Day & Night - An Interesting Variant of Life (part 2/5)
David Bell
SIMPLE OSCILLATORS
------------------
Many simple oscillators have been found in Day & Night by Dean Hickerson
and Mark D. Niemiec, and a few were found by myself. (I have not credited
each oscillator explicitly.) Representative examples of these are shown in
the following figures. Some of these oscillators are extensible and can be
made larger. Several of these oscillators have uses as part of larger
constructions.
O........O......O.O......O..
.OO.....OO.......O........OO
.O......O......OO.O.....OO..
................O.........O.
[Figure 28. Some period 2 oscillators]
.............................O.............O.............OO.............OOO..
.OO.OO.......OO..OO........O...O........................................OOO..
O.O.O.O.....O.O..O.O.......O.O.O.........OOOOO.........OOOOOO.........OOO.OOO
O.O.O.O.....O.O..O.O.....OOOOOOOOO.....OOOO.OOOO.....OOOO..OOOO.......OO...OO
.OO.OO.......OO..OO........O.O.O.......OOOOOOOOO.....OOOOOOOOOO.......OOO.OOO
.........................................OO.OO.........OO..OO...........OOO..
........................................................................OOO..
[Figure 29. Some period 3 oscillators]
.........................OOO.......................................
........................O..O.......................................
O.O.....O.O.....O..O....O.O.O...................................OO.
.O.......O.......OO.....OO.O.O...........OOO...OOO...OOO...OOO..OO.
O.O......O.......OO.......O.O.OO.......OOO...OOO...OOO...OOO...OOOO
........O.O.....O..O.......O.O.O.........OOO...OOO...OOO...OOO..OO.
............................O..O................................OO.
............................OOO....................................
[Figure 30. Some period 4 oscillators]
.......................................................O.....
...............O.......................................O.....
...............O...........................O.........OOOOO...
..OO.........OOOOO.........OOOOO.........OOOOO......OOOOOOO..
.OOOOO......OOOOOOO.......OOOOOOO.......OOOOOO......OOOOOOO..
.OOOOO......OOOOOOO.......OOOOOOO.......OOOO......OOOOOOOOOOO
OOOOO........OOOOO.......OOOOOOOOO.....OOOOO........OOOOOOO..
.OOOO........OO.OO........OOOOOOO.......OO..........OOOOOOO..
.OO.......................OO.O.OO.......OO...........OOOOO...
.......................................................O.....
.......................................................O.....
[Figure 31. Some period 5 oscillators]
.....................................................O.......
...................................................OOOOO.....
...................................................OOOOO.....
....................................................OOO......
...................................O...O............OOO......
......................OOO..........OOO..OO.....OO...OOO...OO.
..O..........O........OOO.........O.O..OO......OOOOOOOOOOOOO.
.OOOO.......OOOO......OOO............OO.O.....OOOOOOOOOOOOOOO
OOOOO......OOOOO.........OOO.......O.OO........OOOOOOOOOOOOO.
.OO.........OOOO.........OOO.......OO..O.O.....OO...OOO...OO.
.OO.........OOOO.........OOO......OO..OOO...........OOO......
....................................O...O...........OOO......
...................................................OOOOO.....
...................................................OOOOO.....
.....................................................O.......
[Figure 32. Some period 6 oscillators]
.........................................OO...
.............OO...OO..............OO.OO..OO...
.............OO...OOOO.OO.........OO.OO...O.OO
..............O.OOOOOO.OO..........O......O.OO
..OOO........OO.OOOOO..O...........O.....OO...
...OOO.......OO..OOOO.O.OOO.....OO.OO....O.OOO
O..OO..........O......OO.OO.....OOO.OOOOOO..OO
OOOOOO.........OO.....O..............OOOOO.OO.
OOOOOO..........O....OO...........OO.OOOOOO.O.
.O.OOOO.........OO...O............OO..OOOO..OO
.....OO..........OOO.OO..............OOOOO..OO
................OOOO.OO...............OO......
................OO....................OO......
[Figure 33. Some period 7 oscillators]
....................................O....................O
....O..............OO...............O....................O
..................................OOO...................OO
..O...O..........O....O..............O..................O.
OOOO.OOOO......OOOO..OOOO.............OOO...........O.....
OOOO.OOOO......OOOO..OOOO.............O............OOO....
..OO.OO..........OO..OO...............O..............O....
..........................................................
[Figure 34. Oscillators of periods 8, 8, 13, and 14]
..........OOO......OOO
..........OOO......OOO
..........OOOO.OO.OOOO
............O.OOOO.O..
.O...........OOOOOO...
OO..........OOOOOOOO..
O...........OOOOOOOO..
