# Multi-colored Life

## Color variants

An interesting variation of the Life rules is to use several distinct living state, with each behaving according to Life rules, but, in addition, carrying independent color information. The color of each newly-born bit is determined by the color of its parents. Resulting patterns have exactly the same life/death behavior as in normal Life, but the colors of resulting bits provides interesting additional complexity.

All the coloring variants discussed here are supersets of ordinary Life (i.e. bits whose parents are all of the same color will be born in that same color).

Multi-color (hybrid) objects can exist in all variants. Objects with permanent bits (such as still-lifes, and oscillators with stators) typically have many different color variants, the number of which increases exponentially with the number of stator bits. There are also many oscillators and spaceships with multi-color forms, but these tend to be more fragile, as the general tendency in such rules is for one color to dominate. (For example, in examining all the ways that two gliders of different colors can collide and not die, in less than 1/6 of them does the census includes bits of both colors).

## Immigration (two-color Life)

Immigration is a coloring variant that uses two different symmetrical colors. The sole rule that determines the color of births is simple:

• Each bit's color is the same color as the majority of its parents.

(The name Immigration is used, because this variant was originally designed for a two-player Life-based game with the same name, where players added bits of their own colors to the edge of the field, at which point the bits would immigrate to the field itself. The object was to dominate the field with one's color, at the expense of the opponent's.)

Quad-Life is a coloring variant that uses four different symmetrical colors. The two rules that determines the color of births are:

• If two parents have the same color (i.e. they form a majority), the child has the same color. (This is the same as Life and Immigration).
• If there is no majority (i.e. all 3 parents have different colors), the child is of the fourth color.

Quad-Life is a superset of both normal Life and Immigration. Its most interesting feature is that populations where one color has totally died out can give birth to that same color again. Only if two colors die out (reducing this to Immigration) is such color death necessarily permanent.

Also, in many cases, oscillator and spaceships that include 3 or 4 colors may have higher periods than their normal Life counterparts, cycling through several different color combinations. For example, the 4-color Glider has one side of one color, while the two bits on the other side are of different colors, and cycle through the three different permutations of the two other colors, giving the resulting multi-color Glider a period of 12, rather than the usual 4. Whenever such multi-colored objects have one or more periods that differ from those of the original objects, those periods are shown in parentheses at the end (if known).

## Oct-Life (eight-color Life)

Oct-Life is a coloring variant that uses a sets of four different symmetrical colors, and another set of their complements. Colors in each of the two sets are symmetrical with each other, but not with colors of the opposite set. The rules that determine the color of births are:

• If two parents have the same color (i.e. they form a majority), the child has the same color. (This is the same as Life and Immigration).
• If two parents have opposite colors, they cancel, and the child has the same color as the third parent.
• Otherwise, if one parent has the opposite type from the other two, the child inherits its color.
• Otherwise, when all three parents have the same type, the child is of the opposite type of the fourth color. (This is similar to, but quite different than Quad-Life)

It turns out that these rules can be simplified into a single rule (that is generalized in the following section). If one conceptualizes the color space as a cube with the four primary colors at the non-adjacent even-numbered vertices, and their complements at the respective diametrically opposite odd-numbered vertices, then the child color is chosen in such a way that the sum of the lengths of the minimal paths along the cube edges from each parent to the child is minimized.

Oct-Life is a superset of both normal Life and Immigration, but not of Quad-Life, since the child of three different colors is the opposite of the fourth color. This makes more sense logically. For example, if one chooses physical colors (e.g. red, blue, green and black, and their respective complements teal, brown, purple and white), a child's color is often the one most closely matching a mixing of the parents (e.g. red+blue+green=white, black+white+green=green, etc.). Unfortunately, the colors of resulting Life patterns are nowhere nearly as interesting as those of Quad-Life.

