|
|
|
|
|
further reading | more topics » |
| mjbWorld program | 3D theory |
3D physics |
3D maths |
3D programming | technology |
about site |
sitemap A-Z |
| index | algebra | geometry | calculus | graph theory | statistics | principles | standards |
| index | equations | multi-dim | vector | matrix | complex | clifford | groups |
| common | concepts | 2D | 3D | 4D | 5D |
| index | basis | arithmetic | geometry | functions | transform | code | pauli |
Here is how I generated the tables for this page.
The tables were generated using this program.
The output of this program is shown below. To produce the results the program needs to have an XML input code. At the bottom of this page I have listed this input code.
| a*b | b.e | b.e1 | b.e2 | b.e3 | b.e4 | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e123 | b.e142 | b.e134 | b.e324 | b.e1234 |
| a.e | e | e1 | e2 | e3 | e4 | e12 | e31 | e23 | e41 | e42 | e43 | e123 | e142 | e134 | e324 | e1234 |
| a.e1 | e1 | e | e12 | -e31 | -e41 | e2 | -e3 | e123 | -e4 | e142 | -e134 | e23 | e42 | -e43 | -e1234 | -e324 |
| a.e2 | e2 | -e12 | e | e23 | -e42 | -e1 | e123 | e3 | -e142 | -e4 | e324 | e31 | -e41 | -e1234 | e43 | -e134 |
| a.e3 | e3 | e31 | -e23 | e | -e43 | e123 | e1 | -e2 | e134 | -e324 | -e4 | e12 | -e1234 | e41 | -e42 | -e142 |
| a.e4 | e4 | e41 | e42 | e43 | e | -e142 | -e134 | -e324 | e1 | e2 | e3 | -e1234 | -e12 | -e31 | -e23 | -e123 |
| a.e12 | e12 | -e2 | e1 | e123 | -e142 | -e | e23 | -e31 | -e42 | e41 | -e1234 | -e3 | e4 | e324 | -e134 | e43 |
| a.e31 | e31 | e3 | e123 | -e1 | -e134 | -e23 | -e | e12 | e43 | -e1234 | -e41 | -e2 | -e324 | e4 | e142 | e42 |
| a.e23 | e23 | e123 | -e3 | e2 | -e324 | e31 | -e12 | -e | -e1234 | -e43 | e42 | -e1 | e134 | -e142 | e4 | e41 |
| a.e41 | e41 | e4 | -e142 | e134 | -e1 | e42 | -e43 | -e1234 | -e | -e12 | e31 | -e324 | e2 | -e3 | e123 | e23 |
| a.e42 | e42 | e142 | e4 | -e324 | -e2 | -e41 | -e1234 | e43 | e12 | -e | -e23 | -e134 | -e1 | e123 | e3 | e31 |
| a.e43 | e43 | -e134 | e324 | e4 | -e3 | -e1234 | e41 | -e42 | -e31 | e23 | -e | -e142 | e123 | e1 | -e2 | e12 |
| a.e123 | e123 | e23 | e31 | e12 | e1234 | -e3 | -e2 | -e1 | e324 | e134 | e142 | -e | -e43 | -e42 | -e41 | -e4 |
| a.e142 | e142 | e42 | -e41 | e1234 | -e12 | e4 | e324 | -e134 | e2 | -e1 | -e123 | e43 | -e | e23 | -e31 | -e3 |
| a.e134 | e134 | -e43 | e1234 | e41 | -e31 | -e324 | e4 | e142 | -e3 | -e123 | e1 | e42 | -e23 | -e | e12 | -e2 |
| a.e324 | e324 | e1234 | e43 | -e42 | -e23 | e134 | -e142 | e4 | -e123 | e3 | -e2 | e41 | e31 | -e12 | -e | -e1 |
| a.e1234 | e1234 | e324 | e134 | e142 | e123 | e43 | e42 | e41 | e23 | e31 | e12 | e4 | e3 | e2 | e1 | e |
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1
analysing associativity: table associates
| a*b | b.e | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e1234 |
| a.e | e | e12 | e31 | e23 | e41 | e42 | e43 | e1234 |
| a.e12 | e12 | -e | e23 | -e31 | -e42 | e41 | -e1234 | e43 |
| a.e31 | e31 | -e23 | -e | e12 | e43 | -e1234 | -e41 | e42 |
