## Spiral and polygonal growth in B34568/S15678

(written 6/2/2006)

Introduction
Spiral growth
The c/4 corner
Wickstretchers
Polygonal growth
The c/8 corner
Some stranger patterns
Growth spurts (5c/19 plus c/8)
Growth spurts (2nc/(10n+6) plus c/8)
The 25c/150 corner
Growth spurts (c/4 plus c/8)
Questions

### Introduction

In the rule B34568/S15678, most random patterns that I've looked at stop growing rather quickly, ending up as polygons with horizontal, vertical, and diagonal sides (with slopes 1 and -1); the edges have period 2 and the corners have small periods, typically 4, 6, 8, and 10. But several types of infinite growth can occur, both linear (spirals and wickstretchers) and quadratic (expanding polygons).

Note: Many of the patterns given here take a long time to finish (in one case more than 10^22 generations), and you'll need a program that implements Bill Gosper's hashlife algorithm to run them. I recommend "Golly", by Tomas Rokicki, Andrew Trevorrow, Dave Greene, Alan Hensel, and Jason Summers, which can be found here: Golly

### Spiral growth

There's a structure, the 'plow', which moves along an edge of a rectangle, increasing its size; here it's shown moving west along a northern edge:

```
.......o.o.o.                        ......o.o.o.o
.o.o.o.oooooo                        o.o.o.ooooooo
... ooooooooooooo ...       ->       ... ooooooooooooo ...
ooooooooooooo                        ooooooooooooo

```

The plow works in 64 rules: B356/S5678 with optional B478 and S014. In at least 4 of these rules (B3456/S15678 with optional B8 and S0) the plow can turn a corner, allowing spiral growth. Here's an example that works in all 4 of these rules:

```
#C A plow moves counterclockwise around a square, causing it to expand
#C forever.  Population growth is linear, but every cell in the plane
#C eventually becomes and remains alive.
x = 10, y = 10, rule = B34568/S15678
5bobo\$bobob4o\$10o\$b8o\$10o\$b8o\$9o\$b9o\$9o\$bobobobobo!

```

(There's another type of plow that works in 16 rules, B37/S345678 with optional B8 and S012. In these rules, a plow doesn't turn a corner, but instead bounces back from it:

```
#C 4 plows move back and forth along the edges of a square, causing it
#C to expand forever.  Population growth is linear, but every cell in
#C the plane eventually becomes and remains alive.
x = 18, y = 18, rule = B378/S012345678
12bo\$9b6o\$3bobobob6o\$b15o\$b14o\$16o\$b14o\$b15o\$b14o\$3b14o\$2b15o\$3b14o\$2b
16o\$3b14o\$2b15o\$3b6obobobo\$3b6o\$5bo!

```

Except for the plows, these rules are quite different from B34568/S15678.)

Here are some examples of small patterns which produce up to 12 plows:

```
#C Spiral growth with 1, 2, 3, 4, 5, 6, 8, and 12 plows.  (These crash
#C in gen 2005.)
x = 217, y = 240, rule = B34568/S15678
213bobo\$213bo2bo\$211bo2bo\$212bobo2\$129bo\$128bo2bo\$131bo\$129b3o\$128bo3b
o5\$56bo2b2o\$57bobo\$57b3o\$60bo7\$2bobo\$obobo\$2o\$b2o70\$28b6o\$28b4o\$28bo3b
2o\$28bo2bo\$30b2obo100b4o\$137b3o\$134b2o2b2o\$134b3o\$136b4o104\$15b2obo2b
2o\$15b2obob3o\$16b2o2bo\$21b2o\$15b2o\$17bo2b2o\$15b3obob2o\$15b2o2bob2o16\$
156b4obobo\$159bo3bo\$156bo3bo2bo\$158bo3b2o\$156b2o3bo\$156bo2bo3bo\$156bo
3bo\$156bobob4o!

```

Note that in the 1, 3, 5, 8, and 12-plow cases, there are 'flaws' near some of the corners, places where the usual sawtooth edge is modified. When a plow crosses a flaw, it's slowed down by 1 gen. This change in timing is necessary for an odd number of plows, and is also important for some forms of non-spiral growth. (Flaws aren't needed for 8 or 12 plows; they just happen to be present in these particular patterns.)

