Difference between revisions of "Map"

From LifeWiki
Jump to navigation Jump to search
(→‎Examples of period maps: The 4- was not with the rest of the text)
 
(44 intermediate revisions by 5 users not shown)
Line 1: Line 1:
{{Glossary}}
''Not to be confused with MAP strings, a common method of notating [[non-isotropic cellular automata]].''
''Not to be confused with MAP strings, a common method of notating [[non-isotropic cellular automata]].''


A '''map''' is a visual diagram obtained from a pattern, generally the annotated [[envelope]] of an [[oscillator]], which reveals and highlights specific properties of said pattern.
A '''map''' is a visual diagram obtained from a pattern, generally the annotated [[envelope]] of an [[oscillator]], which reveals and highlights specific properties of said pattern.
== Envelope ==
The envelope of an oscillator is the simplest type of map, showing if a cell was alive during any given point in evolution. The rule [[LifeHistory]] was created specifically to view historical cells; programs such as [[LifeViewer]] have intrinsic functionality for viewing historical cells even for 2-state rules.


== Period map ==
== Period map ==
Line 8: Line 12:
Oscillator period maps generally reserve an important color such as white for cells that oscillate at the full period, and other colors are used for cells which may oscillate at subperiods.
Oscillator period maps generally reserve an important color such as white for cells that oscillate at the full period, and other colors are used for cells which may oscillate at subperiods.


"Period-1 cells" are marked with one of two possible colors: one color is used for [[stator]] cells which are permanently on, and another is used for cells which never turn on at all, i.e. "background" cells. This is an obviously useful distinction to make in rules such as [[Conway's Game of Life]], as the empty void around an oscillator is never worth analysing. However, in other cases such as a rule invariant under [[black-white reversal]] (e.g. [[Day and Night]]) being run on a [[bounded grid]], distinction between "always off" and "always on" cells is often less useful as they function identically to each other.
"Period-1 cells" are marked with one of two possible colors: one color is used for [[stator]] cells which are permanently on, and another is used for cells which never turn on at all, i.e. "background" cells. This is an obviously useful distinction to make in rules such as [[Conway's Game of Life]], as the empty void around an oscillator is never worth analysing. However, in other cases such as a rule invariant under [[black-white reversal]] (e.g. {{rl|Day & Night}}) being run on a [[bounded grid]], distinction between "always off" and "always on" cells is often less useful as they function identically to each other.


By looking at a period map, it is possible to derive information about an oscillator based on the map's appearance. If "background" cells are ignored:
By looking at a period map, it is possible to derive information about an oscillator based on the map's appearance. If "background" cells are ignored:
* A map consisting of only stator cells is a map of a [[still life]].
* A map consisting of only stator cells is a map of a [[still life]].
* A map which contains no stator cells is a map of an oscillator with a [[volatility]] of exactly 1.
* A map which contains no stator cells is a map of an oscillator with a [[volatility]] [[statorless|of exactly 1]].
** If all of the cells in the map are white (or whatever the "maximum period" color is, the [[strict volatility]] is also 1.
** If all of the cells in the map are white (or whatever the "maximum period" color is, the [[strict volatility]] is also 1.
* A map consisting of only stator cells and full-period cells is a [["volmatchstrict" stamp collection|"volmatchstrict" oscillator]] - i.e. its volatility and strict volatility are identical.
* A map consisting of only stator cells and full-period cells is a [["volmatchstrict" stamp collection|"volmatchstrict" oscillator]] - i.e. its volatility and strict volatility are identical.
* A map containing no full-period cells is that of a [[trivial]] oscillator.
* A map containing no full-period cells is that of a [[trivial]] oscillator.


In addition, highly-symmetric period maps often imply that the oscillator has a rich [[kinetic symmetry]]. However, this is not always the case, as there are many mechanisms through which time- and space-asymmetric oscillators can produce highly symmetric maps, often by repeatedly creating and destroying objects through their evolution cycle.
In addition, highly-symmetric period maps often imply that the oscillator has a rich [[kinetic symmetry]]. However, this is not always the case, as there are many mechanisms through which time- and space-asymmetric oscillators can produce highly symmetric maps, often by repeatedly creating and destroying objects through their evolution cycle (as in [[51P384]]).


