Day & Night: The Symmetric Universe

Take any stable pattern in Conway’s Life — a block, a beehive, a spaceship frozen in your imagination — and flip every cell. Make every live cell dead and every dead cell alive. What you have now is mostly a sea of live cells with a few holes cut out: the “negative” of the original pattern.

Run this negative in Conway’s Life, and it immediately explodes. A sea of live cells in Conway’s Life collapses almost instantly — the overcrowding kills everything, and you’re left with sparse debris that bears no relation to the pattern you started with. Alive and dead are not interchangeable in Conway’s Life. Being alive is fundamentally different from being dead.

Now take that same negative pattern and run it in Day & Night — rule B3678/S34678. Something different happens. The negative is stable. If the original was a still life, the negative is a still life. If the original was a spaceship moving east at speed c/4, the negative is a spaceship moving east at speed c/4. If the original was an oscillator with period 8, the negative oscillates at period 8.

The universe looks back at you with perfect symmetry. Day and night are interchangeable.

The Rule: B3678/S34678

Day & Night is defined by:

A dead cell is born if it has 3, 6, 7, or 8 live neighbors.
A live cell survives if it has 3, 4, 6, 7, or 8 live neighbors.

The first thing to notice is that both lists are large. Conway’s Life, with its B3/S23, is a sparse rule — most neighbor counts lead to death. Day & Night is far denser in its birth and survival conditions. A live cell survives unless it has very few neighbors (0, 1, or 2) or exactly 5 — but it dies at those counts. A dead cell is born unless it has 0, 1, 2, or 4 neighbors — it stays dead at those counts.

The second thing to notice is the symmetry. Consider what happens to the rule when you exchange alive and dead throughout. A cell that was dead now acts as if it were alive, and vice versa. Equivalently, a cell’s “alive neighbor count” becomes 8 minus its “dead neighbor count.” (If 3 of 8 neighbors are alive, then 5 are dead; from the dead cell’s perspective, 5 live-cells-that-are-now-dead surround it.)

Under this exchange, birth-at-n in the original rule becomes survival-at-(8-n) in the flipped rule. Check: B3678 maps to survival at 8-3=5, 8-6=2, 8-7=1, 8-8=0 — but those are the death conditions in the original. The survival rule S34678 maps to birth at 8-3=5, 8-4=4, 8-6=2, 8-7=1, 8-8=0 — again the death conditions.

The point is that B3678/S34678 is self-complementary: the rule applied to the flipped grid is the same as the flip of the rule applied to the original grid. This is the precise mathematical statement of the symmetry that gives Day & Night its name.

Who Found It

Day & Night was invented and named by Nathan Thompson in April 1997, the same year he discovered the Day & Night symmetry property. Thompson — who also discovered HighLife three years earlier — was systematically exploring self-complementary rules in the Life-like space, looking for rules whose birth and survival conditions had this inversion property.

The detailed study of the rule was carried out by David I. Bell, who began examining it in November 1997. Bell catalogued many of the rule’s key patterns: still lifes, oscillators, spaceships, guns, and puffers. His work established Day & Night as a genuinely rich rule — not just a mathematical curiosity, but a universe with as much variety of behavior as Conway’s Life itself.

The Consequences of Symmetry

The self-complementary property has immediate structural implications.

Every still life has a dual

If a finite pattern P is a still life in Day & Night — meaning it doesn’t change from generation to generation — then its complement P’ (every cell flipped) is also a still life. The proof is immediate: if P evolves to P under the rule, and the rule is self-complementary, then P’ evolves to P’ under the same rule.

This is genuinely strange. In Conway’s Life, a dense sea of live cells with small holes is not a still life — it explodes. In Day & Night, the “inside-out” version of any stable structure is equally stable. The rule supports two parallel taxonomies of objects: the sparse-on-dark objects (like Conway’s Life), and the dense-on-bright objects (their complements), both coexisting under the same dynamics.

Every spaceship has a dual

The same argument applies to spaceships. If pattern P is a spaceship moving with velocity v = (dx, dy) per period T, then its complement P’ is a spaceship moving with the same velocity v and the same period T.

This produces a phenomenon you won’t find in Conway’s Life: a spaceship traveling through a “sea” of live cells, cutting a moving hole through the density. It looks like the opposite of a normal spaceship, but it is an equally valid spaceship, guaranteed by the symmetry.