O............OOOOOO...
............O.OOOO.O..
..........OOOO.OO.OOOO
..........OOO......OOO
..........OOO......OOO
[Figure 35. Two oscillators of period 16]
......................O............................
....................OOOOO..........................
.................OOOOOOOOOOO.......................
..................OOOOOOOOO................O....O..
...............OOOOOOOOOOOOOOO............OOO..OOO.
.OOO............OOOOOOOOOOOOO............OOOO..OOOO
O.O.O.........OOOOOOOOOOOOOOOOO...........OO.OO.OO.
OOOOO.......OOOOOOOOOOOOOOOOOOOOO...........OOOO...
O.O...........OOOOOOOOOOOOOOOOO.............OOOO...
.OO.............OOOOOOOOOOOOO.............OO.OO.OO.
...............OOOOOOOOOOOOOOO...........OOOO..OOOO
..................OOOOOOOOO...............OOO..OOO.
.................OOOOOOOOOOO...............O....O..
....................OOOOO..........................
......................O............................
[Figure 36. Oscillators of periods 36, 38, and 42]
The smallest period 4 oscillator shown in figure 30 is useful because it
has the ability to destroy a rocket or a p32 ship. This is shown below.
From now on in this article it will be called an "eater".
...................................O..........................
............................O.O.O....OO.........O.............
.............................OOO.OOOOO.O........O.........OO..
.........OOO.O...........OO.O.OOOOOOOOO....OO..O.O..O....OOOOO
.O......O.OOOOOO........O.OOOOOOOOOOOOO..OOOOOOOO..O.O..OOOOOO
OOO.....OOOOOOOO.O......OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO.
.O......O.OOOOOO........O.OOOOOOOOOOOOO..OOOOOOOO..O.O..OOOOOO
.........OOO.O...........OO.O.OOOOOOOOO....OO..O.O..O....OOOOO
.............................OOO.OOOOO.O........O.........OO..
............................O.O.O....OO.........O.............
...................................O..........................
[Figure 37. A period 4 "eater" oscillator destroys a p32 ship and a rocket]
The eater can also destroy a snail, but changes its phase while doing so.
Some other oscillators can destroy a snail without any phase changes.
One of these oscillators emits sparks with period of 8, and is shown below.
.OO.......................
.OO.......................
OOOO.................O.O..
OOO.................O..OOO
.....O..............OOOOOO
OOO.................O..OOO
OOOO.................O.O..
.OO.......................
.OO.......................
[Figure 38. A period 8 oscillator destroys a snail]
Sparks from oscillators can also be used to prevent the creation of the
white dwarf from a nova explosion. This is shown below, where a period 8
oscillator is used to perturb one of the lobes of the nova. The other
three lobes are unaffected.
................OO.
................OO.
..O............OOOO
.OOO...........O.OO
OOOOO...........O..
.OOO...........O.OO
..O............OOOO
................OO.
................OO.
[Figure 39. A period 8 oscillator destroys the white dwarf from a nova]
Besides the period 8 oscillator above, several of the period 3 oscillators
also emit sparks. These sparks can be used to perturb other objects.
But it turns out that good sparks exist in abundance in this universe,
as is shown later.
Many of the oscillators shown above can be created by the collisions of
spaceships. As an example, the interesting period 36 oscillator can be
created by the collision of a period 16 oscillator and a rocket.
O.....................................
OOO...................................
OO....................................
......................................
......................................
...........O..........................
....O.O.O....OO.........O.............
.....OOO.OOOOO.O........O.........OO..
.OO.O.OOOOOOOOO....OO..O.O..O....OOOOO
O.OOOOOOOOOOOOO..OOOOOOOO..O.O..OOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO.
O.OOOOOOOOOOOOO..OOOOOOOO..O.O..OOOOOO
.OO.O.OOOOOOOOO....OO..O.O..O....OOOOO
.....OOO.OOOOO.O........O.........OO..
....O.O.O....OO.........O.............
...........O..........................
[Figure 40. A rocket and period 16 oscillator collide to form a
period 36 oscillator]
RIPPLE OSCILLATORS
------------------
Dean Hickerson discovered several classes of oscillators which emulate a
1-D XOR cellular automaton. In this automaton the universe consists of a
line of cells, where each cell is in one of the states 0 or 1; any two 1s
are an even distance apart. (The 1s are confined to even positions in
even generations and odd positions in odd generations.) Each generation,
the new state of each cell is the XOR of the previous states of its two
neighbors. For a finite universe, you assume that the cells beyond the
end cells are permanently 0. In Day & Night, a diagonal strip of cells
in one of these oscillators changes state in the same manner as the
corresponding 1-D XOR oscillator. It is easy to prove that 1-D XOR
oscillators can have any even period; therefore oscillators of all even
periods exist in Day & Night as well.