## More general colorizations

The single geometric rule used in Oct-Life can be generalized to any geometric figure, and any number of colors, and this works even for other rules with births on numbers of parents other than three. The following rules apply:

• Arrange the colors at the vertices of a regular polygon, polyhedron, polytope, etc. (e.g. Life uses a point, Immigration uses a line, Quad-Life uses a tetrahedron, and Oct-Life uses a cube).
• The behavior of colors will only be symmetrical with respect to all other colors if every vertex on the figure has the same relationship to every other vertex (e.g. this is true of Life, Immigration and Quad-Life, and also any rule with three colors, but not Oct-Life).
• (Non-regular polyhedra, polytopes, etc. like pyramids and Johnson solids can also be used, but the relative symmetry of the colors space is further reduced).
• For each vertex on the figure, compute the sum of the minimal paths from that vertex all parents.
• Choose the vertex with the smallest sum.
• If two or more vertices share the smallest sum, eliminate them all from consideration, and try again.
• If no vertices remain, choose an arbitrary color (typically color #1).

The last rule breaks color symmetry if more than one color is used, and should be avoided if at all possible. It can occur in any of the following situations:

• Rules that permit birth on 0
• GCD (number of colors, any birth condition) exceeds 1
• Any birth-on-n condition where there are n+2 or more totally symmetrical vertices (this is why polygons usually work better than polyhedra or polytopes, even though they have less symmetry).

These place the following restrictions on birth conditions:

• Birth on n: n-dimensional triangular polytope with n+1 vertices is always allowed.
• Birth on 0: Only one color allowed.
• Birth on 1: No restrictions.
• Birth on 2, 4, 8: Odd polygons.
• Birth on 3: 2 colors, even polygons, polyhedra, polytopes without triangular sides.
• Birth on 5: 2 colors, polygons not divisible by 5, polyhedra except dodecahedrons, polytopes.
• Birth on 6: Any polygons not divisible by 2-4.
• Birth on 7: 2 colors, polygons not divisible by 7, polyhedra, polytopes.

Multiple birth conditions combine all restrictions. For example, the rule B34/S34 requires at least 5 colors, and a pentagon works, but a hyper-tetrahedron does not, since birth on 3 would not know which of the two remaining colors to choose.

## Rational colorizations

Rather than having a finite number of discrete color states, another approach is to treat the colors as a spectrum, represented by a rational number (e.g. 0 for black and 1 for white, with many shades of grey inbetween), or a vector of such numbers (e.g. 0,0,0 for black, 1,1,1 for white 1,0,0 for red, 0,1,0 for green, and 0,0,1 for blue). When a child is born, its color is simply the average color of its parents.

Under such rules, colors tend to bleed together, and usually reach an equilibrium. For example, in oscillators with bushings of multiple colors, nearby rotor cells usually approximate rational sums of nearby bushing cells. Oscillators with no bushings (and also spaceships, since they can never have bushings) usually approximate a single blended color.

One kinds of exception to this occurs when one part of the oscillator or spaceship has a colorization totally independent of any other, because other parts may cause deaths, but never contribute births (or if they do, these are typically of short-lived sparks that in turn turn only contribute subsequent deaths). Another kind happens when one part of the oscillator is sandwiched between two or more independently-colored parts - all of which contribute colors much like multi-colored bushings do (e.g. if a toad is hassled by two oscillators of different colors, the bits of the toad will eventually approach rational fractions of the sums of the colors of the two hasslers).

In the same way that it is possible to construct Life machinery that can have populations that are asymptotic to irrational numbers, it should be possible to construct similar mechanisms to generate colorizations that are asymptotic to irrational numbers as well.

## Multiple colorizations

Just as it's possible to add colorization to Life to add an additional but independent data channel to a Life pattern, it's possible to add multiple independent colorizations simultaneously. For example, one could have one Quad-life colorization of red, green, blue and yellow, and another one of square, round, triangular and hexagonal, yielding 16 possible living states.

## Syntheses

In the synthesis files contained here, each file contains methods for synthesizing one or more Immigration and/or Quad-Life colorization of a normal Life. Instead of the normal convention used elsewhere on this site (i.e. color 1 = before state, color 2 = after state, color 3 (where necessary) = useless debris that must be subsequently eliminated), both "before" and "after" images here use the same colors, representing up to four different colors of Life bits. Whenever four colors are not needed (e.g. for two-color Immigration colorizations), colors 2, 3, and sometimes 4 are used, while color 1 is used to represent "don't care" bits (e.g. an incoming glider whose colorization is irrelevant in the final result, because, while it affects the states of other bits, it does not affect their color, and its own bits all die.) This is especially true in the case of two-color (i.e. Immigration) results; gliders of the default color may be replaced by either color.