=== Examples of period maps ===
=== Examples of period maps ===
{| class="wikitable" style="margin-left:auto; margin-right:auto; border:none"
{| class="wikitable" style="margin-left:auto; margin-right:auto; border:none"
! Pattern
! Pattern
Line 32: Line 35:
}}<br>[[Blinker]]
}}<br>[[Blinker]]
! [[File:Blinker period map.png]]
! [[File:Blinker period map.png]]
| style="background-color:#888888" | <span style=color:000000>█</span> p1 off<br><span style=color:#a8a8a8>█</span> p1 on<br><span style=color:#eeeeee>█</span> p2
| style="background-color:#888888" | <span style=color:#000000>█</span> p1 off<br><span style=color:#a8a8a8>█</span> p1 on<br><span style=color:#ffffff>█</span> p2
| A simple example: the periodic cells are shown in white, and the stator in gray.
| A simple example: the periodic cells are shown in white, and the stator in gray.
|-
|-
Line 40: Line 43:
}}<br>[[Tanner's p46]]
}}<br>[[Tanner's p46]]
! [[File:Tanner's p46 period map.png]]
! [[File:Tanner's p46 period map.png]]
| style="background-color:#888888" | <span style=color:000000>█</span> p1 off<br><span style=color:#a8a8a8>█</span> p1 on<br><span style=color:#eeeeee>█</span> p46
| style="background-color:#888888" | <span style=color:#000000>█</span> p1 off<br><span style=color:#a8a8a8>█</span> p1 on<br><span style=color:#ffffff>█</span> p46
| Much the same, but larger. All non-p1 cells oscillate at the full period.
| Much the same, but larger. All non-p1 cells oscillate at the full period.
|-
|-
Line 46: Line 49:
|pname        = 4blocktwinbeesshuttle
|pname        = 4blocktwinbeesshuttle
|viewerconfig = #C [[ AUTOSTART ZOOM 16 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
|viewerconfig = #C [[ AUTOSTART ZOOM 16 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
}}<br>[[Twin bees shuttle]]
}}<br>4-block [[twin bees shuttle]]
! [[File:4-block twin bees shuttle period map.png]]
! [[File:4-block twin bees shuttle period map.png]]
| style="background-color:#888888" | <span style=color:000000>█</span> p1 off<br><span style=color:#a8a8a8>█</span> p1 on<br><span style=color:#ff6666>█</span> p23<br><span style=color:#eeeeee>█</span> p46
| style="background-color:#888888" | <span style=color:#000000>█</span> p1 off<br><span style=color:#a8a8a8>█</span> p1 on<br><span style=color:#1c92cd>█</span> p23<br><span style=color:#ffffff>█</span> p46
| In this case, a "gutteroid" of period-23 cells exists in the middle due to the [[kinetic symmetry]] of the oscillator, which are marked accordingly.
| In this case, a "gutteroid" of period-23 cells exists in the middle due to the [[kinetic symmetry]] of the oscillator, which are marked accordingly.
|-
|-
Line 56: Line 59:
}}<br>[[Pentadecathlon]]
}}<br>[[Pentadecathlon]]
! [[File:Pentadecathlon period map.png]]
! [[File:Pentadecathlon period map.png]]
| style="background-color:#888888" | <span style=color:000000>█</span> p1 off<br><span style=color:#eeeeee>█</span> p15
| style="background-color:#888888" | <span style=color:#000000>█</span> p1 off<br><span style=color:#ffffff>█</span> p15
| The pentadecathlon has no stator cells and all cells oscillate at the full period. As such, all non-background cells are white.
| The pentadecathlon has no stator cells and all cells oscillate at the full period. As such, all non-background cells are white.
|-
|-
Line 64: Line 67:
}}<br>[[Cis-figure eight on pentadecathlon]]
}}<br>[[Cis-figure eight on pentadecathlon]]
! [[File:Cis-figure eight on pentadecathlon period map.png]]
! [[File:Cis-figure eight on pentadecathlon period map.png]]
| style="background-color:#888888" | <span style=color:000000>█</span> p1 off<br><span style=color:#ff6666>█</span> p8<br><span style=color:#29ffff>█</span> p15<br><span style=color:#eeeeee>█</span> p120
| style="background-color:#888888" | <span style=color:#000000>█</span> p1 off<br><span style=color:#0ab87b>█</span> p8<br><span style=color:#1c92cd>█</span> p15<br><span style=color:#ffffff>█</span> p120
| Neither the pentadecathlon nor figure eight have stator cells. They interact here to create one period-120 cell. The volatility is 1, but the strict volatility is very low.
| Neither the pentadecathlon nor figure eight have stator cells. They interact here to create one period-120 cell. The volatility is 1, but the strict volatility is very low.
|-
|-
Line 72: Line 75:
}}<br>[[Trivial p6-1]]
}}<br>[[Trivial p6-1]]
! [[File:Trivial p6-1 period map.png]]
! [[File:Trivial p6-1 period map.png]]
| style="background-color:#888888" | <span style=color:000000>█</span> p1 off<br><span style=color:#a8a8a8>█</span> p1 on<br><span style=color:#ff6666>█</span> p2<br><span style=color:#29ffff>█</span> p3
| style="background-color:#888888" | <span style=color:#000000>█</span> p1 off<br><span style=color:#a8a8a8>█</span> p1 on<br><span style=color:#0ab87b>█</span> p2<br><span style=color:#1c92cd>█</span> p3
| No cell in this period-6 oscillator actually oscillates at period 6, so no white cells are present.
| No cell in this period-6 oscillator actually oscillates at period 6, so no white cells are present.
|}
|}


