S-expression: Difference between revisions

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=== Significance ===

1. Tempering any two consecutive square-particulars S''k'' and S(''k''+1) will naturally imply tempering the ultraparticular between them (S''k''/S(''k''+1)), meaning they are very common implicit commas.

Note that tempering any two consecutive square-particulars S''k'' and S(''k''+1) will naturally imply tempering the ultraparticular between them (S''k''/S(''k''+1)) (meaning they are very common implicit commas), and that tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness.

2. Tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness.

3. Tempering the ultraparticular S''k''/S(''k''+1) along with either the corresponding 1/2-square-particular S''k'' * S(''k''+1) or one of the two corresponding lopsided commas S''k''2 * S(''k''+1) or S''k'' * S(''k''+1)2 implies tempering both of S''k'' and S(''k''+1) individually, and vice versa, so that there is a total of ''five'' equivalences — corresponding to ''five'' infinite families of commas — for every such S''k'' and S(''k''+1). This only gets better if you temper a third consecutive square-particular. This is an abundance of "at a glance" essential tempering information that is fully general so only needs to be learned once, and is the motivation of the use of '''S-expressions'''. (For example, {S16, S17} => {S16 * S17, S16/S17, S162 * S17, S16 * S172}, and any of the two commas in the latter set imply all the other commas too!)

=== Table of ultraparticulars ===

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 === Significance ===
+. Tempering any two consecutive square-particulars S''k'' and S(''k''+1) will naturally imply tempering the ultraparticular between them (S''k''/S(''k''+1)), meaning they are very common implicit commas.
-Note that tempering any two consecutive square-particulars S''k'' and S(''k''+1) will naturally imply tempering the ultraparticular between them (S''k''/S(''k''+1)) (meaning they are very common implicit commas), and that tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness.
+. Tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness.
+. Tempering the ultraparticular S''k''/S(''k''+1) along with either the corresponding 1/2-square-particular S''k'' * S(''k''+1) or one of the two corresponding lopsided commas S''k''<sup>2</sup> * S(''k''+1) or S''k'' * S(''k''+1)<sup>2</sup> implies tempering both of S''k'' and S(''k''+1) individually, and vice versa, so that there is a total of ''five'' equivalences — corresponding to ''five'' infinite families of commas — for every such S''k'' and S(''k''+1). This only gets better if you temper a third consecutive square-particular. This is an abundance of "at a glance" essential tempering information that is fully general so only needs to be learned once, and is the motivation of the use of '''S-expressions'''. (For example, {S16, S17} => {S16 * S17, S16/S17, S16<sup>2</sup> * S17, S16 * S17<sup>2</sup>}, and any of the two commas in the latter set imply all the other commas too!)
 === Table of ultraparticulars ===