| a.e23 | e23 | e31 | -e12 | -e | -e1234 | -e43 | e42 | e41 |
| a.e41 | e41 | e42 | -e43 | -e1234 | -e | -e12 | e31 | e23 |
| a.e42 | e42 | -e41 | -e1234 | e43 | e12 | -e | -e23 | e31 |
| a.e43 | e43 | -e1234 | e41 | -e42 | -e31 | e23 | -e | e12 |
| a.e1234 | e1234 | e43 | e42 | e41 | e23 | e31 | e12 | e |
analysing commutivity: table does not commute: for example: e12*e31 != e31*e12
analysing associativity: table associates
| a*b | b.e | b.e1 | b.e2 | b.e3 | b.e4 | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e123 | b.e142 | b.e134 | b.e324 | b.e1234 |
| a.e | e | e1 | e2 | e3 | e4 | e12 | e31 | e23 | e41 | e42 | e43 | e123 | e142 | e134 | e324 | e1234 |
| a.e1 | e1 | -e | e12 | -e31 | -e41 | -e2 | e3 | e123 | e4 | e142 | -e134 | -e23 | -e42 | e43 | -e1234 | e324 |
| a.e2 | e2 | -e12 | e | e23 | -e42 | -e1 | e123 | e3 | -e142 | -e4 | e324 | e31 | -e41 | -e1234 | e43 | -e134 |
| a.e3 | e3 | e31 | -e23 | e | -e43 | e123 | e1 | -e2 | e134 | -e324 | -e4 | e12 | -e1234 | e41 | -e42 | -e142 |
| a.e4 | e4 | e41 | e42 | e43 | e | -e142 | -e134 | -e324 | e1 | e2 | e3 | -e1234 | -e12 | -e31 | -e23 | -e123 |
| a.e12 | e12 | e2 | e1 | e123 | -e142 | e | -e23 | -e31 | e42 | e41 | -e1234 | e3 | -e4 | -e324 | -e134 | -e43 |
| a.e31 | e31 | -e3 | e123 | -e1 | -e134 | e23 | e | e12 | -e43 | -e1234 | -e41 | e2 | e324 | -e4 | e142 | -e42 |
| a.e23 | e23 | e123 | -e3 | e2 | -e324 | e31 | -e12 | -e | -e1234 | -e43 | e42 | -e1 | e134 | -e142 | e4 | e41 |
| a.e41 | e41 | -e4 | -e142 | e134 | -e1 | -e42 | e43 | -e1234 | e | -e12 | e31 | e324 | -e2 | e3 | e123 | -e23 |
| a.e42 | e42 | e142 | e4 | -e324 | -e2 | -e41 | -e1234 | e43 | e12 | -e | -e23 | -e134 | -e1 | e123 | e3 | e31 |
| a.e43 | e43 | -e134 | e324 | e4 | -e3 | -e1234 | e41 | -e42 | -e31 | e23 | -e | -e142 | e123 | e1 | -e2 | e12 |
| a.e123 | e123 | -e23 | e31 | e12 | e1234 | e3 | e2 | -e1 | -e324 | e134 | e142 | e | e43 | e42 | -e41 | e4 |
| a.e142 | e142 | -e42 | -e41 | e1234 | -e12 | -e4 | -e324 | -e134 | -e2 | -e1 | -e123 | -e43 | e | -e23 | -e31 | e3 |
| a.e134 | e134 | e43 | e1234 | e41 | -e31 | e324 | -e4 | e142 | e3 | -e123 | e1 | -e42 | e23 | e | e12 | e2 |
| a.e324 | e324 | e1234 | e43 | -e42 | -e23 | e134 | -e142 | e4 | -e123 | e3 | -e2 | e41 | e31 | -e12 | -e | -e1 |
| a.e1234 | e1234 | -e324 | e134 | e142 | e123 | -e43 | -e42 | e41 | -e23 | e31 | e12 | -e4 | -e3 | -e2 | e1 | -e |
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1
analysing associativity: table associates
| a*b | b.e | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e1234 |
| a.e | e | e12 | e31 | e23 | e41 | e42 | e43 | e1234 |
| a.e12 | e12 | e | -e23 | -e31 | e42 | e41 | -e1234 | -e43 |
| a.e31 | e31 | e23 | e | e12 | -e43 | -e1234 | -e41 | -e42 |
| a.e23 | e23 | e31 | -e12 | -e | -e1234 | -e43 | e42 | e41 |
| a.e41 | e41 | -e42 | e43 | -e1234 | e | -e12 | e31 | -e23 |
| a.e42 | e42 | -e41 | -e1234 | e43 | e12 | -e | -e23 | e31 |
| a.e43 | e43 | -e1234 | e41 | -e42 | -e31 | e23 | -e | e12 |