=== Example colors ===
=== Example colors ===
The following table lists the cell colors which are used by the period maps [[LifeViewer]] produces. The colors of subperiod cells are generated on the fly (evenly-spaced points around the HSV wheel, with darkening modulo 4 for distinction).
The following table lists the cell colors which are used by the period maps LifeViewer produces. The colors of subperiod cells are generated on the fly (evenly-spaced points around the HSV wheel, with darkening modulo 4 for distinction). (until recently please update)
{| class="wikitable"
{| class="wikitable"
! colspan="2" | Color
! colspan="2" | Color
Line 90: Line 93:
| Period-1 cells which are always on, i.e. stator cells
| Period-1 cells which are always on, i.e. stator cells
|-
|-
| style="background-color:#EEEEEE" |
| style="background-color:#FFFFFF" |
| #EEEEEE
| #FFFFFF
| Cells which oscillate at the full period, i.e. cells in a period-n oscillator that oscillate at period n
| Cells which oscillate at the full period, i.e. cells in a period-n oscillator that oscillate at period n (period maps only)
|-
|-
| style="background-color:#808080" |
| style="background-color:#808080" |
Line 115: Line 118:
The [[Nakano]] utility, created in response to Oscillizer no longer working, is also capable of producing period maps.
The [[Nakano]] utility, created in response to Oscillizer no longer working, is also capable of producing period maps.


As of December 2022, LifeViewer's ''Identify'' functionality also produces period maps for oscillators up to a period of 4194304 (2<sup>22</sup>), provided sufficient memory is available; as of April 2023, all supported 2-state rulespaces can have period maps created.
Since December 2022, LifeViewer's ''Identify'' functionality also produces period maps for oscillators; from April 2023 onward, all supported 2-state rulespaces can have period maps created. From February 2025 onward, any period of oscillator can have a period map created, provided sufficient memory is available.


== Frequency map ==
== Frequency map ==
A '''frequency map''' of an oscillator displays the constituent cells of said oscillator as to highlight how often specific cells are active. For an oscillator of period n, a cell will be assigned a different color depending on how many of those n generations the cell spends in an alive state.
''Not to be confused with an object's [[Frequency class]], a measure of an object's predicted commonness in natural or seminatural soups''
 
A '''frequency map''' of an oscillator displays the constituent cells of said oscillator as to highlight how often specific cells are active. For an oscillator of period n, a cell will be assigned a different color depending on how many of those n generations the cell spends in an alive state. It is highly likely nontrivial oscillators which are "Omnifrequent" (i.e. at least one oscillator of every period n will have a frequency map covering every possible value from 1 until n) exist for all periods, though only periods 19 and 31 have no implied proof or explicit examples as of April {{year|2026}} (see omnifrequency project thread link below).


Unlike period maps, frequency maps distinguish between the two period-1 cell types by definition; a "background" cell which never comes alive will be assigned a value of 0, whereas a stator cell will be assigned a value of n. They do not, however, distinguish between subperiods; a period-2 cell in a period-4 oscillator will be classified identically to a period-4 cell which is on for two generations, then off for two generations.
Unlike period maps, frequency maps distinguish between the two period-1 cell types by definition; a "background" cell which never comes alive will be assigned a value of 0, whereas a stator cell will be assigned a value of n. They do not, however, distinguish between subperiods; a period-2 cell in a period-4 oscillator will be classified identically to a period-4 cell which is on for two generations, then off for two generations.


For low-period oscillators, frequency maps can be seen visually simply by running the oscillator at a high enough speed, since cells which are on less often will look more like the background color, whereas those on more often will look closer to the alive color. However, this ability quickly disintegrates with higher periods.
For low-period oscillators, frequency maps can be seen visually simply by running the oscillator at a high enough speed, since cells which are on less often will look more like the background color, whereas those on more often will look closer to the alive color. However, this ability quickly disintegrates with higher periods, especially on displays with lower refresh rates.


