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{{Glossary}}
{{Glossary}}
'''Period multiplier''' is a term commonly used for a [[pulse divider]], because dividing the number of [[signal]]s in a regular stream by ''N'' necessarily multiplies the [[period]] by ''N''. The term "period multiplier" can be somewhat misleading in this context, because most such circuits can accept input streams that are not strictly [[periodic]].
A '''period multiplier''' is a [[conduit]] that can accept an uniformly spaced [[period]]-''p'' [[stream]] (typically consisting of [[glider]]s, [[spaceship]]s or [[Herschel]]s) and produce an output stream with period ''pn'', for some positive integer {{nowrap|''n'' > 1}}.


Reactions have also been found to period double or period triple the output of some [[rake]]s to create high-period rakes in a relatively small space (i.e., an exponential increase in period for a linear increase in size).
The term "period multiplier" is often used for a [[pulse divider]], because dividing the number of [[signal]]s in a regular stream by ''n'' necessarily multiplies the [[period]] by ''n''. The term "period multiplier" can be somewhat misleading in this context, because most such circuits can accept input streams where the signals are not uniformly spaced.


For [[Herschel]] signals and [[glider gun]]s, a number of small period doubler, tripler, and quadrupler mechanisms are known. For example, the following [[conduit]] produces one output [[glider]] after accepting four input [[B-heptomino]]es, or four Herschels if a conduit such as [[F117]] is prepended that includes the same [[BFx59H]] converter.
For [[Herschel]] signals and [[glider gun]]s, a number of small [[Stable conduit|stable]] 2×, 3×, and 4× [[Pulse divider|pulse-dividing]] mechanisms are known. For example, the first one of the following [[conduit]]s produces one output [[glider]] after accepting four input Herschels from [[B-heptomino]]es passing a [[BFx59H]] converter. For a period-doubling conduit, see [[L156]].


{{EmbedViewer
{| style="margin:auto; width:300px"
|{{EmbedViewer
|position    = center
|pname        = nw31periodquadrupler
|caption      = An [[NW31]] 4× pulse divider
|viewerconfig = #C [[ THUMBSIZE 2 WIDTH 640 HEIGHT 500 ]]
|}}
|-
|{{EmbedViewer
|position    = center
|position    = center
|pname        = periodmultiplier
|pname        = l112periodquadrupler
|caption      = An NW31 4x period multiplier.
|caption      = An [[L112]] 4× pulse divider
|viewerconfig = #C [[ THUMBSIZE 2 WIDTH 640 HEIGHT 640 ]]
|viewerconfig = #C [[ THUMBSIZE 2 WIDTH 640 HEIGHT 500 ZOOM 8 ]]
}}
|}}
{{EmbedViewer
|-
|{{EmbedViewer
|position    = center
|position    = center
|pname        = l112pm
|pname        = fx70periodquadrupler
|caption      = An L112 4x period multiplier.
|caption      = An Fx70 4× pulse divider requiring a subsequent [[dependent conduit]]. It can be turned to a non-period multiplying conduit with [[Jubjub reflector]].
|viewerconfig = #C [[ THUMBSIZE 2 WIDTH 640 HEIGHT 640 ]]
|viewerconfig = #C [[ THUMBSIZE 2 WIDTH 640 HEIGHT 500 ]]
}}
|}}
|}
 
Additional examples for [[glider stream]]s are [[semi-Snark]], [[tremi-Snark]], [[quadri-Snark]],  [https://conwaylife.com/forums/viewtopic.php?p=167612#p167612 unnamed x5 reflector], [https://conwaylife.com/forums/viewtopic.php?p=167642#p167642 unnamed x12 reflector], [https://conwaylife.com/forums/viewtopic.php?p=167615#p167615 unnamed x11 and x(6n+4) reflectors], and [[crystal]]-based devices requiring same input [[parity]].<ref name="post121838" /> {{AS OF|October 2023}}, stable and elementary
reactions have also been found to double or triple the output period of some [[rake]]s in a relatively small space, enabling an exponential increase in period for a linear increase in size.<ref>{{filename|./c2-extended/var-exp*.lif}} in {{CiteSummersPattern}}</ref>
 
