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.

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

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!)

3. Tempering the ultraparticular S''k''/S({{nowrap|''k'' + 1}}) along with either the corresponding 1/2-square-particular {{nowrap|S''k'' * S(''k'' + 1)}} or one of the two corresponding lopsided commas {{nowrap|S''k''2 * S(''k'' + 1)}} or {{nowrap|S''k'' * S(''k'' + 1)2}} implies tempering both of S''k'' and S({{nowrap|''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({{nowrap|''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, {{nowrap|{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.
+. Tempering any two consecutive square-particulars S''k'' and S({{nowrap|''k'' + 1}}) will naturally imply tempering the ultraparticular between them, {{sfrac|S''k''|S(''k'' + 1)}}, meaning they are very common implicit commas.
 . 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!)
+. Tempering the ultraparticular S''k''/S({{nowrap|''k'' + 1}}) along with either the corresponding 1/2-square-particular {{nowrap|S''k'' * S(''k'' + 1)}} or one of the two corresponding lopsided commas {{nowrap|S''k''<sup>2</sup> * S(''k'' + 1)}} or {{nowrap|S''k'' * S(''k'' + 1)<sup>2</sup>}} implies tempering both of S''k'' and S({{nowrap|''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({{nowrap|''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, {{nowrap|{S16, S17} &rarr; {{(}}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 ===