S-expression: Difference between revisions

(One intermediate revision by the same user not shown)

Line 3:

== Quick rules of S-expressions ==

[[File:S(expression).svg|thumb|Here is a summarized infographic of what many of these S-expression types do to a harmonic series segment when tempered out.]]

As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply when they are [[tempering out|tempered out]]. The linked sections provide deeper information into each comma family.

As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply is equated when they are [[tempering out|tempered out]]. The linked sections provide deeper information into each comma family.

* [[#Sk (square-particulars)|Square-particulars]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''2|''k''2 − 1}}. Tempering out S''k'' equates {{sfrac|''k'' + 1|''k''}} with {{sfrac|''k''|''k'' − 1}} and splits {{sfrac|''k'' + 1|''k'' − 1}} in two.

Line 10:

* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out splits {{sfrac|''k'' + 2|''k'' − 1}} into {{pars|{{sfrac|''k'' + 1|''k''}}}}3.

* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out splits {{sfrac|''k'' + 3|''k'' − 1}} into {{pars|{{sfrac|''k'' + 2|''k''}}}}2.

* [[#Ck and Cpk (cube-particulars)|Cube-particulars]]: '''C''k''''' and '''Cp''k''''', superparticular fractions of the form {{sfrac|''k''3|''k''3 − 1}} and {{sfrac|''k''3 + 1|''k''3}}, respectively.

== S''k'' (square-particulars) ==

Line 1,636:

=== Significance ===

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

# Tempering out any two consecutive ultraparticulars will imply tempering out the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]], which is their 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 S''k'' notation to achieve that compactness.~~{{clarify}}~~

# Tempering out any two consecutive ultraparticulars will imply tempering out the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]], which is their 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 S''k'' notation to achieve that compactness.

# Tempering out 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 {{nowrap|S(''k'' + 1)}}. This only gets better~~{{clarify}}~~ if you temper out a third consecutive square-particular. This is an abundance of "at-a-glance" essential tempering information that is fully general, so it only needs to be learned once, and is another 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.

# Tempering out 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 {{nowrap|S(''k'' + 1)}}. This only gets better if you temper out a third consecutive square-particular. This is an abundance of "at-a-glance" essential tempering information that is fully general, so it only needs to be learned once, and is another 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 ===

@@ Line 3: / Line 3: @@
 == Quick rules of S-expressions ==
 [[File:S(expression).svg|thumb|Here is a summarized infographic of what many of these S-expression types do to a harmonic series segment when tempered out.]]
-As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply when they are [[tempering out|tempered out]]. The linked sections provide deeper information into each comma family.
+As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply is equated when they are [[tempering out|tempered out]]. The linked sections provide deeper information into each comma family.
 * [[#Sk (square-particulars)|Square-particulars]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. <br>Tempering out S''k'' equates {{sfrac|''k'' + 1|''k''}} with {{sfrac|''k''|''k'' − 1}} and splits {{sfrac|''k'' + 1|''k'' − 1}} in two.
@@ Line 10: / Line 10: @@
 * [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out splits {{sfrac|''k'' + 2|''k'' − 1}} into {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.
 * [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out splits {{sfrac|''k'' + 3|''k'' − 1}} into {{pars|{{sfrac|''k'' + 2|''k''}}}}<sup>2</sup>.
 * [[#Ck and Cpk (cube-particulars)|Cube-particulars]]: '''C''k''''' and '''Cp''k''''', superparticular fractions of the form {{sfrac|''k''<sup>3</sup>|''k''<sup>3</sup> − 1}} and {{sfrac|''k''<sup>3</sup> + 1|''k''<sup>3</sup>}}, respectively.
 == S''k'' (square-particulars) ==
@@ Line 1,636: / Line 1,636: @@
 === Significance ===
 # Tempering out any two consecutive square-particulars S''k'' and S({{nowrap|''k'' + 1}}) will naturally imply tempering out the ultraparticular between them, {{sfrac|S''k''|S(''k'' + 1)}}, meaning they are very common implicit commas.
-# Tempering out any two consecutive ultraparticulars will imply tempering out the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]], which is their 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 S''k'' notation to achieve that compactness.{{clarify}}
+# Tempering out any two consecutive ultraparticulars will imply tempering out the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]], which is their 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 S''k'' notation to achieve that compactness.
-# Tempering out 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 {{nowrap|S(''k'' + 1)}}. This only gets better{{clarify}} if you temper out a third consecutive square-particular. This is an abundance of "at-a-glance" essential tempering information that is fully general, so it only needs to be learned once, and is another motivation of the use of S-expressions. For example, {{nowrap|{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 out 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 {{nowrap|S(''k'' + 1)}}. This only gets better if you temper out a third consecutive square-particular. This is an abundance of "at-a-glance" essential tempering information that is fully general, so it only needs to be learned once, and is another motivation of the use of S-expressions. For example, {{nowrap|{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 ===