I haven't seen more than 12 plows occur naturally, but we can construct patterns with as many as we want. E.g. here 32 of them make a square grow and rotate:

```
#C Square with 32 plows grows and slowly rotates, its sides becoming
#C asymptotically parallel to the axes.
x = 71, y = 71, rule = B34568/S15678
58bobo\$50bobobobob4o\$42bobobobob14o\$34bobobobob21o\$26bobobobob30o\$18bo
bobobob37o\$10bobobobob46o\$2bobobobob53o\$2b62o\$b62o\$65o\$b63o\$65o\$2b62o\$
b64o\$2b62o\$b64o\$2b62o\$b65o\$2b63o\$b65o\$3b62o\$2b64o\$3b62o\$2b64o\$3b62o\$2b
65o\$3b63o\$2b65o\$4b62o\$3b64o\$4b62o\$3b64o\$4b62o\$3b65o\$4b63o\$3b65o\$5b62o\$
4b64o\$5b62o\$4b64o\$5b62o\$4b65o\$5b63o\$4b65o\$6b62o\$5b64o\$6b62o\$5b64o\$6b
62o\$5b65o\$6b63o\$5b65o\$7b62o\$6b64o\$7b62o\$6b64o\$7b62o\$6b65o\$7b63o\$6b65o\$
8b62o\$7b62obo\$8b53obobobobo\$7b46obobobobo\$8b37obobobobo\$7b30obobobobo\$
8b21obobobobo\$7b14obobobobo\$9b4obobobobo\$10bobo!

```

There are actually two ways that a plow can turn a corner, depending on the phase of the corner when the plow arrives. The resulting positions for the plow after the turn differ by 2 units. When two successive plows turn a corner for the first time, the distance between them increases by either 0, 2, or 4 units. But after the turn, the distance between them is always congruent to 2 (mod 4), and at subsequent turns their distance increases by either 0 or 4 units. Furthermore, if a sequence of plows, whose successive distances are already 2 (mod 4), turns a corner, then the increases in distance alternate between 0 and 4.

When we add the results of 4 corners, there are 3 possible outcomes. Either the distances between successive plows all increase by 8, or half of them increase by 16 while the others remain the same, or half increase by 4 while the others increase by 12:

```  ... 0  4  0  4 ...         ... 0  4  0  4 ...         ... 0  4  0  4 ...
+ ... 0  4  0  4 ...       + ... 0  4  0  4 ...       + ... 0  4  0  4 ...
+ ... 4  0  4  0 ...       + ... 0  4  0  4 ...       + ... 0  4  0  4 ...
+ ... 4  0  4  0 ...       + ... 0  4  0  4 ...       + ... 4  0  4  0 ...
------------------         ------------------       --------------------
... 8  8  8  8 ...         ... 0 16  0 16 ...         ... 4 12  4 12 ...

```

With an odd number of plows, the result of 2 complete trips around the rectangle is an increase of 16 in all distances between successive plows, so the plows become more and more evenly spaced around the rectangle. With an even number N, we can either end up with N evenly spaced plows or N/2 evenly spaced pairs of close plows, or the distances between successive plows can be in the ratio 1 : 3 : 1 : 3 : ... Here are examples of all 3 possibilities:

```
#C The 3 possible asymptotic spacings between 2, 4, and 6 plows.
#C (These crash in gen 1168.)
x = 189, y = 165, rule = B34568/S15678
4bo166bo\$80bob2o87bobo\$b2obo75bo4bo87bo\$80b2o2bobo\$2ob2o76bob2obo84b3o
bo67\$2bo\$2b2o168bo2bo\$b5o76bobo86b2o\$5o77bo2bo85b2ob2o\$2b2o76bo2bo87b
4o\$3bo77bobo87b4o75\$91bobo\$83bobobobob4o\$81b15o89bobo\$80b15o80bobobobo
bob2o\$81b15o75bobob13o\$80b15o77b15o\$81b16o74b17o\$80b16o77b14o\$82b15o
75b17o\$4o77b15o77b15o\$3b3o76b15o75b17o\$2o2b2o75b13obo77b11obob2o\$3o78b
obobobobobobo79bobobobobobo\$2b4o76b2o89bo!

