Definitions of Life Terms

Life | Terms | Object | Pseudo-object | Still-life | Eater | Rock | Pseudo-still-life | Oscillator | Sparker | Pseudo-oscillator | Spaceship | Pseudo-spaceship | Flotilla | Gun | Puffer | Breeder | Inductor | Constellation | Methuselah

Life

Life is a cellular automata invented by professor John Horton Conway in the early 1970s.

The universe is an infinitely large array of square cells. Each cell can be in one of two distinct states: dead or alive, empty or full, 0 or 1, etc. Each cell is adjacent to eight other neighboring cells.

In every generation, a cell changes state according to its current state and the state of its eight neighbors:

• If a cell is dead, and has exactly three living neighbors, it becomes alive (a birth).
• If a cell is dead, and has less than three or more than three living neighbors, it remains dead.
• If a cell is alive, and has two or three living neighbors, it remains alive.
• If a cell is alive, and has less than two or more than three living neighbors, it dies.

All cells change state simultaneously. By successively repeating this operation, patterns evolve.

To avoid confusion, on these pages, the term cell refers to a cell as an entity that can be in one of several different states, while the term bit always refers to a cell that is alive, and is generally used for counting purposes. For example, a 12-bit still-life is a still-life that has exactly 12 living cells.

Most of the other definitions on this page apply not only to Life, but also to many other cellular automata with different states, neighborhoods or transition rules.

Patterns are usually characterized by their behavior:

• Type I (still-lifes): these patterns have no births and no deaths, and never change.
• Type II (oscillators): these patterns evolve through a fixed sequence of states and eventually return to their original state..
• Type III (spaceships): these patterns evolve through a fixed sequence of states and eventually return to their original state, but translated in space, moving at a constant velocity.
• Type IV: these patterns constantly increase in population:
  • Type IVa (guns): these are oscillators that emit spaceships each cycle.
  • Type IVb (puffers): these are spaceships that leave behind still-lifes, oscillators, and/or spaceships each cycle.
  • Type IVc (breeders): these patterns increase their population quadratically:
    • MMM (moving/moving/moving) breeders: puffers that emit puffers that emit spaceships
    • MMS (moving/moving/still) breeders: puffers that emit puffers that emit still-lifes and/or oscillators
    • MSM (moving/still/moving) breeders: puffers that emit guns that emit spaceships
    • SMM (still/moving/moving) breeders: guns that shoot puffers that emit spaceships
  • Other gun/puffer combinations that expand at other rates
• Type V (unstable): these patterns evolve through a sequence of states that never return to the original state. Small patterns that last a long time before stabilizing are called methuselahs.

Terms

The following terms are used frequently when discussing Life and other cellular automata. These are all basic terms that require little explanation. Terms with more complex definitions are described separately in subsequent sections.

• Agar:This is a pattern that repeats infinitely in two dimensions. It is equivalent to Life on a torus, with (for example), the left edge of the cylinder connected to the right edge, and the top edge connected to the bottom edge.
• Bit:A living cell.
• Cell:A single location in a Life field. It may be in any one of several different states (e.g. dead or alive.)
• Composite period:For oscillators and other repeating patterns, if none of the cells oscillate with full period, and the full period is just the LCM (lowest common multiple) of smaller sub-sections, the period is composite (rather than true), and the oscillator is usually not interesting in and of itself, except that it forms a trivial example of a period that may not otherwise be obtainable.
• Egg:For puffers and guns, these are extra objects left behind after the original puffer or gun returns to its initial configuration.
• Generation:The number of Life steps that have been applied to the initial configuration of cells.
• Heat:This is a measure of the number of a pattern's activity. It is defined as the average number of cells that change state each generation. Due to its definition, heat is always rational. (For non-repeating patterns, it may be measured asymptotically, e.g. lim_x→∞(heat of first x generations). For unstable patterns like methuselahs, this becomes the heat of the final census. For expanding patterns like still-life-making puffers and still wick-stretchers, the growing parts have no effect on the heat. For other expanding patterns like guns, other puffers and breeders, the heat is infinite.)
• Modulus: Some repeating patterns re-form themselves multiple times during their period, subject to rotation or reflection. Oscillators that rotate 90 degrees do this four times, while those that rotate 180 degrees, or that are reflected through an axis of symmetry, as well as spaceships with glide symmetry (i.e. gliders) do so twice. The modulus (or mod) is the length of time it takes to do this, which is always the period divided by 1, 2, or 4. (This can also have other values for rules with non-rectangular geometries. For example, with a hexagonal grid, it can be period divided by 1, 2, 3 or 6).
• Period:For repeating patterns, this is the number of generations after which the configuration of life cells returns to the same state as that of the initial configuration, or a reasonable facsimile. (For spaceships and puffers, this will be translated in space; for guns and puffers, this will also include eggs.)
• Population:This is the number of bits in a pattern.
• Temperature:This is a measure of a pattern's volatility. It is defined as heat divided by average population (or 0 for an empty field). The minimum value is 0, for still-lifes (where no cells change), and the maximum is 1, for phoenixes (where all cells change). Most patterns lie somewhere between these two extremes. Unlike heat, temperature is well-defined even for expanding patterns, and is asymptotic to the temperature of the most rapidly expanding parts.
• True period:For oscillators and other repeating patterns, if one or more cells oscillate with full period, the period is true, and the oscillator is considered non-trivial.
• Wick:This is a pattern that repeats infinitely in one dimension. It is equivalent to Life on a cylinder, with (for example), the left edge of the cylinder connected to the right edge.