Frequency maps are a useful utility for finding potential [[sparker]]s; low-scoring cells at the edges of a pattern's [[envelope]] are often sparks which have utility in hassling other objects.
Frequency maps are a useful utility for finding potential [[sparker]]s; low-scoring cells at the edges of a pattern's [[envelope]] are often sparks which have utility in hassling other objects.
For periods 1 and 2, the frequency map of a given oscillator will be identical to its period map.


=== Examples of frequency maps ===
=== Examples of frequency maps ===
{| class="wikitable" style="margin-left:auto; margin-right:auto; border:none"
{| class="wikitable" style="margin-left:auto; margin-right:auto; border:none"
! Pattern
! Pattern
Line 141: Line 147:
! [[File:Caterer period map.png]]
! [[File:Caterer period map.png]]
! [[File:Caterer frequency map.png]]
! [[File:Caterer frequency map.png]]
| style="background-color:#888888" | <span style=color:000000>█</span> 0/3<br><span style=color:#ff6666>█</span> 1/3<br><span style=color:#29ffff>█</span> 2/3<br><span style=color:#a8a8a8>█</span> 3/3
| style="background-color:#888888" | <span style=color:#000000>█</span> 0/3<br><span style=color:#ff6666>█</span> 1/3<br><span style=color:#29ffff>█</span> 2/3<br><span style=color:#a8a8a8>█</span> 3/3
| As 3 is a prime number, no subperiods are possible other than 1. However, there are still two types of p3 cell, which are highlighted by the frequency map.
| As 3 is a prime number, no subperiods are possible other than 1. However, there are still two types of p3 cell, which are highlighted by the frequency map.
|-
|-
Line 150: Line 156:
! [[File:Mazing period map.png]]
! [[File:Mazing period map.png]]
! [[File:Mazing frequency map.png]]
! [[File:Mazing frequency map.png]]
| style="background-color:#888888" | <span style=color:000000>█</span> 0/4<br><span style=color:#ff6666>█</span> 1/4<br><span style=color:#29ff29>█</span> 2/4<br><span style=color:#0000eb>█</span> 3/4
| style="background-color:#888888" | <span style=color:#000000>█</span> 0/4<br><span style=color:#ff6666>█</span> 1/4<br><span style=color:#29ff29>█</span> 2/4<br><span style=color:#0000eb>█</span> 3/4
| Note how the period map does not distinguish between the types of period-4 cells, whereas the frequency map does not distinguish between p2 and p4 cells.
| Note how the period map does not distinguish between the types of period-4 cells, whereas the frequency map does not distinguish between p2 and p4 cells.
|-
|-
Line 159: Line 165:
! [[File:Pentadecathlon period map.png]]
! [[File:Pentadecathlon period map.png]]
! [[File:Pentadecathlon frequency map.png]]
! [[File:Pentadecathlon frequency map.png]]
| style="background-color:#888888" | <span style=color:000000>█</span> 0/15<br><span style=color:#ff6666>█</span> 1/15<br><span style=color:#ffb829>█</span> 2/15<br><span style=color:#9ceb00>█</span> 3/15<br><span style=color:#00ad00>█</span> 4/15<br><span style=color:#66ffcc>█</span> 5/15<br><span style=color:#29b8ff>█</span> 6/15<br><span style=color:#0000eb>█</span> 7/15<br><span style=color:#7400ad>█</span> 8/15<br><span style=color:#ff66cc>█</span> 9/15
| style="background-color:#888888" | <span style=color:#000000>█</span> 0/15<br><span style=color:#ff6666>█</span> 1/15<br><span style=color:#ffb829>█</span> 2/15<br><span style=color:#9ceb00>█</span> 3/15<br><span style=color:#00ad00>█</span> 4/15<br><span style=color:#66ffcc>█</span> 5/15<br><span style=color:#29b8ff>█</span> 6/15<br><span style=color:#0000eb>█</span> 7/15<br><span style=color:#7400ad>█</span> 8/15<br><span style=color:#ff66cc>█</span> 9/15
| The frequency map of a pentadecathlon reveals a complex interior structure which is completely impossible to see on a period map.
| The frequency map of a pentadecathlon reveals a complex interior structure which is completely impossible to see on a period map.
|}
|}


=== Support ===
=== Support ===
No known programs currently support the creation of frequency maps from input oscillators.
As of build 1250, LifeViewer can now create frequency maps for oscillators.
 
== Heat map ==
A '''heat map''' of an oscillator displays which cells in the oscillator change state the most in a period cycle. Cells which remain in the same state for the most time will be mapped to a lower value, and those which undergo the most changes will be mapped to a higher value.
 
For periods up to and including 4, assuming the rule is 2-state, the heat map of an oscillator will be identical to its period map, whereas for periods 5 and above they may differ.
 