| a.e1234 | e1234 | -e43 | -e42 | e41 | -e23 | e31 | e12 | -e |
analysing commutivity: table does not commute: for example: e12*e31 != e31*e12
analysing associativity: table associates
| a*b | b.e | b.e1 | b.e2 | b.e3 | b.e4 | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e123 | b.e142 | b.e134 | b.e324 | b.e1234 |
| a.e | e | e1 | e2 | e3 | e4 | e12 | e31 | e23 | e41 | e42 | e43 | e123 | e142 | e134 | e324 | e1234 |
| a.e1 | e1 | -e | e12 | -e31 | -e41 | -e2 | e3 | e123 | e4 | e142 | -e134 | -e23 | -e42 | e43 | -e1234 | e324 |
| a.e2 | e2 | -e12 | -e | e23 | -e42 | e1 | e123 | -e3 | -e142 | e4 | e324 | -e31 | e41 | -e1234 | -e43 | e134 |
| a.e3 | e3 | e31 | -e23 | e | -e43 | e123 | e1 | -e2 | e134 | -e324 | -e4 | e12 | -e1234 | e41 | -e42 | -e142 |
| a.e4 | e4 | e41 | e42 | e43 | e | -e142 | -e134 | -e324 | e1 | e2 | e3 | -e1234 | -e12 | -e31 | -e23 | -e123 |
| a.e12 | e12 | e2 | -e1 | e123 | -e142 | -e | -e23 | e31 | e42 | -e41 | -e1234 | -e3 | e4 | -e324 | e134 | e43 |
| a.e31 | e31 | -e3 | e123 | -e1 | -e134 | e23 | e | e12 | -e43 | -e1234 | -e41 | e2 | e324 | -e4 | e142 | -e42 |
| a.e23 | e23 | e123 | e3 | e2 | -e324 | -e31 | -e12 | e | -e1234 | e43 | e42 | e1 | -e134 | -e142 | -e4 | -e41 |
| a.e41 | e41 | -e4 | -e142 | e134 | -e1 | -e42 | e43 | -e1234 | e | -e12 | e31 | e324 | -e2 | e3 | e123 | -e23 |
| a.e42 | e42 | e142 | -e4 | -e324 | -e2 | e41 | -e1234 | -e43 | e12 | e | -e23 | e134 | e1 | e123 | -e3 | -e31 |
| a.e43 | e43 | -e134 | e324 | e4 | -e3 | -e1234 | e41 | -e42 | -e31 | e23 | -e | -e142 | e123 | e1 | -e2 | e12 |
| a.e123 | e123 | -e23 | -e31 | e12 | e1234 | -e3 | e2 | e1 | -e324 | -e134 | e142 | -e | -e43 | e42 | e41 | -e4 |
| a.e142 | e142 | -e42 | e41 | e1234 | -e12 | e4 | -e324 | e134 | -e2 | e1 | -e123 | e43 | -e | -e23 | e31 | -e3 |
| a.e134 | e134 | e43 | e1234 | e41 | -e31 | e324 | -e4 | e142 | e3 | -e123 | e1 | -e42 | e23 | e | e12 | e2 |
| a.e324 | e324 | e1234 | -e43 | -e42 | -e23 | -e134 | -e142 | -e4 | -e123 | -e3 | -e2 | -e41 | -e31 | -e12 | e | e1 |
| a.e1234 | e1234 | -e324 | -e134 | e142 | e123 | e43 | -e42 | -e41 | -e23 | -e31 | e12 | e4 | e3 | -e2 | -e1 | e |
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1
analysing associativity: table associates
| a*b | b.e | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e1234 |
| a.e | e | e12 | e31 | e23 | e41 | e42 | e43 | e1234 |
| a.e12 | e12 | -e | -e23 | e31 | e42 | -e41 | -e1234 | e43 |
| a.e31 | e31 | e23 | e | e12 | -e43 | -e1234 | -e41 | -e42 |
| a.e23 | e23 | -e31 | -e12 | e | -e1234 | e43 | e42 | -e41 |
| a.e41 | e41 | -e42 | e43 | -e1234 | e | -e12 | e31 | -e23 |
| a.e42 | e42 | e41 | -e1234 | -e43 | e12 | e | -e23 | -e31 |
| a.e43 | e43 | -e1234 | e41 | -e42 | -e31 | e23 | -e | e12 |
| a.e1234 | e1234 | e43 | -e42 | -e41 | -e23 | -e31 | e12 | e |
analysing commutivity: table does not commute: for example: e12*e31 != e31*e12
analysing associativity: table associates
| a*b | b.e | b.e1 | b.e2 | b.e3 | b.e4 | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e123 | b.e142 | b.e134 | b.e324 | b.e1234 |