For example, the line of 16 cells below becomes its mirror image after
5 generations, so it has period 10:
gen 0: 0101010001000000
gen 1: 1000001010100000
gen 2: 0100010000010000
gen 3: 1010101000101000
gen 4: 0000000101000100
gen 5: 0000001000101010
The following figure shows examples of the most versatile class of these
oscillators. The first emulates the p10 shown above; the second has
period 62 and a rotor of size 10. (The "rotor" of any oscillator is the
set of all cells that change state at some time.) The presence or absence
of "steps" along the diagonal represents the cells of the 1-D automaton,
where the presence of a step is the state 1 and an absence of a step is
the state 0. Each generation, the cells making up the steps are present
if and only if exactly one of their two adjacent steps was present in the
previous generation (except at the two ends).
...OO..........................OO.............
...OO..........................OO.............
..OOOO........................OOOO............
.O....OO.....................O....O...........
.OO..O.O.....................OO..O.OO.........
OO..O.O.OO..................OO..O.O.O.........
.O.OO..O.O...................O.OO..O.O........
.O.OO...O.OO.................O.OO...O.O.......
OO.......O.O................OO.......O.O......
OO........O.O...............OO........O.O.....
...........O.O.........................O.O....
............O.OO........................O.O...
.............O.O.........................O.O..
..............O.O.....................OOO..OOO
...............O.O....................OO...OOO
................O.O......................O.O..
.................O.O................OOOOOOO...
..................O.O...............OO..O.....
...................O.O........................
................OOO..OOO......................
................OO...OOO......................
...................O.O........................
..............OOOOOOO.........................
..............OO..O...........................
[Figure 41. Period 10 and period 62 1-D XOR cellular automaton emulators (DH)]
If the number of active cells being emulated is 2 (mod 3), then there is a
second class of oscillator which also emulates the 1-D XOR cellular automaton.
The rotor in this case is along the diagonal which runs between the two
diagonal rows of white dwarfs. The following example shows a period 30
oscillator with a rotor size of 14.
...........O.........
.........OOOOO.......
...O.....OOOOO.......
.OOOOO..OOOOOOO......
.OOOOO...OOOOO.......
OOOOOOO.OOOOOO.......
.OOOOO.....O.....O...
.OOOOO.........OOOOO.
...O.....O.....OOOOO.
.......OOOOO..OOOOOOO
.......OOOOO...OOOOO.
......OOOOOOO..OOOOO.
.......OOOOO.....O...
.......OOOOO.........
.........O.....O.....
.............OOOOO...
.............OOOOO...
............OOOOOOO..
.............OOOOO...
.............OOOOO...
...............O.....
[Figure 42. Period 30 1-D XOR cellular automaton emulator (DH)]
In some cases, the rotor can be bent, and can make use of both classes of
oscillators, as in the figure below.
.........................O.....
.......................OOOOO...
.......................OOOOO...
......................OOOOOOO..
...OO..................OOOOO...
...OO..................OOOOO...
..OOOO.............O.....O.....
.O....OO.........OOOOO.........
.OO..O.O.........OOOOO.....O...
OO..O.O.O.......OOOOOOO..OOOOO.
.O.OO..O.O.......OOOOO.O.OOOOO.
.O.OO...O.O......OOOOO..OOOOOOO
OO.......O.O.......O.....OOOOO.
OO........O.O............OOOOO.
...........O.O.......O.....O...
............O.O....OOOOO.......
.........OOO.O.O...OOOOO.......
.........OO.O.O.O.OOOOO........
.............O..OO...OO........
.........OOOOO.O...O...........
.........OO.O.O.OOOOOOO........
...............OO.O..OO........
[Figure 43. Period 16 1-D XOR cellular automaton emulator with a
bent rotor (DH)]
In rare cases, the rotor can be bent in more than one place, as in the
oscillators below.
..O......O..................OO.OO....
OOOOO..OOOOO................OOOO.O.OO
OOOOO..OOOOO............OO...OO..OOOO
..O..OO..O..............OOOOO.OO..OO.
...O.OO.O.........OO...OOOOO....OO.OO
..OO.O..OO......OOOO...OOOOO....O..O.
...O....O.......OOO......O.......O.OO
..O......O.......OO.............OO.OO
..OO....OO.......OOO.......O.....O...
OOO......OOO........O....OOOOO..OO.OO
OOOO....OOOO.......O.O.O.OOOOO..OO.OO
..OO....OO........O.O.O.OOOOO........
..................OO..OO...OO........