David Bell’s rocket — a period-40 spaceship he catalogued in Day & Night — occurs frequently in random evolutions and has a complement that is also a period-40 spaceship of the same shape. When you study Day & Night long enough, you start to see patterns traveling through bright fields as commonly as patterns traveling through dark fields.

Random soups converge in pairs

Start with a random initial configuration in Day & Night. Let it evolve. After many generations, it will typically settle into a complex but stable or near-stable configuration. Now consider the complement of that same initial configuration — every cell flipped. The complement will settle into the complement of whatever the original settled into.

This means that if you run Day & Night’s “thermal equilibrium” distributions — the typical long-run configurations from random starts at various densities — you will find that the density-50% starting configuration (equal numbers of live and dead cells) is self-complementary: its statistics are invariant under the alive/dead exchange. The behavior of the rule at 50% density is, in a precise sense, independent of which state you call “alive.”

Patterns in Day & Night

Day & Night supports a rich catalog of objects, many with no analogs in Conway’s Life.

The still lifes in Day & Night include dense structures — clusters of live cells where high neighbor counts (6, 7, 8) satisfy the survival condition — that would be impossible in Conway’s Life. There are also large “holes” in live backgrounds, maintained by the birth-at-8 condition: a dead cell with 8 live neighbors (completely surrounded) is born, filling the hole.

The spaceships include both low-density objects (like Conway’s gliders) and high-density objects (dense blobs cutting through live backgrounds). Bell catalogued the rocket (period 40) as one of the most commonly occurring spaceships, useful as a building block for more complex engineered structures like guns and reflectors.

The general behavior of random soups in Day & Night resembles Conway’s Life in texture — chaotic initial growth, local stabilization, a mixture of still lifes and oscillators in the eventual “ash” — but the objects that survive are different, and the density of the ash is typically higher (closer to 50% live cells) than in Conway’s Life.

What Day & Night Reveals About Generic Rule Space

The self-complementary property of Day & Night is not just aesthetically pleasing. It reveals something about the structure of the space of all Life-like rules.

There are very few self-complementary rules in the 262,144-rule Life-like space. To be self-complementary, a rule must satisfy: n is in the birth list if and only if (8-n) is in the death conditions (i.e., not in the survival list). This is a strong constraint. Counting: the number of self-complementary rules is $2^4 = 16$ for the four-bit free choices (counting the pairs {n, 8-n} independently). Most of these 16 rules are trivial — everything dies, or everything lives. Day & Night is one of the very few that has rich behavior.

This rarity is interesting. It says that the alive/dead symmetry is not a generic property of Life-like rules — it is exceptional. Conway’s Life, which treats alive and dead asymmetrically, is representative of the vast majority of rules. Day & Night’s symmetry is a mathematical accident: one of 16 rules with a special property, and one of probably 2-3 of those 16 with non-trivial dynamics.

The deeper implication is that most interesting properties of Life-like rules do not arise from mathematical symmetry. Conway found interesting dynamics by ignoring symmetry entirely, working empirically. The fact that you can also find interesting dynamics by imposing a strong symmetry constraint — and that the result looks qualitatively similar, with the same kinds of objects and the same kinds of long-run behavior — suggests that the richness is structural rather than coincidental. Complexity emerges in many different ways from the same framework.

Comparison with Conway’s Life

Day & Night and Conway’s Life are superficially different but structurally similar. Both are rich: both support gliders, oscillators, still lifes, guns, puffers, and Turing-complete computation. Both produce chaotic initial evolution that eventually resolves into stable configurations. Both have been studied enough to have extensive pattern catalogs.

The differences are in character:

Day & Night’s objects tend to be denser. Because survival is possible at higher neighbor counts (up to 8), stable configurations can be tightly packed in ways Conway’s Life doesn’t support.
Day & Night’s dynamics are somewhat faster to stabilize. Random configurations often settle into their final state more quickly than in Conway’s Life.
Day & Night has a parallel universe: every object has a dual, the complement. There is always another way to see every pattern — from the inside, as it were.

Conway’s asymmetry — the fact that alive and dead are not interchangeable — turns out to be a feature rather than a bug, at least aesthetically. It makes Conway’s Life feel like a universe with a definite “ground state” (the empty grid), from which complexity grows. Day & Night’s symmetry makes it feel like a universe where there is no natural ground state: alive and dead are equally valid starting points, equally valid backgrounds for structure.

Neither perspective is superior. They are different mathematical characters, both capable of producing the same fundamental kinds of emergent behavior.