In 2-state rules, cell heats can only be even; for 3 states and above, a cell can never have a heat of 1.
 
=== Examples of heat maps ===
{| class="wikitable" style="margin-left:auto; margin-right:auto; border:none"
! Pattern
! Period map
! Frequency map
! Heat map
! Key
! Description
|-
! {{EmbedViewer
|pname        = statorlessp5
|viewerconfig = #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
}}<br>[[Statorless p5]]
! [[File:Statorless p5 period map.png]]
! [[File:Statorless p5 frequency map.png]]
! [[File:Statorless p5 heat map.png]]
| style="background-color:#888888" | <span style=color:#66ff66>█</span> 2<br><span style=color:#ff6666>█</span> 4
| 5 is prime, so non-stator subperiods cannot exist, however there are two distinct cell heats present.<br>Note how cells of different heats can have the same frequency.
|}
 
=== Support ===
{{AS OF|September 2025}}, [[Oscilloscope]] is the only known program which creates heat maps.
 
== Signature map ==
A '''signature map''' of an oscillator is a generalized map of every constituent periodic cell's "signature" - i.e. the sequence of generations it is on versus off in. Cyclic permutations of each signature are classified as identical. A signature map can be considered as containing the information of 3both period maps, frequency maps and heat maps.
 
For periods 1, 2 and 3, assuming that the rule is 2-state, the signature map of an oscillator will be identical to its frequency map.
 
=== Examples of signature maps ===
 
{| class="wikitable" style="margin-left:auto; margin-right:auto; border:none"
! Pattern
! Period map
! Frequency map
! Signature map
! Frequency key
! Signature key
! Description
|-
! {{EmbedViewer
|pname        = mazing
|viewerconfig = #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
}}<br>[[Mazing]]
! [[File:Mazing period map.png]]
! [[File:Mazing frequency map.png]]
! [[File:Mazing signature map.png]]
| style="background-color:#888888" | <span style=color:#000000>█</span> 0/4<br><span style=color:#ff6666>█</span> 1/4<br><span style=color:#29ff29>█</span> 2/4<br><span style=color:#0000eb>█</span> 3/4
| style="background-color:#888888" | <span style=color:#000000>█</span> 0000<br><span style=color:#ff6666>█</span> 0001<br><span style=color:#29ff29>█</span> 0011<br><span style=color:#ddaa46>█</span> 0101<br><span style=color:#0000eb>█</span> 0111
| The signature map allows us to see the different frequencies for each cell, while also highlighting the two period-2 cells differently.
|-
! {{EmbedViewer
|pname        = fumarole
|viewerconfig = #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
}}<br>[[Fumarole]]
! [[File:Fumarole period map.png]]
! [[File:Fumarole frequency map.png]]
! [[File:Fumarole signature map.png]]
| style="background-color:#888888" | <span style=color:#000000>█</span> 0/5<br><span style=color:#ff6666>█</span> 1/5<br><span style=color:#94ff29>█</span> 2/5<br><span style=color:#00ebeb>█</span> 3/5<br><span style=color:#5700ad>█</span> 4/5<br><span style=color:#a8a8a8>█</span> 5/5
| style="background-color:#888888" | <span style=color:#000000>█</span> 00000<br><span style=color:#ff6666>█</span> 00001<br><span style=color:#94ff29>█</span> 00011<br><span style=color:#e0b000>█</span> 00101<br><span style=color:#00ebeb>█</span> 01011<br><span style=color:#5700ad>█</span> 01111<br><span style=color:#a8a8a8>█</span> 11111
| 5 is a prime number, containing no nontrivial subperiods as a result. We can still, however, use this map to distinguish cells of identical frequency.
|-
! {{EmbedViewer
|pname        = pseudobarberpole
|viewerconfig = #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
}}<br>[[Pseudo-barberpole]]
! [[File:Pseudo-barberpole period map.png]]
! [[File:Pseudo-barberpole frequency map.png]]
! [[File:Pseudo-barberpole signature map.png]]
| style="background-color:#888888" | <span style=color:#000000>█</span> 0/5<br><span style=color:#ff6666>█</span> 1/5<br><span style=color:#29ff29>█</span> 2/5<br><span style=color:#0000eb>█</span> 3/5<br><span style=color:#a8a8a8>█</span> 5/5
| style="background-color:#888888" | <span style=color:#000000>█</span> 00000<br><span style=color:#ff6666>█</span> 00001<br><span style=color:#29ff29>█</span> 00011<br><span style=color:#ddaa46>█</span> 00101<br><span style=color:#0000eb>█</span> 00111<br><span style=color:#6010c0>█</span> 01011<br><span style=color:#a8a8a8>█</span> 11111
| The pseudo-barberpole has cells with signature 00111, which the fumarole does not. It does not, however, have any frequency 4/5 cells.
|}
 
=== Support ===
As of v1.5.0 (October 3, 2025), [[Oscilloscope]] can produce signature maps of oscillators. [https://conwaylife.com/forums/viewtopic.php?p=149232#p149232 A script] by [[Paul Callahan]] can be used to output a textual summary for some objects.
 
