The 256 Cellular Automata Rules of Wolfram correspond to the 256 basis elements of the 256-dimensional Clifford Algebra of 16x16 Real Matrices, Cl(8) and Cl(1,7), and its discrete counterpart, as well as to the 256 Odu of IFA.
To visualize the correspondence, write the rule numbers in binary notation, put them with the pictures of the 30-level actions of each of the 256 rules as shown on pages 55 and 56 of Wolfram's book A New Kind of Science, and organize them with respect to the
graded structure of Cl(8). Here is 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 exchanging 0s and 1s in the binary numbers.
Note that:
are related to the Spinors and Primitive Idempotents of Cl(0,8);
correspond to the 4 dimensions of physical spacetime;
correspond to the 4 dimensions of internal symmetry space;
Here are all 28 rules for each of grades 2 and 6.
Note that:
after dimensional reduction to 4-dimensional physical spacetime, correspond to the 8 generators of color force SU(3), whose root vector diagram is illustrated above;
after dimensional reduction to 4-dimensional physical spacetime, correspond to the 3 generators of weak force SU(2);
after dimensional reduction to 4-dimensional physical spacetime, correspond to the 1 generator of electromagnetic U(1);
after dimensional reduction to 4-dimensional physical spacetime, correspond to the 16 generators of Gravity/Higgs/phase U(2,2). One of them
corresponds to the propagator phase U(1) while the other 15 correspond to the Conformal Group SU(2,2) = Spin(2,4) whose root vector diagram
is a 12-vertex cuboctahedron (the other 3 bivectors corresponding to the 3 generators of the Cartan Subalgebra).
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 grade indicated:
Grade: 0 1 2 3 4 5 6 7 8 000 00000000 001 00000001 002 00000010 003 00000011 004 00000100 005 00000101 006 00000110 007 00000111 008 00001000 009 00001001 010 00001010 011 00001011 012 00001100 013 00001101 014 00001110 015 00001111 016 00010000 017 00010001 018 00010010 019 00010011 020 00010100 021 00010101 022 00010110 023 00010111 024 00011000 025 00011001 026 00011010 027 00011011 028 00011100 029 00011101 030 00011110 031 00011111 032 00100000 033 00100001 034 00100010 035 00100011 036 00100100 037 00100101 038 00100110 039 00100111 040 00101000 041 00101001 042 00101010 043 00101011 044 00101100 045 00101101 046 00101110 047 00101111 048 00110000 049 00110001 050 00110010 051 00110011 052 00110100 053 00110101 054 00110110 055 00110111 056 00111000 057 00111001 058 00111010 059 00111011 060 00111100 061 00111101 062 00111110 063 00111111 064 01000000 065 01000001 066 01000010 067 01000011 068 01000100 069 01000101 070 01000110 071 01000111 072 01001000 073 01001001 074 01001010 075 01001011 076 01001100 077 01001101 078 01001110 079 01001111 080 01010000 081 01010001 082 01010010 083 01010011 084 01010100 085 01010101 086 01010110 087 01010111 088 01011000 089 01011001 090 01011010 091 01011011 092 01011100 093 01011101 094 01011110 095 01011111 096 01100000 097 01100001 098 01100010 099 01100011 100 01100100 101 01100101 102 01100110 103 01100111 104 01101000 105 01101001 106 01101010 107 01101011 108 01101100 109 01101101 110 01101110 111 01101111 112 01110000 113 01110001 114 01110010 115 01110011 116 01110100 117 01110101 118 01110110 119 01110111 120 01111000 121 01111001 122 01111010 123 01111011 124 01111100 125 01111101 126 01111110 127 01111111 128 10000000 129 10000001 130 10000010 131 10000011 132 10000100 133 10000101 134 10000110 135 10000111 136 10001000 137 10001001 138 10001010 139 10001011 140 10001100 141 10001101 142 10001110 143 10001111 144 10010000 145 10010001 146 10010010 147 10010011 148 10010100 149 10010101 150 10010110 151 10010111 152 10011000 153 10011001 154 10011010 155 10011011 156 10011100 157 10011101 158 10011110 159 10011111 160 10100000 161 10100001 162 10100010 163 10100011 164 10100100 165 10100101 166 10100110 167 10100111 168 10101000 169 10101001 170 10101010 171 10101011 172 10101100 173 10101101 174 10101110 175 10101111 176 10110000 177 10110001 178 10110010 179 10110011 180 10110100 181 10110101 182 10110110 183 10110111 184 10111000 185 10111001 186 10111010 187 10111011 188 10111100 189 10111101 190 10111110 191 10111111 192 11000000 193 11000001 194 11000010 195 11000011 196 11000100 197 11000101 198 11000110 199 11000111 200 11001000 201 11001001 202 11001010 203 11001011 204 11001100 205 11001101 206 11001110 207 11001111 208 11010000 209 11010001 210 11010010 211 11010011 212 11010100 213 11010101 214 11010110 215 11010111 216 11011000 217 11011001 218 11011010 219 11011011 220 11011100 221 11011101 222 11011110 223 11011111 224 11100000 225 11100001 226 11100010 227 11100011 228 11100100 229 11100101 230 11100110 231 11100111 232 11101000 233 11101001 234 11101010 235 11101011 236 11101100 237 11101101 238 11101110 239 11101111 240 11110000 241 11110001 242 11110010 243 11110011 244 11110100 245 11110101 246 11110110 247 11110111 248 11111000 249 11111001 250 11111010 251 11111011 252 11111100 253 11111101 254 11111110 255 11111111
Frank D. (Tony) Smith, Jr. - May 2004 - Click here for pdf version.
(June 2004 correction to SU(3) diagram suggested by Michael Gibbs)
Tony Smith's Home Page - Here is some more about Cellular Automata.