Map

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 an oscillator, which reveals and highlights specific properties of said pattern.

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 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.

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.

Example colors

The following table lists the cell colors which are used by the period maps LifeViewer produces:

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
	#EEEEEE	Cells which oscillate at the full period, i.e. cells in a period-n oscillator that oscillate at period n
	#808080	Used to mark cells at the edge of a bounded grid
	#606060	Used to distinguish 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

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		█ 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		█ 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		█ 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$2bo4bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ AUTOSTART ZOOM 32 THUMBNAIL THUMBSIZE 4 WIDTH 600 HEIGHT 600 ]] (click above to open LifeViewer) RLE: here Plaintext: here Pentadecathlon		█ p1 off █ p15	The pentadecathlon has no stator cells and all cells oscillate at the full period. As such, all non-background cells are white.
x = 14, y = 10, rule = B3/S23 bo6b2o$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		█ 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		█ p1 off █ p1 on █ p2 █ p3	No cell in this period-6 oscillator actually oscillates at period 6, so no white cells are present.

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.

As of December 2022, LifeViewer's Identify functionality also produces period maps for oscillators up to a period of 4194304 (2²²), provided sufficient memory is available; as of April 2023, all supported 2-state rulespaces can have period maps created.