| a.e | e | e1 | e2 | e3 | e4 | e12 | e31 | e23 | e41 | e42 | e43 | e123 | e142 | e134 | e324 | e1234 |
| a.e1 | e1 | -e | e12 | -e31 | -e41 | -e2 | e3 | e123 | e4 | e142 | -e134 | -e23 | -e42 | e43 | -e1234 | e324 |
| a.e2 | e2 | -e12 | -e | e23 | -e42 | e1 | e123 | -e3 | -e142 | e4 | e324 | -e31 | e41 | -e1234 | -e43 | e134 |
| a.e3 | e3 | e31 | -e23 | -e | -e43 | e123 | -e1 | e2 | e134 | -e324 | e4 | -e12 | -e1234 | -e41 | e42 | e142 |
| a.e4 | e4 | e41 | e42 | e43 | -e | -e142 | -e134 | -e324 | -e1 | -e2 | -e3 | -e1234 | e12 | e31 | e23 | e123 |
| a.e12 | e12 | e2 | -e1 | e123 | -e142 | -e | -e23 | e31 | e42 | -e41 | -e1234 | -e3 | e4 | -e324 | e134 | e43 |
| a.e31 | e31 | -e3 | e123 | e1 | -e134 | e23 | -e | -e12 | -e43 | -e1234 | e41 | -e2 | e324 | e4 | -e142 | e42 |
| a.e23 | e23 | e123 | e3 | -e2 | -e324 | -e31 | e12 | -e | -e1234 | e43 | -e42 | -e1 | -e134 | e142 | e4 | e41 |
| a.e41 | e41 | -e4 | -e142 | e134 | e1 | -e42 | e43 | -e1234 | -e | e12 | -e31 | e324 | e2 | -e3 | -e123 | e23 |
| a.e42 | e42 | e142 | -e4 | -e324 | e2 | e41 | -e1234 | -e43 | -e12 | -e | e23 | e134 | -e1 | -e123 | e3 | e31 |
| a.e43 | e43 | -e134 | e324 | -e4 | e3 | -e1234 | -e41 | e42 | e31 | -e23 | -e | e142 | -e123 | e1 | -e2 | e12 |
| a.e123 | e123 | -e23 | -e31 | -e12 | e1234 | -e3 | -e2 | -e1 | -e324 | -e134 | -e142 | e | -e43 | -e42 | -e41 | e4 |
| a.e142 | e142 | -e42 | e41 | e1234 | e12 | e4 | -e324 | e134 | e2 | -e1 | e123 | e43 | e | e23 | -e31 | e3 |
| a.e134 | e134 | e43 | e1234 | -e41 | e31 | e324 | e4 | -e142 | -e3 | e123 | e1 | e42 | -e23 | e | e12 | e2 |
| a.e324 | e324 | e1234 | -e43 | e42 | e23 | -e134 | e142 | e4 | e123 | e3 | -e2 | e41 | e31 | -e12 | e | e1 |
| a.e1234 | e1234 | -e324 | -e134 | -e142 | -e123 | e43 | e42 | e41 | e23 | e31 | e12 | -e4 | -e3 | -e2 | -e1 | e |
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1
analysing associativity: table associates
| a*b | b.e | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e1234 |
| a.e | e | e12 | e31 | e23 | e41 | e42 | e43 | e1234 |
| a.e12 | e12 | -e | -e23 | e31 | e42 | -e41 | -e1234 | e43 |
| a.e31 | e31 | e23 | -e | -e12 | -e43 | -e1234 | e41 | e42 |
| a.e23 | e23 | -e31 | e12 | -e | -e1234 | e43 | -e42 | e41 |
| a.e41 | e41 | -e42 | e43 | -e1234 | -e | e12 | -e31 | e23 |
| a.e42 | e42 | e41 | -e1234 | -e43 | -e12 | -e | e23 | e31 |
| a.e43 | e43 | -e1234 | -e41 | e42 | e31 | -e23 | -e | e12 |
| a.e1234 | e1234 | e43 | e42 | e41 | e23 | e31 | e12 | e |
analysing commutivity: table does not commute: for example: e12*e31 != e31*e12
analysing associativity: table associates
| a*b | b.e | b.e1 | b.e2 | b.e3 | b.e4 | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e123 | b.e142 | b.e134 | b.e324 | b.e1234 |
| a.e | e | e1 | e2 | e3 | e4 | e12 | e31 | e23 | e41 | e42 | e43 | e123 | e142 | e134 | e324 | e1234 |
| a.e1 | e1 | e | e12 | -e31 | -e41 | e2 | -e3 | e123 | -e4 | e142 | -e134 | e23 | e42 | -e43 | -e1234 | -e324 |