```

Plows can also survive a collision, if conditions are right:

```
#C 2 and 4 plows repeatedly bounce off (or pass through) each other.
#C (These crash in gen 2385.)
x = 90, y = 4, rule = B34568/S15678
4o80bob2obo\$4o80bo4bo\$o2bo80bo4bo\$o2bo80bob2obo!

```

### The c/4 corner

Many examples of non-spiral growth, both linear and quadratic, involve a corner that grows orthogonally at c/4; here 4 of them fill the plane:

```
#C Asymptotic shape is an octagon with vertices (in gen t) at (t/4, 0),
#C (t/5, t/5), and their rotations.  Population is asymptotic to t^2/5.
#C Each of the corners along the axes grows at c/4, sending plows in
#C both perpendicular directions.  These plows meet along the lines y=x
#C and y=-x, creating the other 4 corners.
x = 5, y = 5, rule = B34568/S15678
b3o\$5o\$5o\$5o\$b3o!

```

In the pattern below, one c/4 corner fills 1/4 of the plane, specifically the region -x < y < x. But it's a bit complicated: 4/5 of the time there are 3 c/4 corners, the large one growing to the east, and smaller ones growing north and south. When a northward corner meets the line y=x, it stops growing and a horizontal edge starts growing to the east. When that line meets the northward plows from the eastward corner, a new northward corner forms and the process repeats. Although the asymptotic shape is simply a quadrilateral, two of its corners grow arbitrarily large.

```
#C c/4 corner fills 1/4 of the plane
#C Asymptotic shape is a convex quadrilateral, with vertices at (0,0),
#C (t/5, -t/5), (t/4, 0), and (t/5, t/5).  At the top and bottom corners,
#C the horizontal distance between the edges fluctuates between
#C 2 sqrt(t/5)  and  8/5 sqrt(t/5);  the horizontal distance fluctuates
#C between 0 and  2/5 sqrt(t/5).
x = 5, y = 5, rule = B34568/S15678
5o\$o2bo\$o2bo\$o2bo\$5o!

```

The northward c/4 corner can also be stopped by a flaw on the horizontal edge, rather than by reaching the slope 1 diagonal; as the corner grows northward, its westward plows pull the flaw northeast until the flaw reaches the corner and stops its growth:

```
#C c/4 corner fills 1/4 of the plane
x = 32, y = 9, rule = B34568/S15678
4bobobobobobobobobobo2bobob2o\$2b30o\$2b29o\$32o\$32o\$b31o\$b30o\$b31o\$3bobo
bobobobobobobobobobobob2o!

```

### Wickstretchers

The reactions in which c/4 corners start and stop growing can be used to construct infinitely many speeds of diagonal wickstretchers, as in the example below:

```
#C 50c/256 wickstretcher
#C c/4 corners grow south and east, stopping when they reach flaws
#C and forming new corners when the last of their plows reach the
#C southeast corner.
x = 74, y = 76, rule = B34568/S15678
60b4o\$60b4o\$58b8o\$58b8o\$56b12o\$56b12o\$54b16o\$54b15o\$52b18o\$52b17o\$50b
20o\$50b19o\$48b22o\$48b21o\$46b24o\$46b23o\$44b26o\$44b25o\$42b27o\$42b28o\$40b
29o\$40b30o\$38b31o\$38b32o\$36b33o\$36b34o\$34b35o\$34b36o\$32b37o\$32b38o\$30b
39o\$30b40o\$28b41o\$28b42o\$26b43o\$26b44o\$24b45o\$24b46o\$22b47o\$22b48o\$20b
49o\$20b50o\$18b51o\$18b52o\$16b53o\$16b54o\$14b55o\$14b57o\$12b58o\$12b59o\$10b
60o\$10b61o\$8b62o\$8b63o\$6b64o\$6b65o\$4b66o\$4b68o\$2b69o\$2b70o\$71o\$73o\$2b
70o\$2b71o\$4bobobobobob2obobob51o\$21bobobobobob43o\$31bobob38o\$35bobob
35o\$39bobob31o\$43bobob27o\$47bobob22o\$51bobob19o\$55bobob14o\$59bobob5obo
bo\$63bo3bo\$65bo!