Object

An object is a pattern consisting of one or more closely-connected bits. In order to provide an unambiguous method of distinguish objects from non-objects, the definition of "close" is defined precisely:

Any bit (i.e. living cell) forms an object.

If two objects contains any bits that are neighbors of each other, they are a single connected object. In Life, this includes any bits adjacent to each other either orthogonally or diagonally.

If two unconnected objects interact with each other so that they cause births that would otherwise not occur, or suppress births that would otherwise occur, they form a single spatially-connected object.

If two or more unconnected objects are adjacent to each other in such a way that some empty cells change from being sterile due to under-population to being sterile due to over-population, but they do not affect each other, they do not automatically form an object; these form a pseudo-object.

If three or more unconnected objects are adjacent to each other in such a way that there exists any way of partitioning these objects into two distinct sets, such that the combined objects in one set do not affect the combined objects in the other set, these objects do not automatically form a single object, even if there is some partition where they do affect one another. They form a pseudo-object. The third example, found by Pavel Podgoretsky, was the first pseudo-object where the pieces must be partitioned into more than two distinct sets. Gabriel Nivasch subsequently found the next two examples. The fourth example is a smaller one with similar properties, and the fifth one requires partitioning into four distinct sets. The Four Color Map Theorem can show that no more than four distinct sets will ever be required.

When dealing with periodic objects over their entire lifespan, such as oscillators and spaceships, if any components form a single object together in any generation, they are considered to form a single object in all generations. (This only applies when considering all generations as a whole; when only considering a single generation at a time, this rule is not applicable.)

The above also applies, even if the interaction between individual components is completely trivial, either forming attachments between dying bits, or creating additional births that die unobtrusively, or suppressing births that would otherwise die unobtrusively.

Pseudo-object

A pseudo-object is a pattern that, although not quite meeting the criteria necessary to be a single object, is nevertheless closely connected.

If two unconnected objects are adjacent to one another in such a way that any empty cells between then do not cause new births, but have a neighborhood that could form a birth if some if its cells were removed (in Life, this means four or more neighbors), the objects combine to form a pseudo-object. (See previous section for a more rigorous description).

Still-life

A still-life is an object where all bits survive, and no new births are formed. For example:

Eater

An eater is a still-life (or sometimes an pseudo-still-life or constellation) that can eat other objects, or parts of other objects, which may temporarily damage or destroy the eater, but the eater later re-appears unchanged. For example:

Rock

A rock is a still-life (or sometimes a pseudo-still-life or oscillator) that is an eater, but with the additional property that its own behavior is never altered in the process. For example:

Pseudo-still-life

A pseudo-still-life is a pseudo-object where all bits survive, and no new births are formed. For example:

Oscillator

An oscillator is an object that, after several generations, returns to its original state. Objects that return to their original state in 1 generation are still-lifes, and are not considered oscillators.