== Phase coloring ==
Resemble a time-lapsed image of one period of an oscillator's progress: a spectrum of colors is lined up with the time axis, and the alive cells at a certain time are given a  certain color that matches their position on said color-time spectrum. This shows what activity is present in what areas at a given phase without the need to show individual phases or an animation.
 
== External Links ==
 
=== Forum threads ===
 
{{LinkForumThread|f=2|t=7026|title=The Omnifrequency Project}}

Latest revision as of 18:48, 14 April 2026

Not to be confused with MAP strings, a common method of notating non-isotropic cellular automata.

A map is a visual diagram obtained from a pattern, generally the annotated envelope of an oscillator, which reveals and highlights specific properties of said pattern.

Envelope

The envelope of an oscillator is the simplest type of map, showing if a cell was alive during any given point in evolution. The rule LifeHistory was created specifically to view historical cells; programs such as LifeViewer have intrinsic functionality for viewing historical cells even for 2-state rules.

Period map

A period map of an oscillator displays different constituent cells of said oscillator differently, usually in different colors, as to highlight which constituent cells oscillate at certain periods.

Oscillator period maps generally reserve an important color such as white for cells that oscillate at the full period, and other colors are used for cells which may oscillate at subperiods.

"Period-1 cells" are marked with one of two possible colors: one color is used for stator cells which are permanently on, and another is used for cells which never turn on at all, i.e. "background" cells. This is an obviously useful distinction to make in rules such as Conway's Game of Life, as the empty void around an oscillator is never worth analysing. However, in other cases such as a rule invariant under black-white reversal (e.g. Day & Night) being run on a bounded grid, distinction between "always off" and "always on" cells is often less useful as they function identically to each other.

By looking at a period map, it is possible to derive information about an oscillator based on the map's appearance. If "background" cells are ignored:

  • A map consisting of only stator cells is a map of a still life.
  • A map which contains no stator cells is a map of an oscillator with a volatility of exactly 1.
    • If all of the cells in the map are white (or whatever the "maximum period" color is, the strict volatility is also 1.
  • A map consisting of only stator cells and full-period cells is a "volmatchstrict" oscillator - i.e. its volatility and strict volatility are identical.
  • A map containing no full-period cells is that of a trivial oscillator.

In addition, highly-symmetric period maps often imply that the oscillator has a rich kinetic symmetry. However, this is not always the case, as there are many mechanisms through which time- and space-asymmetric oscillators can produce highly symmetric maps, often by repeatedly creating and destroying objects through their evolution cycle (as in 51P384).

Examples of period maps

Pattern Map Key Description
3o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 64 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

Blinker 		Blinker period map.png █ p1 off
█ p1 on
█ p2 		A simple example: the periodic cells are shown in white, and the stator in gray.
x = 13, y = 26, rule = B3/S23 2b2o9b$2bo10b$3bo9b$2b2o9b$13b$9b2o2b$9bo3b$10bo2b$9b2o2b$b2o10b$b2o6b 2o2b$o7bobo2b$b2o6bo3b$b2o7b3o$12bo$13b$13b$13b$13b$13b$13b$b2o10b$b2o 2b2o6b$5bobo5b$7bo5b$7b2o4b! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 16 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

Tanner's p46 		Tanner's p46 period map.png █ p1 off
█ p1 on
█ p46 		Much the same, but larger. All non-p1 cells oscillate at the full period.
x = 29, y = 11, rule = B3/S23 17b2o$2o15bobo7b2o$2o17bo7b2o$17b3o4$17b3o$2o17bo7b2o$2o15bobo7b2o$17b 2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 16 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

4-block twin bees shuttle 		4-block twin bees shuttle period map.png █ p1 off
█ p1 on
█ p23
█ p46 		In this case, a "gutteroid" of period-23 cells exists in the middle due to the kinetic symmetry of the oscillator, which are marked accordingly.
2bo4bo2b$2ob4ob2o$2bo4bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

Pentadecathlon 		Pentadecathlon period map.png █ p1 off
█ p15 		The pentadecathlon has no stator cells and all cells oscillate at the full period. As such, all non-background cells are white.
x = 14, y = 10, rule = B3/S23 bo6b2o$bo6b2obo$obo9bo$bo7bo$bo8bob2o$bo10b2o$bo$obo$bo$bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

