Douglas Blumeyer's RTT How-To: Difference between revisions
Cmloegcmluin (talk | contribs) m →multi(co)vectors: improve formatting |
Cmloegcmluin (talk | contribs) m →multivectors: consolidating table |
||
| Line 1,153: | Line 1,153: | ||
An important observation to make about multivectors and multicovectors is that — for a given temperament — they are always the same length. This may surprise you, since the rank and nullity for a temperament are often different, and the length of the multicovector comes from the rank while the length of the multivector comes from the nullity. But there’s a simple explanation for this. In either case, the length is not directly equal to the rank or nullity, but to the dimensionality choose the rank or nullity. And there’s a pattern to combinations that can be visualized in the symmetry of rows of Pascal’s triangle: <span><math>{d \choose n}</math></span> is always equal to <span><math>{d \choose {d - n}}</math></span>, or in other words, <span><math>{d \choose n}</math></span> is always equal to <span><math>{d \choose r}</math></span>. Here are some examples: | An important observation to make about multivectors and multicovectors is that — for a given temperament — they are always the same length. This may surprise you, since the rank and nullity for a temperament are often different, and the length of the multicovector comes from the rank while the length of the multivector comes from the nullity. But there’s a simple explanation for this. In either case, the length is not directly equal to the rank or nullity, but to the dimensionality choose the rank or nullity. And there’s a pattern to combinations that can be visualized in the symmetry of rows of Pascal’s triangle: <span><math>{d \choose n}</math></span> is always equal to <span><math>{d \choose {d - n}}</math></span>, or in other words, <span><math>{d \choose n}</math></span> is always equal to <span><math>{d \choose r}</math></span>. Here are some examples: | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ '''Table 5a.''' Multi(co)vector prime combinations | |+'''Table 5a.''' Multi(co)vector prime combinations | ||
! | ![math]d[/math] | ||
!<math>{d \choose r}</math> | ![math][r or n][/math] | ||
!<math>{d \choose n}</math> | ![math]d - [r or n][/math] | ||
!<math>{d \choose {[r or n]}}</math> | |||
!<math>{d \choose {d - [r or n]}}</math> | |||
!count | |||
|- | |- | ||
|3 | |||
|2 | |||
|1 | |||
|(2,3) (2,5) (3,5) | |(2,3) (2,5) (3,5) | ||
|(2) (3) (5) | |(2) (3) (5) | ||
|3 | |||
|- | |- | ||
| | |4 | ||
|3 | |3 | ||
| | |1 | ||
|(2,3,5) (2,3,7) (2,5,7) (3,5,7) | |(2,3,5) (2,3,7) (2,5,7) (3,5,7) | ||
|(2) (3) (5) (7) | |(2) (3) (5) (7) | ||
|4 | |4 | ||
|- | |- | ||
|5 | |||
|3 | |||
|2 | |||
| | |||
|(2,3,5) (2,3,7) (2,3,11) (2,5,7) (2,5,11) (2,7,11) (3,5,7) (3,5,11) (3,7,11) (5,7,11) | |(2,3,5) (2,3,7) (2,3,11) (2,5,7) (2,5,11) (2,7,11) (3,5,7) (3,5,11) (3,7,11) (5,7,11) | ||
|(2,3) (2,5) (2,7) (2,11) (3,5), (3,7) (3,11) (5,7) (5,11) (7,11) | |(2,3) (2,5) (2,7) (2,11) (3,5), (3,7) (3,11) (5,7) (5,11) (7,11) | ||
|10 | |10 | ||
|} | |} | ||
Each set of one side corresponds to a set in the other side which has the exact opposite elements. | Each set of one side corresponds to a set in the other side which has the exact opposite elements. | ||