................OO....OOO............
................OO...OOOO............
.....................OO..............
[Figure 44. Period 14 and 28 1-D XOR cellular automaton emulators
with bent rotors (DH)]
Dean Hickerson found that a staggered boundary with a slope of 1/2 can be
perturbed by sparks to generate ripples along the boundary, which then
returns to its original state. The staggered boundary is symmetrical with
respect to Day and Night, and so the ripples can be created from either side
of the boundary, and can travel in either direction. It is even possible
for ripples to be traveling in both directions at the same time and to
pass though each other. The ripples move with a period of 3 at 2/3 the
speed of light.
The following figure shows multiple ripples being generated by a period 13
oscillator, and traveling to the upper right along the boundary. The ends
of the object are designed to absorb the ripples without damage.
.......................OO.OO.OO
.......................OO.OO.OO
........................O..O...
........................OO.O...
.O....................OO.OO.OO.
O.O.................OOOO....OO.
.OOO..............OOOOOOO.OO...
..O.O...........OOOOOOOOO.OO...
...OOO........OOOOOOOOO........
....O.O.....OOOOOOOOO..........
.....O....OOOOOOOOO............
........OOOOOOOOO..............
......OOOOOOOOO................
....OOOOOOOOO..................
..OOOOOOOOO....................
..OOOOOOO......................
....O.O........................
[Figure 45. Ripples along the boundary between Day and Night]
It turns out that an external oscillator is not required for creating
repeated ripples along the staggered boundary. The ripples can be
reflected by 180 degrees at each end of an object to create an oscillator
which can have any period of the form 28+6N. The following figure shows
a period 70 oscillator of this type. Unfortunately, the oscillator cannot
support more than one ripple on the boundary.
.............................OO.OO...
.............................OO.OO...
...........................OO..O.....
...........................OO..O.....
..........................OOOOOOO.O..
..........................OO.O...OOOO
................................OOOOO
................................O.O..
..............................OO.....
............................OOO......
..........................OOOOOO.....
........................OOOOOOOO.....
........OO............OOOOOOO.O......
......OOOO..........OOOOOOO.....O....
OO..OOOOO.........OOOOOOOOO....OOOO..
OOOOOOOOO.......OOOOOOOOOO.....OOOO..
......O.......OOOOOOOO..OO.......O...
..OO........OOOOOOOO.............O...
..OO.......OOOOOOO............OO.OO..
OOOOO...OOOOOOOO..............OO.OO..
OOOO..OOOOOOOO.......................
..O.OOOOOOOO.........................
....OOOOOO...........................
...OOO.O.............................
..O..................................
OOOO.OO..............................
OOOO.OO..............................
.OO..................................
.OO..................................
[Figure 46. A period 70 ripple oscillator (DH)]
A slight modification of the ripple oscillator creates a double ripple
injector, which produces a pair of ripples 8 generations apart every
28+6N generations. The following period 34 oscillator demonstrates this
modification.
.............................OO.OO...
.............................OO.OO...
...........................OO..O.....
...........................OO..O.....
..........................OOOOOOO.O..
..........................OO.O...OOOO
....................OO..........OOOOO
..................OOOO..........O.O..
................OOOOO.........OO.....
................OOOOO.......OOO......
..................O......OOOOOOO.....
........................OOOOOOOO.....
......................OOOOOOO.O......
....................OOOOOOO.....O....
..................OOOOOOOOO....OOOO..
................OOOOOOOOOO.....OOOO..
..............OOOOOOOO..OO.......O...
..OO........OOOOOOOO.............O...
..OO......OOOOOOOO............OO.OO..
OOOOO...OOOOOOOO..............OO.OO..
OOOO..OOOOOOOO.......................
..O.OOOOOOOO.........................
....OOOOOO...........................
...OOO.O.............................
..O..................................
OOOO.OO..............................
OOOO.OO..............................
.OO..................................
.OO..................................
[Figure 47. A period 34 ripple duplicating oscillator (DH)
The ripples on the staggered boundary can affect objects adjacent to the
boundary. Because of this, a signal can be sent across the boundary so
that a normal area can affect an inverted area, and vice versa. The
following construction shows how a rocket traveling in inverted space
is destroyed by a period 8 oscillator, and in so doing causes a ripple
on the boundary between Day and Night which in turn destroys a passing
rocket traveling in normal space.