Terms used to describe electric motors can also describe parts of oscillators:

• The rotor consists of the cells of the oscillator that change from generation to generation.
• The stator consists of the cells of the oscillator that never change (and also includes vital dead cells).
• The bushing is the part of the stator that borders on the rotor, and affects its function.
• The casing is the part of the stator that not touch the rotor, and that is used to support the bushing.
• Sparks are dying bits that are emitted by an oscillator that have no effect on its function, but that are sometimes useful for other purposes, such as modifying other oscillators.
• Eggs are thrown off like sparks, but do not die, so they must normally be removed or suppressed by some other means.

Oscillators are often classified by their form or function:

• A flip-flop is a period 2 oscillator where all rotor bits die from underpopulation.
• An on-off is a period 2 oscillator where all rotor bits die from over-population. Some period 2 oscillators combine both flip-flop and on-off components.
• A phoenix is an oscillator where every living bit dies in every generation. Most such oscillators are period 2.
• A pulsator is a larger-period oscillator with the rotor on the outside, and usually little or no stator.
• An inductor is a pulsator with a line of symmetry that is used to suppress births.
• A billiard table configuration (BTC) is an oscillator with the stator on the outside totally surrounding the rotor.
• A shuttle is a larger-period pulsator with no stator, where the oscillator moves from place to place during the course of its oscillation, typically being rotated or reflected in the process.
• A sparker is an oscillator that throws off sparks that are useful for building other oscillators, etc.
• A hassler is an oscillator whose rotor creates projections that can affect other oscillators without being damaged.
• An eater-bound oscillator has an object that is attacked by one or more still-life eaters that affect its behavior. Common eaters are the block, tub, and eater, although many more complex eaters have been designed for specialized purposes.
• An sparked oscillator is has an object that is attacked by one or more sparkers that affect its behavior.
• An hassled oscillator is an object whose behavior is altered by one or more other objects, including eaters, sparkers, or hasslers.
• An babbling brook is an oscillator whose rotor consists of a string of single cells connected end to end. (A muttering moat is similar, with the ends joined together to form a closed loop). These oscillators are equivalent to one-dimensional cellular automata, and easy to model. Unfortunately, due to the constraints imposed by Life, only a small number of non-trivial variants of such oscillators can exist, because it is usually quite difficult to create bushing cells that remain stable in every state of the rotor.

Sparker

A sparker is a special kind of oscillator that throws off sparks, i.e. dying bits at the edge of the object, or a part of the object at the edge that protrudes. These can be useful to provide momentary alterations of neighborhoods of cells in another object, to either cause additional births, or suppress unwanted ones. There are several categories of sparks that are commonly seen:

• Bit: a single detached bit at an edge.
• Domino: two orthogonally connected detached bits at an edge, parallel to that edge.
• Delayed domino: a domino spark where one bit appears first, and the other is added later.
• Fountain bit: a single bit at an edge, with no other bits in the two rows behind it.
• Fountain domino: two orthogonally connected bits at an edge, with no other bits in the two rows behind it.
• Diagonal bit: a single detached bits at a corner.
• Diagonal domino: two diagonally connected detached bits at a corner.
• Vertical domino: two detached back-to-back bits at an edge, perpendicular to that edge. Oscillators that make these are called pipsquirters.
• Finger: like a vertical domino, except attached orthogonally to something else at the back.
• Thumb: like a bit, but attached diagonally to something else.

Pseudo-oscillator

A pseudo-oscillator is a pseudo-object that, after several generations, returns to its original state. Pseudo-objects that return to their original state in 1 generation are pseudo-still-lifes, and are not considered pseudo-oscillators.

Spaceship

A spaceship is an object that, after several generations, returns to its original state, but translated in space. Objects that return to their original state without moving are oscillators, and are not considered spaceships. Many spaceships also produce sparks. Sparks at the back of the spaceship are called tails parks, while those at the sides are called belly sparks.

Pseudo-spaceship

A pseudo-spaceship is an pseudo-object that, after several generations, returns to its original state, but translated in space. Pseudo-objects that return to their original state without moving are pseudo-oscillators, and are not considered pseudo-spaceships.

Spaceship Flotilla (Flotilla)

An spaceship flotilla (or just flotilla) is a collection of two or more spaceships traveling in tandem. Some of the spaceships may escort mechanisms that are not able to travel by themselves, or they may interact trivially with each other. Many flotillae are themselves technically just spaceships or pseudo-spaceships; however, it is usually easier to discuss them separately. Flotillae that are not objects (i.e. multiple independent spaceships traveling in tandem, separated by a small distance) are not detailed on these pages, because there are infinitely many of them.

