Kite's thoughts on pergens: Difference between revisions
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===Simplifying a "squared" EI=== | ===Simplifying a "squared" EI=== | ||
Consider an uninflected EI of AA1. AA1 is "squared" in the sense that AA1 = A1 + A1, thus both numbers in its ratio are square numbers. If inflected by an even number of arrows, it would obviously be an invalid EI, for if v<sup>4</sup>AA1 = 0¢, then so does vvA1, and v<sup>4</sup>AA1 could be replaced with vvA1. So the number of arrows must be odd. | Consider an uninflected EI of AA1. AA1 is "squared" in the sense that AA1 = A1 + A1, thus both numbers in its ratio are square numbers. If inflected by an even number of arrows, it would obviously be an invalid EI, for if v<sup>4</sup>AA1 = 0¢, then so does vvA1, and v<sup>4</sup>AA1 could be replaced with vvA1. So the number of arrows must be odd. | ||
Consider an EI of v<sup>3</sup>AA1. The pergen is (P8, P4/3). Here are the [[twin squares]]. | Consider an EI of v<sup>3</sup>AA1. The pergen is (P8, P4/3). Here are the [[twin squares]]. | ||
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</math> | </math> | ||
Note that while the EI has become simpler, the generator has become more complex in that it is dup not up. This is remedied by adding the EI to it. Changes are in red: | |||
<math> | |||
\begin{array} {rrr} | |||
P8 \\ | |||
downmajor 2nd \\ | |||
vvvA1 \\ | |||
\end{array} | |||
\left[ \begin{array} {rrr} | |||
1 & 0 & 0 \\ | |||
{\color {Red}-3} & {\color {Red}2} & {\color {Red}-1} \\ | |||
\hline | |||
-11 & 7 & -3 \\ | |||
\end{array} \right] | |||
\longleftrightarrow | |||
\left[ \begin{array} {rrr} | |||
1 & 2 & 1 \\ | |||
0 & -3 & {\color {Red}-7} \\ | |||
\hline | |||
{\color {Red}0} & {\color {Red}1} & {\color {Red}2} \\ | |||
\end{array} \right] | |||
</math> | |||
Following this procedure, it's always possible to simplify a squared (or cubed, etc.) EI. | |||
===Arrow commas=== | ===Arrow commas=== | ||