x = 78, y = 79, rule = B3678/S34678
53boo$53boo$54boboo$53boob4o$3bobbobbobbobbobbobbobbobbobbobbobbobbobb
o10boobb3o$b44o3booboo3boo$b44o3b4ob3o$46o8boo$b45o4boob4o$b26obbobb
15o3boob4o$25o4bo4b13o7b4o$b24o3bobo3b14o6b4o$b26obbobb16obbo4b4o$53o
bb4o$b59o$60o$b60o$28o4b29o$b24obboo4b29o$b23obo9b27o$11ob4obbobb4o9bo
b27o$b5ob3obo11boo10bob26o$b6o5boo23b27o$6ob3obo11boo10bob27o$b10ob4o
bbobb4o9bob27o$b23obo9b28o$25obboo4b31o$b27o4b31o$b62o$64o$b62o$b62o$
64o$b62o$b62o$3bobbobbobbobbobbobbobbobbobbobbobbobbobbobbobbobbobbobb
obbo4$71b3o$70bobobo$69bob3obo$69b7o$69b7o$68b9o$68bob5obo$69bob3obo$
71b3o$70bobobo$71b3o$70b5o$70b5o$69b7o$69b7o$69b7o$67boob5oboo$69bobbo
bbo$72bo$71b3o$71b3o$70b5o$70b5o$71b3o$72bo$69bob3obo$70b5o$70bo3bo$
71b3o$71b3o$70bobobo$70bobobo$71bobo$$69bobbobbo$68bob5obo$67b11o$69b
7o$71b3o$70booboo!
[Figure 48. A rocket from inverse space destroying a rocket in
normal space (DH)]
SPARK OSCILLATORS
-----------------
Dean Hickerson discovered a large class of amazing oscillators which are
extremely useful. They can produce periodic sparks at a large distance
from the main oscillator body, and can have very high periods. Such spark
oscillators make many constructions such as rocket guns possible because
of the ease with which they can perturb things.
The smallest spark oscillators have already been shown. These are two of
the period 3 oscillators in figure 29. Period 3 spark oscillators occur
naturally in many reactions. A useful spark oscillator which is slightly
larger is shown below and shows more about how they work.
.......O.......
...............
.......O.......
...............
.......O.......
..OOOOO.OOOOO..
OOOOOOOOOOOOOOO
OOOOOOO.OOOOOOO
..OOOOOOOOOOO..
..OOOOO.OOOOO..
....OOOOOOO....
....OOOOOOO....
......O.O......
[Figure 49. A period 8 spark oscillator (DH)]
These spark oscillators work by "bubbling" along the boundary between
Day and Night, where the state of the cells on both sides of the boundary
are inverses of each other. Occasionally, a long row of cells is in the
same state, and then a wave of rows of cells reaches out from the oscillator
to form sparks such as the ones shown above. The bowl shape of the
oscillator is simply due to minimizing of the area of ON cells which contains
the inverted sparks, and is designed to stabilize the boundary between
Day and Night.
Many of the spark oscillators have a companion oscillator where one column
is doubled so as to form a two bit spark instead of a one bit spark.
For the period 8 oscillator, this companion oscillator is the following.
.......OO.......
................
.......OO.......
................
.......OO.......
..OOOOO..OOOOO..
OOOOOOOOOOOOOOOO
OOOOOOO..OOOOOOO
..OOOOOOOOOOOO..
..OOOOO..OOOOO..
....OOOOOOOO....
....OOOOOOOO....
......O..O......
[Figure 50. A period 8 spark oscillator with two-bit sparks (DH)]
Soon after discovering these oscillators, Dean discovered that the bowl
shape could be avoided and only a thin 5 cell thick body needs to be present.
The sparks then can shoot out from both sides of the oscillator. This
improvement is shown below and in most later representations of the spark
oscillators. Except for the corners, such oscillators are symmetrical
around their middle row.
........O........
.................
........O........
.................
........O........
...OOOOO.OOOOO...
.OOOOOOOOOOOOOOO.
.OOOOOOO.OOOOOOO.
OOOOOOOOOOOOOOOOO
OO.OOOOO.OOOOO.OO
........O........
.................
........O........
.................
........O........
[Figure 51. A period 8 spark oscillator whose body is only 5 cells thick (DH)]
This thin-body style does have the slight disadvantage that it is two rows
thicker than the bowl-shaped style. In some applications where the oscillator
is near other objects, the bowl-shaped style must be used instead.
Many different and useful periods have been discovered for these spark
oscillators. Periods such as 3, 8, 14, 20, 30, 40, 46, 48, 50, 64, 106,
240, 256, 320, 440, and many others have been found. Some oscillators
with periods in the many millions have been found. It is believed that
arbitrarily large periods are possible. But only a few of these oscillators
have sparks that stick out far enough to be really useful, and some
oscillators send out multiple sparks to the same point during their period,
thus reducing their usefulness. It is not known which periods are possible,
or how to easily find those oscillators with good sparks. Some large prime
periods have been found. On the other hand, Dean Hickerson has shown that
spark oscillators are impossible for periods 4, 6, or 7.