```

If we move both flaws 2 units farther from the southeast corner, we increase the period by 10 and the distance that the wickstretcher grows per period by 2, increasing the speed to 52c/266. (If we move just the east-edge flaw 2 units north, then the pattern still grows diagonally until about gen 5840, at which point the last plows on both edges reach the corner too close together; the pattern stops growing in gen 10556.)

We can also move both flaws closer to the corner to decrease the period and speed; in this way we obtain wickstretchers of speed 2nc/(10n+6) for all n >= 4. We can also adjust the relative timing of the south and east edges; each must follow the other by 23+10k gens for some k >= 0. And by changing the lengths of the edges west and north of the flaws, we can make such wickstretchers arbitrarily wide.

Here are examples of the different timings of wickstretchers with n from 0 to 5. In these, the flaws have been replaced by the diagonal edges, to give examples with minimum width (I think).

```
#C 12 wickstretchers, with speeds 8c/46, 10c/56, 12c/66 (2 forms),
#C 14c/76 (2 forms), 16c/86 (3 forms), and 18c/96 (3 forms).
x = 139, y = 298, rule = B34568/S15678
134b2o\$94bobobo7bo3bo3bo3bobobo11b2o\$132b6o\$94bo3bo6bo4bo3bo3bo13b6o\$
130b9o\$94bobobo5bo5bobobo3bobobo7b8o\$128b11o\$94bo3bo4bo10bo3bo3bo5b10o
\$128b11o\$94bobobo3bo11bo3bobobo5b10o\$130b8o\$130bobobobobo\$134bo8\$134b
2o\$134b2o\$132b6o\$88bo3bobobo7bo3bobobo3bobobo11b6o\$130b9o\$88bo3bo3bo6b
o4bo7bo13b8o\$128b11o\$88bo3bo3bo5bo5bobobo3bobobo7b10o\$126b13o\$88bo3bo
3bo4bo10bo3bo3bo5b12o\$126b13o\$88bo3bobobo3bo7bobobo3bobobo5b12o\$128bob
8o\$130bobobobobo\$134bo10\$106b2o26b2o\$106b2o26b2o\$104b6o22b6o\$104b6o22b
6o\$102b10o18b9o\$58bo3bobobo7bo3bobobo3bobobo11b10o18b8o\$100b13o15b11o\$
58bo7bo6bo4bo7bo13b12o16b10o\$98b15o13b13o\$58bo3bobobo5bo5bobobo3bobobo
7b14o14b12o\$96b17o11b15o\$58bo3bo8bo6bo3bo3bo3bo5b16o12b14o\$96b17o11b
15o\$58bo3bobobo3bo7bobobo3bobobo5b16o12b14o\$98b15o13b12o\$98bob12o14bob
ob5obobo\$100bobob8o18bo3bo\$104bobobobobo19bo\$108bo38\$102b2o30b2o\$102b
2o30b2o\$100b6o26b6o\$100b6o26b6o\$98b10o22b9o\$98b10o22b8o\$96b13o19b11o\$
52bo3bo3bo7bo3bobobo3bobobo11b12o20b10o\$94b15o17b13o\$52bo3bo3bo6bo8bo
3bo13b14o18b12o\$92b17o15b15o\$52bo3bobobo5bo9bo3bobobo7b16o16b14o\$90b
19o13b17o\$52bo7bo4bo10bo3bo3bo5b18o14b16o\$90b19o13b17o\$52bo7bo3bo11bo
3bobobo5b18o14b16o\$92b17o15bob12o\$92bobob12o18bobob5obobo\$96bobob8o22b
o3bo\$100bobobobobo23bo\$104bo44\$62b2o34b2o34b2o\$62b2o34b2o34b2o\$60b6o
30b6o30b6o\$60b6o30b6o30b6o\$58b10o26b10o26b9o\$58b10o26b10o26b8o\$56b13o
23b13o23b11o\$56b12o24b12o24b10o\$54b15o21b15o21b13o\$10bo3bobobo7bo3bobo
bo3bobobo11b14o22b14o22b12o\$52b17o19b17o19b15o\$10bo3bo10bo4bo3bo3bo13b
16o20b16o20b14o\$50b19o17b19o17b17o\$10bo3bobobo5bo5bobobo3bobobo7b18o
18b18o18b16o\$48b21o15b21o15b19o\$10bo3bo3bo4bo6bo3bo3bo3bo5b20o16b20o
16b18o\$48b21o15b21o15b19o\$10bo3bobobo3bo7bobobo3bobobo5b20o16b20o16b
18o\$50bob17o17b19o17b16o\$52bobob12o18bob16o18bobob9obobo\$56bobob8o20bo
bob12o22bobobobobo\$60bobobobobo23bobob5obobo25bo\$64bo31bo3bo\$98bo85\$
54b2o38b2o38b2o\$54b2o38b2o38b2o\$52b6o34b6o34b6o\$52b6o34b6o34b6o\$50b10o
30b10o30b9o\$50b10o30b10o30b8o\$48b13o27b13o27b11o\$48b12o28b12o28b10o\$
46b15o25b15o25b13o\$46b14o26b14o26b12o\$44b17o23b17o23b15o\$o3bobobo7bo3b
obobo3bobobo11b16o24b16o24b14o\$42b19o21b19o21b17o\$o3bo3bo6bo4bo3bo3bo
13b18o22b18o22b16o\$40b21o19b21o19b19o\$o3bobobo5bo5bobobo3bobobo7b20o
20b20o20b18o\$38b23o17b23o17b21o\$o3bo3bo4bo10bo3bo3bo5b22o18b22o18b20o\$
38b23o17b23o17b21o\$o3bobobo3bo7bobobo3bobobo5b22o18b22o18b20o\$40bobob
17o19b21o19bob16o\$44bobob12o20bobob16o22bobob9obobo\$48bobob8o24bobob
12o26bobobobobo\$52bobobobobo27bobob5obobo29bo\$56bo35bo3bo\$94bo!

