User:Contribution/Successive superparticular complementary pair
For each pair of superparticular ratios [math]\displaystyle{ {s1}/{s2} }[/math] and [math]\displaystyle{ {s2}/{s3} }[/math], there exists a ratio [math]\displaystyle{ {a}/{b} }[/math] such that [math]\displaystyle{ {s1}/{s2} }[/math] and [math]\displaystyle{ {s2}/{s3} }[/math] are [math]\displaystyle{ {a}/{b} }[/math] complementary; it is observed that [math]\displaystyle{ a−b=1 }[/math] or [math]\displaystyle{ a−b=2 }[/math]. In other words, for each ratio [math]\displaystyle{ a/b }[/math] where [math]\displaystyle{ a−b=1 }[/math] or [math]\displaystyle{ a−b=2 }[/math], there exists a pair of superparticular ratios [math]\displaystyle{ {s1}/{s2} }[/math] and [math]\displaystyle{ {s2}/{s3} }[/math] that are [math]\displaystyle{ {a}/{b} }[/math] complementary.
Bellow is a table that show for equal divisions of [math]\displaystyle{ a/b }[/math] the cent error in the mapping of superparticular ratios [math]\displaystyle{ {s1}/{s2} }[/math] and [math]\displaystyle{ {s2}/{s3} }[/math] that are [math]\displaystyle{ a/b }[/math] complementary.
We can observe a converging sequence and pattern for low errors: 5, 7, 12; then 7, 9, 16; then 9, 11, 20; then 11, 13, 24; then 13, 15, 28; then 15, 17, 32; then 17, 19, 36; then 19, 21, 40; then 21, 23, 44; etc. --
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Todo: Table
Show the table for 3/1, 2/1, 5/3, 3/2, 7/5, 4/3, 9/7, 5/4, 11/9, 6/5. Highlight the pattern in bold. |
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Todo: Finish the article and move
When the article is finished, move it to the main root |