The following figures show some useful examples of the spark oscillators.
Note that most of these come in two-bit spark versions in addition to the
single-bit spark versions shown here. Also notice that not all of them are
symmetrical. Some spark oscillators are flipped across a vertical axis
after 1/2 of the full period. This means that they produce two sets of
good sparks during their period, and so might need placing in the correct
one of the two possible positions in order to be useful for a particular
reaction. Some spark oscillators show no symmetry at all during their period.
..........O..........
.....................
..........O..........
.....................
..........O..........
.....OO.......OO.....
..........O..........
...OOOOOOO.OOOOOOO...
.OOOO..OOOOOOO..OOOO.
.OOOOOOOOO.OOOOOOOOO.
OOOOO..OOOOOOO..OOOOO
OO.OOOOOOO.OOOOOOO.OO
..........O..........
.....OO.......OO.....
..........O..........
.....................
..........O..........
.....................
..........O..........
[Figure 52. A period 20 spark oscillator (DH)]
.................O.................
...................................
.................O.................
...................................
.................O.................
...................................
.................O.................
...................................
.................O.................
............OOOO...OOOO............
...OOOOOOOOO....OOO....OOOOOOOOO...
.OOOOOOOOOOOOOOOO.OOOOOOOOOOOOOOOO.
.OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO.
OOOOOOOOOOOOOOOOO.OOOOOOOOOOOOOOOOO
OO.OOOOOOOOO....OOO....OOOOOOOOO.OO
............OOOO...OOOO............
.................O.................
...................................
.................O.................
...................................
.................O.................
...................................
.................O.................
...................................
.................O.................
[Figure 53. A period 30 spark oscillator]
.......................O.......................
...............................................
.......................O.......................
...............................................
.......................O.......................
...............................................
.......................O.......................
....OO..OO..OO......OO...OO......OO..OO..OO....
...O..OO..OO..OOOOOO..OOO..OOOOOO..OO..OO..O...
.OOOOOOOOOOOOOOOOOOOOOO.OOOOOOOOOOOOOOOOOOOOOO.
.OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO.
OOOOOOOOOOOOOOOOOOOOOOO.OOOOOOOOOOOOOOOOOOOOOOO
OO.O..OO..OO..OOOOOO..OOO..OOOOOO..OO..OO..O.OO
....OO..OO..OO......OO...OO......OO..OO..OO....
.......................O.......................
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[Figure 54. A period 40 spark oscillator]
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.......OOO...................OOO.......
....OO.....OOOO....O....OOOO.....OO....
...O..OOOOO....OOOO.OOOO....OOOOO..O...
.OOOOOO...OOOOOOOOOOOOOOOOOOO...OOOOOO.
.OOOOOOOOOOOOOOOOOO.OOOOOOOOOOOOOOOOOO.
OOOOOOO...OOOOOOOOOOOOOOOOOOO...OOOOOOO
OO.O..OOOOO....OOOO.OOOO....OOOOO..O.OO
....OO.....OOOO....O....OOOO.....OO....
.......OOO...................OOO.......
...................O...................
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...................O...................
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...................O...................
[Figure 55. A period 48 spark oscillator (DH)]
..........O.....................
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..........O.....................
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..........O.....................
.....OO.......OOOO...OO.........
..........O........O....OO......
...OOOOOOO.OOOOOOOO.OOOO..OOO...
.OOOO..OOOOOOO....OOO..OOOOOOOO.
.OOOOOOOOO.OOOOOOOOOOOOOOOOOOOO.
OOOOO..OOOOOOO....OOO..OOOOOOOOO
OO.OOOOOOO.OOOOOOOO.OOOO..OOO.OO
..........O........O....OO......
.....OO.......OOOO...OO.........
..........O.....................
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[Figure 56. A period 64 spark oscillator]
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.....OO...................OO.....
........OOOO....O....OOOO........
...OOOOO....OOOO.OOOO....OOOOO...
.OOOO..OOOOOOOOOOOOOOOOOOO..OOOO.
.OOOOOOOOOOOOOOO.OOOOOOOOOOOOOOO.
OOOOO..OOOOOOOOOOOOOOOOOOO..OOOOO
OO.OOOOO....OOOO.OOOO....OOOOO.OO
........OOOO....O....OOOO........
.....OO...................OO.....
................O................
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................O................
[Figure 57. A period 106 spark oscillator (DH)]
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...............O......OO........................
....OOOO....OO...OOOO....OOOOOOOOOO.OOOO..OO....