| a.e2 | e2 | -e12 | 0 | e23 | -e42 | 0 | e123 | 0 | -e142 | 0 | e324 | 0 | 0 | -e1234 | 0 | 0 |
| a.e3 | e3 | e31 | -e23 | e | -e43 | e123 | e1 | -e2 | e134 | -e324 | -e4 | e12 | -e1234 | e41 | -e42 | -e142 |
| a.e4 | e4 | e41 | e42 | e43 | e | -e142 | -e134 | -e324 | e1 | e2 | e3 | -e1234 | -e12 | -e31 | -e23 | -e123 |
| a.e12 | e12 | -e2 | 0 | e123 | -e142 | 0 | e23 | 0 | -e42 | 0 | -e1234 | 0 | 0 | e324 | 0 | 0 |
| a.e31 | e31 | e3 | e123 | -e1 | -e134 | -e23 | -e | e12 | e43 | -e1234 | -e41 | -e2 | -e324 | e4 | e142 | e42 |
| a.e23 | e23 | e123 | 0 | e2 | -e324 | 0 | -e12 | 0 | -e1234 | 0 | e42 | 0 | 0 | -e142 | 0 | 0 |
| a.e41 | e41 | e4 | -e142 | e134 | -e1 | e42 | -e43 | -e1234 | -e | -e12 | e31 | -e324 | e2 | -e3 | e123 | e23 |
| a.e42 | e42 | e142 | 0 | -e324 | -e2 | 0 | -e1234 | 0 | e12 | 0 | -e23 | 0 | 0 | e123 | 0 | 0 |
| a.e43 | e43 | -e134 | e324 | e4 | -e3 | -e1234 | e41 | -e42 | -e31 | e23 | -e | -e142 | e123 | e1 | -e2 | e12 |
| a.e123 | e123 | e23 | 0 | e12 | e1234 | 0 | -e2 | 0 | e324 | 0 | e142 | 0 | 0 | -e42 | 0 | 0 |
| a.e142 | e142 | e42 | 0 | e1234 | -e12 | 0 | e324 | 0 | e2 | 0 | -e123 | 0 | 0 | e23 | 0 | 0 |
| a.e134 | e134 | -e43 | e1234 | e41 | -e31 | -e324 | e4 | e142 | -e3 | -e123 | e1 | e42 | -e23 | -e | e12 | -e2 |
| a.e324 | e324 | e1234 | 0 | -e42 | -e23 | 0 | -e142 | 0 | -e123 | 0 | -e2 | 0 | 0 | -e12 | 0 | 0 |
| a.e1234 | e1234 | e324 | 0 | e142 | e123 | 0 | e42 | 0 | e23 | 0 | e12 | 0 | 0 | e2 | 0 | 0 |
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1
analysing associativity: table associates
| a*b | b.e | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e1234 |
| a.e | e | e12 | e31 | e23 | e41 | e42 | e43 | e1234 |
| a.e12 | e12 | 0 | e23 | 0 | -e42 | 0 | -e1234 | 0 |
| a.e31 | e31 | -e23 | -e | e12 | e43 | -e1234 | -e41 | e42 |
| a.e23 | e23 | 0 | -e12 | 0 | -e1234 | 0 | e42 | 0 |
| a.e41 | e41 | e42 | -e43 | -e1234 | -e | -e12 | e31 | e23 |
| a.e42 | e42 | 0 | -e1234 | 0 | e12 | 0 | -e23 | 0 |
| a.e43 | e43 | -e1234 | e41 | -e42 | -e31 | e23 | -e | e12 |
| a.e1234 | e1234 | 0 | e42 | 0 | e23 | 0 | e12 | 0 |
analysing commutivity: table does not commute: for example: e12*e31 != e31*e12
analysing associativity: table associates
| a*b | b.e | b.e1 | b.e2 | b.e3 | b.e4 | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e123 | b.e142 | b.e134 | b.e324 | b.e1234 |
| a.e | e | e1 | e2 | e3 | e4 | e12 | e31 | e23 | e41 | e42 | e43 | e123 | e142 | e134 | e324 | e1234 |
| a.e1 | e1 | -e | e12 | -e31 | -e41 | -e2 | e3 | e123 | e4 | e142 | -e134 | -e23 | -e42 | e43 | -e1234 | e324 |
| a.e2 | e2 | -e12 | 0 | e23 | -e42 | 0 | e123 | 0 | -e142 | 0 | e324 | 0 | 0 | -e1234 | 0 | 0 |
| a.e3 | e3 | e31 | -e23 | e | -e43 | e123 | e1 | -e2 | e134 | -e324 | -e4 | e12 | -e1234 | e41 | -e42 | -e142 |
| a.e4 | e4 | e41 | e42 | e43 | e | -e142 | -e134 | -e324 | e1 | e2 | e3 | -e1234 | -e12 | -e31 | -e23 | -e123 |
| a.e12 | e12 | e2 | 0 | e123 | -e142 | 0 | -e23 | 0 | e42 | 0 | -e1234 | 0 | 0 | -e324 | 0 | 0 |