The lists of objects (and, indeed, the specific multi-color forms synthesized) are not intended to be exhaustive; instead, they are merely small samplings of what is possible. The number of possible multi-color forms increases exponentially. For example, for an asymmetric n-bit still-life or pseudo-still-life, the number of distinct non-trivial k-color colorizations is kn/k! - (k-1)n/(k-1)!. The number of possible symmetrical objects is smaller for combinatorical reasons. The number of possible oscillators and spaceships is also usually much smaller because, while stators can have the same number of variations as still-lifes, rotors and spaceships are much more restricted, as color information is frequently lost with cell births and deaths.

For some of the smaller objects, the syntheses show syntheses for all 2-color variants (or these can be deduced from examples shown); these syntheses are marked with a *. If they also show all 4-color variants, they are marked with **.

Larger objects consisting of multiple distinct parts can often be trivially hybridized by constructing different independent parts from components of different colors.

The stamp collections each show a single colorization of each object. Most of the syntheses include more than one colorization, so the costs in gliders frequently show ranges, as some colorizations are more expensive to synthesize than others.

## Running Patterns

The patterns on this page can be run by Golly, but require the following rule files to first be downloaded and installed. They can either be installed for all users (in the Rules directory directly under where the Golly application is installed), or in the current user's personal Rule directory. (To find out where this is, run Golly, select File/Preferences..., look at the Control tab. It is displayed right next to the Your Rules... button.

Golly versions 2.5 and later can use the single rule file (Life4c.rule).

Earlier versions of Golly require two files: the rule table (Life4c.table) and the corresponding color table (Life4c.colors).

## 5 Multi-color pseudo-still-lifes

In most cases, multi-color pseudo-still-lifes (and other pseudo-objects) are not very interesting, since most of them are formed either from synthesizing two objects simultaneously back-to-back, or adding one object to a previously-synthesized object. In either case, it is usually trivial to make one piece of one color, and the other piece of a different color. Cases like the ones shown above, where pseudo-objects form spontaneously, are more interesting.

## 28 Multi-color Oscillators

 Blinker** [82] Toad [3-4] (P2, P4, P8) Clock [4-9] (P2, P4, P8) Beacon [3] Bipole [12] (P2, P4, P6) Tripole [8] (P2, P4, P8) Quad- pole [6-10] (P2, P4, P8) 12-bit P2 #1 [18] Fox; Gnome [12] Phoenix [10] Spark Coil [4-10] Fore and back; Complem- entary blinker [20] Caterer [11] Jam [8] (P3, P6, P12, P18) Two eaters; Eater eating eater [8-11] (P3, P6, P9, P18) Block on up cuphook w/ tail [8-13] Mold [5-13] (P4, P12) Mazing [5] Mono- gram; Initials; JHC [15] A for all [14] Octagon II [10-22] Pentoad between two eaters [12] Unix [7] (P6, P18) Figure-8; Big beacon [12] Blocker [6-7] Tumbler [6-10] Penta- decath- lon [4] Queen bee shuttle between cis blocks [6-7]

## 7 Multi-color spaceships

There are actually four different basic arrangements of escort ships for the Schick ship, but only one is shown here, as the engine only cares about the colors of the tail sparks. All the examples shown here show escort ships whose tail sparks are a single color. Schick ships with multi-colored escort ships are also possible.

The Turtle cannot be synthesized, but it is included here because of its interesting hybrid colorization.

## 10 Multi-color constellations

 Two blocks offset (4, 1) [2-8] Two point- to- point boats [3-5] Four skewed blocks [2] Honeyfarm [4] Bakery [4] Blockade (via Lumps of muck) [2] Two blinkers offset (4, 0) [3] Loaf and blinker [2] Traffic light [5-9] Interchange [2]

In most cases, multi-color constellations are not very interesting, since most of them are formed either from synthesizing two or more objects simultaneously back-to-back, or adding one object to a previously-synthesized object or constellation. In either case, it is usually trivial to make one piece of one color, and the other piece of a different color. Cases like the ones shown above, where pseudo-objects form spontaneously, are more interesting.

See also: Life objects sorted by: counts, frequency of occurrence, cost in gliders, name, size in bits, or type.