"Period multiplication" may also refer to a technique of producing a stream of period ''pn'' from two or more input streams with periods ''p'' (i.e. colliding glider streams), or a way of stabilizing an [[engine]] so that the period of the reaction is multiplied by a positive integer {{nowrap|''n'' > 1}}, for example the usage of a [[half-bakery]]/[[Ship-tie]] to restore a [[ship]].
 
{{EmbedViewer
{{EmbedViewer
|pname        = p38honeyfarmhassler6
|viewerconfig = #C [[ THUMBSIZE 2 ZOOM 12 HEIGHT 480 WIDTH 480 GPS 19 THEME Book AUTOSTART STARTFROM 38 ]]
|position    = center
|position    = center
|pname        = fx70pm
|caption      = p38, found by [[Period1GliderGun]] on February 22, {{year|2024}}, based on a partial by iNoMed.<ref name="post178793"/> The honey farm regions oscillate with period 19, but this stabilization doubles the period. Nevertheless, a variant, [[Goban]] is known that is period-19.
|caption      = An Fx70 4x period multiplier.
|style        = width:300px;
|viewerconfig = #C [[ THUMBSIZE 2 WIDTH 640 HEIGHT 640 ]]
|apgcode      = xp38_g8o653y3ok46wj9q4og4cz11x653y81y3oge2zxg8ox4357y47561zw23xg08o0ok8y08oy1653wokczy311w1246w311y4653
}}
}}


Also see the [[L156]] for a period-doubling one.
== See also ==
* [[Pulse divider]]
* [[Filter]]
* [[Toggle circuit]]
* [[LCM oscillator]]


See [[semi-Snark]] and [[tremi-Snark]] for additional examples using [[glider stream]]s. As of the end of {{year|2017}} no [[stable]] period-multiplying [[elementary conduit]]s are known for a multiplication factor of five or higher, though it is easy to construct [[composite conduit|composite]] ones.
==References==
<references>
<ref name="post121838">{{LinkForumThread
|format = ref
|p      = 121838
|title  = Re: Crystal Programming
|author = Sphenocorona
|date  = February 5, 2021
}}</ref>
<ref name="post178793">{{LinkForumThread
|format = ref
|p      = 178793
|title  = Re: Oscillator Discussion Thread
|author = Anivec
|date  = February 23, 2024
}}</ref>


==External links==
</references>
{{LinkLexicon|lex_p.htm#periodmultiplier}}
 
== External links ==
* {{LinkLexicon|lex_p.htm#periodmultiplier}}
* {{LinkForumThread|f=2|t=1097|title=I'm looking for period multiplicating Herschel Tracks.}}
* {{cite web
|url  = https://conwaylife.com/forums/search.php?keywords=period+multiplier
|title = period multiplier
|work  = ConwayLife.com
}}

Latest revision as of 22:12, 13 September 2025

A period multiplier is a conduit that can accept an uniformly spaced period-p stream (typically consisting of gliders, spaceships or Herschels) and produce an output stream with period pn, for some positive integer n > 1.

The term "period multiplier" is often used for a pulse divider, because dividing the number of signals in a regular stream by n necessarily multiplies the period by n. The term "period multiplier" can be somewhat misleading in this context, because most such circuits can accept input streams where the signals are not uniformly spaced.

For Herschel signals and glider guns, a number of small stable 2×, 3×, and 4× pulse-dividing mechanisms are known. For example, the first one of the following conduits produces one output glider after accepting four input Herschels from B-heptominoes passing a BFx59H converter. For a period-doubling conduit, see L156.