| a.e31 | e31 | -e3 | e123 | -e1 | -e134 | e23 | e | e12 | -e43 | -e1234 | -e41 | e2 | e324 | -e4 | e142 | -e42 |
| a.e23 | e23 | e123 | 0 | e2 | -e324 | 0 | -e12 | 0 | -e1234 | 0 | e42 | 0 | 0 | -e142 | 0 | 0 |
| a.e41 | e41 | -e4 | -e142 | e134 | -e1 | -e42 | e43 | -e1234 | e | -e12 | e31 | e324 | -e2 | e3 | e123 | -e23 |
| a.e42 | e42 | e142 | 0 | -e324 | -e2 | 0 | -e1234 | 0 | e12 | 0 | -e23 | 0 | 0 | e123 | 0 | 0 |
| a.e43 | e43 | -e134 | e324 | e4 | -e3 | -e1234 | e41 | -e42 | -e31 | e23 | -e | -e142 | e123 | e1 | -e2 | e12 |
| a.e123 | e123 | -e23 | 0 | e12 | e1234 | 0 | e2 | 0 | -e324 | 0 | e142 | 0 | 0 | e42 | 0 | 0 |
| a.e142 | e142 | -e42 | 0 | e1234 | -e12 | 0 | -e324 | 0 | -e2 | 0 | -e123 | 0 | 0 | -e23 | 0 | 0 |
| a.e134 | e134 | e43 | e1234 | e41 | -e31 | e324 | -e4 | e142 | e3 | -e123 | e1 | -e42 | e23 | e | e12 | e2 |
| a.e324 | e324 | e1234 | 0 | -e42 | -e23 | 0 | -e142 | 0 | -e123 | 0 | -e2 | 0 | 0 | -e12 | 0 | 0 |
| a.e1234 | e1234 | -e324 | 0 | e142 | e123 | 0 | -e42 | 0 | -e23 | 0 | e12 | 0 | 0 | -e2 | 0 | 0 |
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1
analysing associativity: table associates
| a*b | b.e | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e1234 |
| a.e | e | e12 | e31 | e23 | e41 | e42 | e43 | e1234 |
| a.e12 | e12 | 0 | -e23 | 0 | e42 | 0 | -e1234 | 0 |
| a.e31 | e31 | e23 | e | e12 | -e43 | -e1234 | -e41 | -e42 |
| a.e23 | e23 | 0 | -e12 | 0 | -e1234 | 0 | e42 | 0 |
| a.e41 | e41 | -e42 | e43 | -e1234 | e | -e12 | e31 | -e23 |
| a.e42 | e42 | 0 | -e1234 | 0 | e12 | 0 | -e23 | 0 |
| a.e43 | e43 | -e1234 | e41 | -e42 | -e31 | e23 | -e | e12 |
| a.e1234 | e1234 | 0 | -e42 | 0 | -e23 | 0 | e12 | 0 |
analysing commutivity: table does not commute: for example: e12*e31 != e31*e12
analysing associativity: table associates
| a*b | b.e | b.e1 | b.e2 | b.e3 | b.e4 | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e123 | b.e142 | b.e134 | b.e324 | b.e1234 |
| a.e | e | e1 | e2 | e3 | e4 | e12 | e31 | e23 | e41 | e42 | e43 | e123 | e142 | e134 | e324 | e1234 |
| a.e1 | e1 | 0 | e12 | -e31 | -e41 | 0 | 0 | e123 | 0 | e142 | -e134 | 0 | 0 | 0 | -e1234 | 0 |
| a.e2 | e2 | -e12 | 0 | e23 | -e42 | 0 | e123 | 0 | -e142 | 0 | e324 | 0 | 0 | -e1234 | 0 | 0 |
| a.e3 | e3 | e31 | -e23 | e | -e43 | e123 | e1 | -e2 | e134 | -e324 | -e4 | e12 | -e1234 | e41 | -e42 | -e142 |
| a.e4 | e4 | e41 | e42 | e43 | e | -e142 | -e134 | -e324 | e1 | e2 | e3 | -e1234 | -e12 | -e31 | -e23 | -e123 |
| a.e12 | e12 | 0 | 0 | e123 | -e142 | 0 | 0 | 0 | 0 | 0 | -e1234 | 0 | 0 | 0 | 0 | 0 |
| a.e31 | e31 | 0 | e123 | -e1 | -e134 | 0 | 0 | e12 | 0 | -e1234 | -e41 | 0 | 0 | 0 | e142 | 0 |
| a.e23 | e23 | e123 | 0 | e2 | -e324 | 0 | -e12 | 0 | -e1234 | 0 | e42 | 0 | 0 | -e142 | 0 | 0 |
| a.e41 | e41 | 0 | -e142 | e134 | -e1 | 0 | 0 | -e1234 | 0 | -e12 | e31 | 0 | 0 | 0 | e123 | 0 |
| a.e42 | e42 | e142 | 0 | -e324 | -e2 | 0 | -e1234 | 0 | e12 | 0 | -e23 | 0 | 0 | e123 | 0 | 0 |