...O....OOOO..OOO....OOOO..........O....OO..O...
.OOOOOOOOOOOOOO.OOOOOO..OOOOOOOOOOOOOOOOOOOOOOO.
.OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO.
OOOOOOOOOOOOOOO.OOOOOO..OOOOOOOOOOOOOOOOOOOOOOOO
OO.O....OOOO..OOO....OOOO..........O....OO..O.OO
....OOOO....OO...OOOO....OOOOOOOOOO.OOOO..OO....
...............O......OO........................
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[Figure 58. A period 240 spark oscillator]
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.....OO............O......OO.............
........OOOO....OO...OOOO........OOOO....
...OOOOO....OOOO..OOO....OOOOOOOO....O...
.OOOO..OOOOOOOOOOOO.OOOOOO..OOOOOOOOOOOO.
.OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO.
OOOOO..OOOOOOOOOOOO.OOOOOO..OOOOOOOOOOOOO
OO.OOOOO....OOOO..OOO....OOOOOOOO....O.OO
........OOOO....OO...OOOO........OOOO....
.....OO............O......OO.............
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[Figure 59. A period 256 spark oscillator (DH)]
The following period 440 spark oscillator shoots sparks that reach out 25
cells from the base line!
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...........................OO.......OO...........................
....OOOOOOOO..OO..OOOO..........O..........OOOO..OO..OOOOOOOO....
...O........OO..OO....OOOOOOOOOO.OOOOOOOOOO....OO..OO........O...
.OOOOOOOOOOOOOOOOOOOOOOOOOO..OOOOOOO..OOOOOOOOOOOOOOOOOOOOOOOOOO.
.OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO.OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO.
OOOOOOOOOOOOOOOOOOOOOOOOOOO..OOOOOOO..OOOOOOOOOOOOOOOOOOOOOOOOOOO
OO.O........OO..OO....OOOOOOOOOO.OOOOOOOOOO....OO..OO........O.OO
....OOOOOOOO..OO..OOOO..........O..........OOOO..OO..OOOOOOOO....
...........................OO.......OO...........................
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[Figure 60. A period 440 spark oscillator]
There is a way to shrink the size occupied by spark oscillators if they
only have one large spike, such as in the period 440 oscillator. This is
to add a "slab" of inert cells below the oscillator that will inhibit some
of the sparks on that side. The use of the slab is shown below.
......OO.................................................OO......
...OOO..OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO..OOO...
.OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO.
.OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO.
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
OO.OOO..OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO..OOO.OO
......OO.................................................OO......
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..................O.O.O.O.O.O.O.O.O.O.O.O.O.O.O..................
................OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO................
................OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO................
..................O..O..O..O..O.O.O..O..O..O..O..................
[Figure 61. A "slab" inhibits some of the sparks from a period 440 oscillator]
Sometimes the "slab" which inhibits the sparks can be internal to the
oscillator itself, and is made up of the boundary of the "bowl" shaped
boundary of the oscillator. An example of this is shown in the following
figure which shows the smallest form of the period 8 spark oscillator.
........O........
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........O........
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........O........
OO.OOOOO.OOOOO.OO
OOOOOOOOOOOOOOOOO
.OOOOOOO.OOOOOOO.
.OOOOOOOOOOOOOOO.
...OOOOO.OOOOO...
.....O.....O.....
[Figure 62. A period 8 spark oscillator with spark inhibitors on the boundary]
The previous spark oscillators were shown mostly because they are useful
in having a reasonably low period and large sparks. The spark oscillators
shown next don't have much use, but are shown to demonstrate the large
periods that the spark oscillators can produce.
The following spark oscillator has period 207991871, which is the largest
period spark oscillator that has been constructed. This period also happens
to be prime. This oscillator shoots sparks 32 cells out from the base line
and is shown in the generation which has those sparks.
x = 112, y = 69, rule = B3678/S34678
35boo$$35boo$$35boo$$35boo$$35boo$$35boo$$35boo$$35boo$$35boo$$35boo$$
35boo$$35boo$$35boo$$35boo$$35boo32boo12b14o$76boo$9boo10boo12boo6boo
10boo12boo12b14o$12b4o8b4o4boo4b4o8b4o4boo6boo4boobboobboo16b10o$oob9o
4b8o4b4obb4o4b8o4b4obb6obb4obboobboobb16o10boboo$9obb10obb12obb6obb10o
bb12obb12o14b15o$b75obb33o$b8obb10obb12obb6obb10obb12obb12o14b14o$3b9o
4b8o4b4obb4o4b8o4b4obb6obb4obboobboobb16o10bo$12b4o8b4o4boo4b4o8b4o4b
oo6boo4boobboobboo16b10o$9boo10boo12boo6boo10boo12boo12b14o$76boo$35b
oo32boo12b14o$$35boo$$35boo$$35boo$$35boo$$35boo$$35boo$$35boo$$35boo
$$35boo$$35boo$$35boo$$35boo$$35boo$$35boo!