#C Period multiplier example based on NW31 x = 44, y = 29, rule = B3/S23 19bo$19b3o$22bo$11b2o8b2o$12bo$12bobo$b2o10b2o$2bo$bo$b2o3$42b2o$42b2o 2$o3b2o$2o2b2o$b2o$2o9$32b2o$32b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 WIDTH 640 HEIGHT 500 ]]
An NW31 4× pulse divider
(click above to open LifeViewer)
RLE: here Plaintext: here
x = 43, y = 49, rule = B3/S23 34b2o$34bo$32b3o16$32b2o$23b2o7b2o$24bo$24bobo$19b2o4b2o$b2o17bo20b2o$ 2bo17bobo18bo$bo19b2o16bobo$b2o36b2o6$o3b2o$2o2b2o$b2o$2o2$35b2o$35bo$ 36b3o$38bo5$21b2o$22bo$21bo$21b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 WIDTH 640 HEIGHT 500 ZOOM 8 ]]
An L112 4× pulse divider
(click above to open LifeViewer)
RLE: here Plaintext: here
x = 44, y = 27, rule = B3/S23 20bo$19bobo$20bo2$18b5o$17bo4bo$16bo2bo$13bo2bob2o13b2o$b2o9bobobo5bo 10b2o$2bo10bo2bo4bobo$bo14b2o2bo2bo$b2o18b2o2$40bob2o$38b3ob2o$38bo$ 38bo$o3b2o$2o2b2o$b2o$2o5$33b2o$33b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 WIDTH 640 HEIGHT 500 ]]
An Fx70 4× pulse divider requiring a subsequent dependent conduit. It can be turned to a non-period multiplying conduit with Jubjub reflector.
(click above to open LifeViewer)
RLE: here Plaintext: here

Additional examples for glider streams are semi-Snark, tremi-Snark, quadri-Snark, unnamed x5 reflector, unnamed x12 reflector, unnamed x11 and x(6n+4) reflectors, and crystal-based devices requiring same input parity.[1] As of October 2023, stable and elementary reactions have also been found to double or triple the output period of some rakes in a relatively small space, enabling an exponential increase in period for a linear increase in size.[2]

"Period multiplication" may also refer to a technique of producing a stream of period pn from two or more input streams with periods p (i.e. colliding glider streams), or a way of stabilizing an engine so that the period of the reaction is multiplied by a positive integer n > 1, for example the usage of a half-bakery/Ship-tie to restore a ship.

x = 34, y = 23, rule = B3/S23 4b2o13b2o$3bobo10bo2bobo$3b2o9b3o5bo2b2o$b2o10bo6b2obo2bo$obo10b2o4bo bob2o$2o4b2o12bo$5bobo22b2o$5b2o23bo$10b3o8b3o4bobo$9bo3bo6bo3bo3b2o$ 8bo5bo4bo5bo$8bo5bo4bo5bo$8bo5bo4bo5bo$4b2o3bo3bo6bo3bo$3bobo4b3o8b3o $3bo23b2o$2b2o22bobo$13bo12b2o4b2o$9b2obobo4b2o10bobo$7bo2bob2o6bo10b 2o$7b2o2bo5b3o9b2o$12bobo2bo10bobo$13b2o13b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 12 HEIGHT 480 WIDTH 480 GPS 19 THEME Book AUTOSTART STARTFROM 38 ]]
p38, found by Period1GliderGun on February 22, 2024, based on a partial by iNoMed.[3] The honey farm regions oscillate with period 19, but this stabilization doubles the period. Nevertheless, a variant, Goban is known that is period-19.
(click above to open LifeViewer)
RLE: here Plaintext: here
Catagolue: here

See also

References

  1. ↑ Sphenocorona (February 5, 2021). Re: Crystal Programming (discussion thread) at the ConwayLife.com forums
  2. ↑ ./c2-extended/var-exp*.lif in Jason Summers' pattern collections.
  3. ↑ Anivec (February 23, 2024). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
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