| a.e43 | e43 | -e134 | e324 | e4 | -e3 | -e1234 | e41 | -e42 | -e31 | e23 | -e | -e142 | e123 | e1 | -e2 | e12 |
| a.e123 | e123 | 0 | 0 | e12 | e1234 | 0 | 0 | 0 | 0 | 0 | e142 | 0 | 0 | 0 | 0 | 0 |
| a.e142 | e142 | 0 | 0 | e1234 | -e12 | 0 | 0 | 0 | 0 | 0 | -e123 | 0 | 0 | 0 | 0 | 0 |
| a.e134 | e134 | 0 | e1234 | e41 | -e31 | 0 | 0 | e142 | 0 | -e123 | e1 | 0 | 0 | 0 | e12 | 0 |
| a.e324 | e324 | e1234 | 0 | -e42 | -e23 | 0 | -e142 | 0 | -e123 | 0 | -e2 | 0 | 0 | -e12 | 0 | 0 |
| a.e1234 | e1234 | 0 | 0 | e142 | e123 | 0 | 0 | 0 | 0 | 0 | e12 | 0 | 0 | 0 | 0 | 0 |
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1
analysing associativity: table associates
| a*b | b.e | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e1234 |
| a.e | e | e12 | e31 | e23 | e41 | e42 | e43 | e1234 |
| a.e12 | e12 | 0 | 0 | 0 | 0 | 0 | -e1234 | 0 |
| a.e31 | e31 | 0 | 0 | e12 | 0 | -e1234 | -e41 | 0 |
| a.e23 | e23 | 0 | -e12 | 0 | -e1234 | 0 | e42 | 0 |
| a.e41 | e41 | 0 | 0 | -e1234 | 0 | -e12 | e31 | 0 |
| a.e42 | e42 | 0 | -e1234 | 0 | e12 | 0 | -e23 | 0 |
| a.e43 | e43 | -e1234 | e41 | -e42 | -e31 | e23 | -e | e12 |
| a.e1234 | e1234 | 0 | 0 | 0 | 0 | 0 | e12 | 0 |
analysing commutivity: table does not commute: for example: e31*e23 != e23*e31
analysing associativity: table associates
| a*b | b.e | b.e1 | b.e2 | b.e3 | b.e4 | b.e12 | b.e31 | b.e23 | b.e41 | b.e42 | b.e43 | b.e123 | b.e142 | b.e134 | b.e324 | b.e1234 |
| a.e | e | e1 | e2 | e3 | e4 | e12 | e31 | e23 | e41 | e42 | e43 | e123 | e142 | e134 | e324 | e1234 |
| a.e1 | e1 | 0 | e12 | -e31 | -e41 | 0 | 0 | e123 | 0 | e142 | -e134 | 0 | 0 | 0 | -e1234 | 0 |
| a.e2 | e2 | -e12 | 0 | e23 | -e42 | 0 | e123 | 0 | -e142 | 0 | e324 | 0 | 0 | -e1234 | 0 | 0 |
| a.e3 | e3 | e31 | -e23 | 0 | -e43 | e123 | 0 | 0 | e134 | -e324 | 0 | 0 | -e1234 | 0 | 0 | 0 |
| a.e4 | e4 | e41 | e42 | e43 | 0 | -e142 | -e134 | -e324 | 0 | 0 | 0 | -e1234 | 0 | 0 | 0 | 0 |
| a.e12 | e12 | 0 | 0 | e123 | -e142 | 0 | 0 | 0 | 0 | 0 | -e1234 | 0 | 0 | 0 | 0 | 0 |
| a.e31 | e31 | 0 | e123 | 0 | -e134 | 0 | 0 | 0 | 0 | -e1234 | 0 | 0 | 0 | 0 | 0 | 0 |
| a.e23 | e23 | e123 | 0 | 0 | -e324 | 0 | 0 | 0 | -e1234 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| a.e41 | e41 | 0 | -e142 | e134 | 0 | 0 | 0 | -e1234 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| a.e42 | e42 | e142 | 0 | -e324 | 0 | 0 | -e1234 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| a.e43 | e43 | -e134 | e324 | 0 | 0 | -e1234 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| a.e123 | e123 | 0 | 0 | 0 | e1234 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| a.e142 | e142 | 0 | 0 | e1234 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| a.e134 | e134 | 0 | e1234 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| a.e324 | e324 | e1234 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| a.e1234 | e1234 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1
analysing associativity: table associates
To produce the results the program needs to have an XML input code listed here:
<classDef>
<outputTable type="product" format="html" name="g 4,0,0" analyse="on" enableLabels="on">
<mathTypeMulti name="a" type="4" sign="0" zero="0" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 4,0,0" analyse="on" enableLabels="on">
<mathTypeMulti name="a" type="4" sign="0" zero="0" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 3,1,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="1" zero="0" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 3,1,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="1" zero="0" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 2,2,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="3" zero="0" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 2,2,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="3" zero="0" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 0,4,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="15" zero="0" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 0,4,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="15" zero="0" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 3,0,1" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="0" zero="2" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 3,0,1" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="0" zero="2" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 2,1,1" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="1" zero="2" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 2,1,1" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="1" zero="2" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 2,0,2" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="0" zero="3" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 2,0,2" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="0" zero="3" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 0,0,4" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="0" zero="15" subAlgebra="all"/>
</outputTable>
</classDef>
|
metadata block |
|
| see also: |
|
| Correspondence about this page | |
|
Book Shop - Further reading. Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them. |
|
|
Commercial Software Shop Where I can, I have put links to Amazon for commercial software, not directly related to the software project, but related to the subject being discussed, click on the appropriate country flag to get more details of the software or to buy it from them. |
|
Can you help? Please send me any improvements to here. I would appreciate ideas to make the pages more useful including error correction, ideas for new pages, improvements to wording. It helps if you quote the full URL of the page. |
|
|
Terminology and Notation Specific to this page here: |
|
|
program I am working on a project which uses these principles, if you would like to help me with this you are welcome to join in, here: |
|
This site may have errors. Don't use for critical systems.
Copyright (c) 1998-2008 Martin John Baker - All rights reserved - privacy policy.
