Difference between revisions of "Map"
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=== Support === | === Support === | ||
As of build 1250, LifeViewer can now create frequency maps for oscillators. | |||
== Signature map == | == Signature map == | ||
Revision as of 17:46, 21 February 2025
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 |
|---|---|---|---|
Blinker |
|
█ p1 off █ p1 on █ p2 |
A simple example: the periodic cells are shown in white, and the stator in gray. |
Tanner's p46 |
|
█ p1 off █ p1 on █ p46 |
Much the same, but larger. All non-p1 cells oscillate at the full period. |
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. |
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. |
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. |
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. |
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).
| 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 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.
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.
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.
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 |
|---|---|---|---|---|
Caterer |
|
|
█ 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. |
Mazing |
|
|
█ 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. |
Pentadecathlon |
|
|
█ 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.
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 both period maps and frequency 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 |
|---|---|---|---|---|---|---|
Mazing |
|
|
|
█ 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. |
Fumarole |
|
|
|
█ 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. |
Pseudo-barberpole |
|
|
|
█ 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
No known programs currently support the creation of signature maps from input oscillators, however a script by Paul Callahan can be used to output a textual summary for some objects.
Domain 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.