Cis-figure eight on pentadecathlon 		Cis-figure eight on pentadecathlon period map.png █ p1 off
█ p8
█ p15
█ p120 		Neither the pentadecathlon nor figure eight have stator cells. They interact here to create one period-120 cell. The volatility is 1, but the strict volatility is very low.
x = 27, y = 18, rule = B3/S23 11b2o$11bo6b2o$8b2obo6bo2bo$8bobob2ob2obob2o2bo$bo8bo2bobo2bo2bobobo$b obo5b2o2bobobob2o2bo2bo$bobobo8b2obobob2ob2o$7bobob3o3bo4b2o$2bo3bo4bo b2o8bo$4bo3b3o2b2ob3obobob2o$2o2bo6bo10bobo$2bo11b2ob2o3bobo$8bob5o3bo 4bo$8b2o3bo2bo2bo$11b4ob3o$8b3o4bo$8bo2bo2bo2b2o$10b2o3b2obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 16 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

Trivial p6-1 		Trivial p6-1 period map.png █ p1 off
█ p1 on
█ p2
█ p3 		No cell in this period-6 oscillator actually oscillates at period 6, so no white cells are present.

Example colors

The following table lists the cell colors which are used by the period maps LifeViewer produces. The colors of subperiod cells are generated on the fly (evenly-spaced points around the HSV wheel, with darkening modulo 4 for distinction). (until recently please update)

Color Usage
#000000 Period-1 cells which are always off, i.e. are not "part" of the oscillator, but are still within its bounding box
#A8A8A8 Period-1 cells which are always on, i.e. stator cells
#FFFFFF Cells which oscillate at the full period, i.e. cells in a period-n oscillator that oscillate at period n (period maps only)
#808080 Used to mark cells at the edge of a bounded grid
#606060 Used to distinguish off-deathforcer cells from other cells that do not oscillate
All other colors Used for cells in an oscillator that do not oscillate at the full period, i.e. they oscillate at a subperiod
#505050 This color is reserved for grid lines, as well as the outer borders of very large maps

Support

The first well-known implementation of period maps was by Oscillizer, which allowed for these to be displayed for valid sufficiently-low-period oscillators specifically in Conway's Game of Life.

The Nakano utility, created in response to Oscillizer no longer working, is also capable of producing period maps.

Since December 2022, LifeViewer's Identify functionality also produces period maps for oscillators; from April 2023 onward, all supported 2-state rulespaces can have period maps created. From February 2025 onward, any period of oscillator can have a period map created, provided sufficient memory is available.

Frequency map

Not to be confused with an object's Frequency class, a measure of an object's predicted commonness in natural or seminatural soups

A frequency map of an oscillator displays the constituent cells of said oscillator as to highlight how often specific cells are active. For an oscillator of period n, a cell will be assigned a different color depending on how many of those n generations the cell spends in an alive state. It is highly likely nontrivial oscillators which are "Omnifrequent" (i.e. at least one oscillator of every period n will have a frequency map covering every possible value from 1 until n) exist for all periods, though only periods 19 and 31 have no implied proof or explicit examples as of April 2026 (see omnifrequency project thread link below).

Unlike period maps, frequency maps distinguish between the two period-1 cell types by definition; a "background" cell which never comes alive will be assigned a value of 0, whereas a stator cell will be assigned a value of n. They do not, however, distinguish between subperiods; a period-2 cell in a period-4 oscillator will be classified identically to a period-4 cell which is on for two generations, then off for two generations.

For low-period oscillators, frequency maps can be seen visually simply by running the oscillator at a high enough speed, since cells which are on less often will look more like the background color, whereas those on more often will look closer to the alive color. However, this ability quickly disintegrates with higher periods, especially on displays with lower refresh rates.

Frequency maps are a useful utility for finding potential sparkers; low-scoring cells at the edges of a pattern's envelope are often sparks which have utility in hassling other objects.

For periods 1 and 2, the frequency map of a given oscillator will be identical to its period map.