Spaceship Gun (Gun)

A spaceship gun (or gun) is an oscillator that creates eggs that are spaceships that can safely escape. The population of a gun increases linearly with time.

Guns that emit puffer trains are SMM breeders, and have populations that increase quadratically with time.

Glider guns (i.e. guns that emit gliders) can be constructed for all periods 56 and higher, and guns of many lower periods are known. Guns that emit glider streams of any period 14 and up can be constructed, but even though the glider stream has a period as low as 14, the the gun itself typically has a period that is some larger multiple of the stream period.

For example, one can construct a gun that shoots gliders 14 generations apart using base period 30 guns by creating a slow period 210 gun by using several base guns and a reaction that divide the period by seven, then using many such gun to create and align 15 period 210 glider streams in the same lane.

Since the three simple c/2 orthogonal spaceships can be constructed from gliders, guns (also known as factories) for these spaceships of the same periods can be constructed. Light-weight spaceship streams must at least period 24. Middle-weight spaceship streams must at least period 28. Heavy-weight spaceship streams must at least period 32. (Theoretically, streams as dense as periods 14, 16, and 18 respectively could be created, but there are currently no known synthesis techniques to do so).

Puffer Train (Puffer)

A puffer train (or puffer) is a spaceship that emits debris, that may be anything that does not move with the same velocity as the puffer train, such as still-lifes, oscillators, or spaceships.

• Puffer trains that leave no debris are called clean puffers, and are actually just spaceships, and are generally not considered with other puffers; however, such clean puffers can often be sparked to produce debris, and are thus often form the basis for larger puffers.
• Puffers trains that produce a fixed number of objects of increasing population are called wick-stretchers, as their output may considered to be an ever-growing sub-section of a wick.
• Wick-stretchers that stretch oscillating wicks (as opposed to still-lies) are called taffy-pullers.
• Puffer trains that leave spaceship guns or other puffer trains are breeders, and have populations that increase quadratically with time.
• Breeders that generate a single expanding object are called space-fillers, and their output may be considered to be an ever-growing sub-section of an agar.

Breeder

A breeder is a pattern whose population increases forever, and whose population is greater than a linear function of time. Breeders are generally factories that build producers that produce debris, and their population normally increases quadratically with respect to time. Breeders that produce a fixed number of objects of quadratically-increasing population are called space-fillers. The four categories of breeders are categorized by which of the three parts are stationary:

1. SMM breeder (stationary factory): guns that builds puffer trains that emit spaceships.
2. MSM breeder (stationary producer): puffer trains that build guns that emit spaceships (this is the first type of breeder ever built).
3. MMS breeder (stationary debris): puffer trains that build puffer trains that emit still-lifes and/or oscillators.
4. MMM breeder (nothing stationary): puffer trains that build puffer trains that emit spaceships.

There are also many other hybrid categories possible, including many variations whose population increases at other rates. Dean Hickerson has created many such patterns, with populations of O(√n), O(n log n), etc.

Induction Coil (inductor)

An induction coil (or inductor) is an unstable object that, by itself, would produce several unwanted births on its outside. These births can be suppressed by placing another object beside the induction coil. This birth-suppression is referred to as induction.

This is a sampling of inducted objects (named below), with their corresponding inductors (named above) shown in teal.

Constellation

A constellation is a collection of several periodic objects or pseudo-objects. Since there are so many possible constellations (for example, there are an infinite number of arrangements of two blocks alone), most constellations are not interesting in themselves. Exceptions include constellations that occur naturally, or that have some functional significance, or that can be constructed for a cost less than their individual components.

Methuselah

A methuselah is a small unstable object that lives for an unusually long time before becoming stable. (The name is derived from the biblical patriarch of the same name, who had the longest recorded human life span: 969 years.) In Life, this was originally defined as any object of 8 bits or less that survives at least 50 generations before becoming stable. In colloquial usage, and on these pages, this definition is relaxed somewhat to also include some larger long-lived objects or pseudo-objects, or even patterns that are neither, and focuses on the results (that can be obtained from several predecessors), rather than dwelling on specific predecessors. The key feature methuselahs share is that they start small and run for a disproportionately long time.

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This page was last updated on 2015-02-19.