00101100045                                         00101101046                                         00101110047                                                   00101111048                     00110000049                               00110001050                               00110010051                                         00110011052                               00110100053                                         00110101054                                         00110110055                                                   00110111056                               00111000057                                         00111001058                                         00111010059                                                   00111011060                                         00111100061                                                   00111101062                                                   00111110063                                                             00111111064           01000000065                     01000001066                     01000010067                               01000011068                     01000100069                               01000101070                               01000110071                                         01000111072                     01001000073                               01001001074                     01001010075                                         01001011076                               01001100077                                         01001101078                                         01001110079                                                   01001111080                     01010000081                               01010001082                               01010010083                                         01010011084                               01010100085                                         01010101086                                         01010110087                                                   01010111088                               01011000089                                         01011001090                                         01011010091                                                   01011011092                                         01011100093                                                   01011101094                                                   01011110095                                                             01011111096                     01100000097                               01100001098                               01100010099                                         01100011100                               01100100101                                         01100101102                                         01100110103                                                   01100111104                               01101000105                                         01101001106                                         01101010107                                                   01101011108                                         01101100109                                                   01101101110                                                   01101110111                                                             01101111112                               01110000113                                         01110001114                                         01110010115                                                   01110011116                                         01110100117                                                   01110101118                                                   01110110119                                                             01110111120                                         01111000121                                                   01111001122                                                   01111010123                                                             01111011124                                                   01111100125                                                             01111101126                                                             01111110127                                                                       01111111128           10000000129                     10000001130                     10000010131                               10000011132                     10000100133                               10000101134                               10000110135                                         10000111136                     10001000137                               10001001138                               10001010139                                         10001011140                               10001100141                                         10001101142                                         10001110143                                                   10001111144                     10010000145                               10010001146                               10010010147                                         10010011148                               10010100149                                         10010101150                                         10010110151                                                   10010111152                               10011000153                                         10011001154                                         10011010155                                                   10011011156                                         10011100157                                                   10011101158                                                   10011110159                                                             10011111160                     10100000161                               10100001162                               10100010163                                         10100011164                               10100100165                                         10100101166                                         10100110167                                                   10100111168                               10101000169                                         10101001170                                         10101010171                                                   10101011172                                         10101100173                                                   10101101174                                                   10101110175                                                             10101111176                               10110000177                                         10110001178                                         10110010179                                                   10110011180                                         10110100181                                                   10110101182                                                   10110110183                                                            10110111184                                         10111000185                                                   10111001186                                                   10111010187                                                            10111011188                                                   10111100189                                                            10111101190                                                            10111110191                                                                      10111111192                     11000000193                               11000001194                               11000010195                                         11000011196                               11000100197                                         11000101198                                         11000110199                                                   11000111200                               11001000201                                         11001001202                                         11001010203                                                   11001011204                                         11001100205                                                   11001101206                                                   11001110207                                                             11001111208                               11010000209                                         11010001210                                         11010010211                                                   11010011212                                         11010100213                                                   11010101214                                                   11010110215                                                             11010111216                                         11011000217                                                   11011001218                                                   11011010219                                                             11011011220                                                   11011100221                                                             11011101222                                                             11011110223                                                                       11011111224                               11100000225                                         11100001226                                         11100010227                                                   11100011228                                         11100100229                                                   11100101230                                                   11100110231                                                             11100111232                                         11101000233                                                   11101001234                                                   11101010235                                                             11101011236                                                   11101100237                                                             11101101238                                                             11101110239                                                                       11101111240                                         11110000241                                                   11110001242                                                   11110010243                                                             11110011244                                                   11110100245                                                             11110101246                                                             11110110247                                                                       11110111248                                                   11111000249                                                             11111001250                                                             11111010251                                                                       11111011252                                                             11111100253                                                                       11111101254                                                                       11111110255                                                                                 11111111


 

FrankD. (Tony) Smith, Jr. - May 2004 - Clickhere for pdf version.

(June 2004 correction to SU(3) diagram suggested by MichaelGibbs)


Tony Smith's Home Page - Hereis some more about Cellular Automata.

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