Examples of frequency maps

Pattern Period map Frequency map Key Description
2bo5b$o3b4o$o3bo3b$o7b$3bo4b$b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

Caterer 		Caterer period map.png Caterer frequency map.png █ 0/3
█ 1/3
█ 2/3
█ 3/3 		As 3 is a prime number, no subperiods are possible other than 1. However, there are still two types of p3 cell, which are highlighted by the frequency map.
3b2o2b$bobo3b$o5bo$bo3b2o2$3bobob$4bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

Mazing 		Mazing period map.png Mazing frequency map.png █ 0/4
█ 1/4
█ 2/4
█ 3/4 		Note how the period map does not distinguish between the types of period-4 cells, whereas the frequency map does not distinguish between p2 and p4 cells.
2bo4bo2b$2ob4ob2o$2bo4bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

Pentadecathlon 		Pentadecathlon period map.png Pentadecathlon frequency map.png █ 0/15
█ 1/15
█ 2/15
█ 3/15
█ 4/15
█ 5/15
█ 6/15
█ 7/15
█ 8/15
█ 9/15 		The frequency map of a pentadecathlon reveals a complex interior structure which is completely impossible to see on a period map.

Support

As of build 1250, LifeViewer can now create frequency maps for oscillators.

Heat map

A heat map of an oscillator displays which cells in the oscillator change state the most in a period cycle. Cells which remain in the same state for the most time will be mapped to a lower value, and those which undergo the most changes will be mapped to a higher value.

For periods up to and including 4, assuming the rule is 2-state, the heat map of an oscillator will be identical to its period map, whereas for periods 5 and above they may differ.

In 2-state rules, cell heats can only be even; for 3 states and above, a cell can never have a heat of 1.

Examples of heat maps

Pattern Period map Frequency map Heat map Key Description
x = 15, y = 14, rule = B3/S23 o13bo$b2o9b2o$o4b2ob2o4bo$2obo7bob2o$bobo2bobo2bobo$2bobo5bobo$2b3o5b 3o$2b3o5b3o$2bobo5bobo$bobo2bobo2bobo$2obo7bob2o$o4b2ob2o4bo$b2o9b2o$o 13bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

Statorless p5 		Statorless p5 period map.png Statorless p5 frequency map.png Statorless p5 heat map.png █ 2
█ 4 		5 is prime, so non-stator subperiods cannot exist, however there are two distinct cell heats present.
Note how cells of different heats can have the same frequency.

Support

As of September 2025, Oscilloscope is the only known program which creates heat maps.

Signature map

A signature map of an oscillator is a generalized map of every constituent periodic cell's "signature" - i.e. the sequence of generations it is on versus off in. Cyclic permutations of each signature are classified as identical. A signature map can be considered as containing the information of 3both period maps, frequency maps and heat maps.

For periods 1, 2 and 3, assuming that the rule is 2-state, the signature map of an oscillator will be identical to its frequency map.

Examples of signature maps

Pattern Period map Frequency map Signature map Frequency key Signature key Description
3b2o2b$bobo3b$o5bo$bo3b2o2$3bobob$4bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

Mazing 		Mazing period map.png Mazing frequency map.png Mazing signature map.png █ 0/4
█ 1/4
█ 2/4
█ 3/4 		█ 0000
█ 0001
█ 0011
█ 0101
█ 0111 		The signature map allows us to see the different frequencies for each cell, while also highlighting the two period-2 cells differently.
x = 8, y = 7, rule = B3/S23 2o4b2o$obo2bobo$2bo2bo$bo4bo$bo4bo$bo4bo$3b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

Fumarole 		Fumarole period map.png Fumarole frequency map.png Fumarole signature map.png █ 0/5
█ 1/5
█ 2/5
█ 3/5
█ 4/5
█ 5/5 		█ 00000
█ 00001
█ 00011
█ 00101
█ 01011
█ 01111
█ 11111 		5 is a prime number, containing no nontrivial subperiods as a result. We can still, however, use this map to distinguish cells of identical frequency.
x = 12, y = 12, rule = B3/S23 10b2o$11bo$9bo$7bobo2$5bobo2$3bobo2$2b2o$o$2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

Pseudo-barberpole 		Pseudo-barberpole period map.png Pseudo-barberpole frequency map.png Pseudo-barberpole signature map.png █ 0/5
█ 1/5
█ 2/5
█ 3/5
█ 5/5 		█ 00000
█ 00001
█ 00011
█ 00101
█ 00111
█ 01011
█ 11111 		The pseudo-barberpole has cells with signature 00111, which the fumarole does not. It does not, however, have any frequency 4/5 cells.

Support

As of v1.5.0 (October 3, 2025), Oscilloscope can produce signature maps of oscillators. A script by Paul Callahan can be used to output a textual summary for some objects.

Phase coloring

Resemble a time-lapsed image of one period of an oscillator's progress: a spectrum of colors is lined up with the time axis, and the alive cells at a certain time are given a certain color that matches their position on said color-time spectrum. This shows what activity is present in what areas at a given phase without the need to show individual phases or an animation.

External Links

Forum threads

Retrieved from "https://conwaylife.com/w/index.php?title=Map&oldid=172410"