[Figure 63. A period 207991871 spark oscillator]
Spark oscillators having arbitrarily large periods may result by continuing
the series of spark oscillators shown in the following figure. Here each
spark oscillator has a period twice as large as the previous one. In
particular, the periods are 3*2^(N-1), for N larger than 2. However,
their widths are 2^N + 2, so the oscillators also double in size through
the series. All of the sparks produced by this series of oscillators are
small and similar to each other.
x = 136, y = 63, rule = B3678/S34678
4boobb4o$oobobboo4boboo$16o$b14o$b14o$3bobboo4bo$4boobb4o8$4boobboobb
oobb4o$oobobboobboobboo4boboo$24o$b22o$b22o$3bobboobboobboo4bo$4boobb
oobboobb4o8$4boobboobboobboobboobboobboobb4o$oobobboobboobboobboobboo
bboobboo4boboo$40o$b38o$b38o$3bobboobboobboobboobboobboobboo4bo$4boobb
oobboobboobboobboobboobb4o8$4boobboobboobboobboobboobboobboobboobboobb
oobboobboobboobboobb4o$oobobboobboobboobboobboobboobboobboobboobboobb
oobboobboobboobboo4boboo$72o$b70o$b70o$3bobboobboobboobboobboobboobboo
bboobboobboobboobboobboobboobboo4bo$4boobboobboobboobboobboobboobboobb
oobboobboobboobboobboobboobb4o8$4boobboobboobboobboobboobboobboobboobb
oobboobboobboobboobboobboobboobboobboobboobboobboobboobboobboobboobboo
bboobboobboobboobb4o$oobobboobboobboobboobboobboobboobboobboobboobboo
bboobboobboobboobboobboobboobboobboobboobboobboobboobboobboobboobboobb
oobboobboo4boboo$136o$b134o$b134o$3bobboobboobboobboobboobboobboobboo
bboobboobboobboobboobboobboobboobboobboobboobboobboobboobboobboobboobb
oobboobboobboobboobboo4bo$4boobboobboobboobboobboobboobboobboobboobboo
bboobboobboobboobboobboobboobboobboobboobboobboobboobboobboobboobboobb
oobboobboobb4o!
[Figure 64. Spark oscillators with periods in the series 12, 24, 48,
96, 192, ... (DH)]
Dean Hickerson found that it is easy to program an algorithm which will
simulate the behavior of any spark oscillator, without needing to calculate
the Day & Night universe in full. All that is needed is to track the height
of each column above the Day & Night boundary, and follow some simple rules.
Assign a non-negative integer state to each column on the top boundary row
of the oscillator. For the period 8 oscillator in figure 49, there are 11
columns. (The three columns on each end of the oscillator are not
considered.) The state value indicates the height of that column, which is
the distance from the base row to the furthest spark in that column.
From that spark back down to the base row, the cells alternate ON and OFF.
For the period 8 spark oscillator in the generation shown, the states of
the columns are 00000500000.
To simulate the next generation of the oscillator, consider each internal
column and its two closest neighbors. (The end columns always have state 0.)
Use the following table of the three column states to determine the new
state for the column. Here N represents any integer greater than zero,
and E represents any even integer greater than zero. The 0 and 1 states
are shown literally. Only the combinations shown in the table can exist as
part of the period of a spark oscillator.
neighbor cell neighbor -> new cell
-------- ---- -------- --------
0 0 0 1
1 0 1 1
0 0 N N mod 2
N 0 0 N mod 2
1 0 E 0
E 0 1 0
0 N 0 0
0 N N 0
N N 0 0
N N N N + 1
As an example of this simulation, the columns in the period 8 spark
oscillator go through the following states:
gen 0: 00000500000
gen 1: 01111011110
gen 2: 00220102200
gen 3: 00000000000
gen 4: 01111111110
gen 5: 00222222200
gen 6: 00033333000
gen 7: 01104440110
gen 8: 00000500000
Most of the spark oscillators shown above were discovered by running such a
simulator on various numbers of columns with random legal beginning states.
There are probably many more oscillators to be found having a particular
number of columns. Some column sizes can support many different periods.
Aslo, some sizes can support many oscillators of the same period, a fact
for which no explanation has been found. For example, there are at least 31
period 46 oscillators with a width of 34. There is some structure known
for the spark oscillators, but much about them is still obscure.