```

I've seen 8 different speeds of diagonal wickstretchers occur naturally: 5c/19, 8c/46, 10c/56, 12c/66, 14c/76, 16c/86, 28c/146, and 56c/286. All but the first of these are of the type just described, with n = 4, 5, 6, 7, 8, 14, and 28. The 5c/19 is the most commonly occurring wickstretcher. It also has c/4 corners during part of its cycle, but I don't see a way to generalize it.

```
#C Diagonal wickstretchers formed by random patterns
x = 366, y = 413, rule = B34568/S15678
329bobobo7bo3bo3bobobo2\$329bo10bo4bo3bo3bo7b5o\$365bo\$329bobobo5bo5bo3b
obobo8b4o2\$333bo4bo6bo7bo9bobo2\$329bobobo3bo7bo3bobobo32\$324bobobo7bo
3bo3bo3bobobo2\$324bo3bo6bo4bo3bo3bo13b2obo\$364b2o\$324bobobo5bo5bobobo
3bobobo8bo\$361b5o\$324bo3bo4bo10bo3bo3bo9b3o2\$324bobobo3bo11bo3bobobo
46\$250bo3bobobo7bo3bobobo3bobobo2\$250bo3bo3bo6bo4bo7bo14bobo\$291bo3bo
29b3o34b2obo\$250bo3bo3bo5bo5bobobo3bobobo8b3o31b3obo31bob2o\$291b2ob2o
33bo31bob2o\$250bo3bo3bo4bo10bo3bo3bo9b4o30b4o31bobobo\$325bob3o34bo\$
250bo3bobobo3bo7bobobo3bobobo21\$182bo3bobobo7bo3bobobo3bobobo\$291bob3o
\$182bo7bo6bo4bo7bo13bob2o63bo2b2o\$223b3obo63bo2bo\$182bo3bobobo5bo5bobo
bo3bobobo12bo63b3o\$223bo67b5o\$182bo3bo8bo6bo3bo3bo3bo8bob3o2\$182bo3bob
obo3bo7bobobo3bobobo23\$177bo3bo3bo7bo3bobobo3bobobo2\$177bo3bo3bo6bo8bo
3bo12bobo\$218b4obo\$177bo3bobobo5bo9bo3bobobo8bo2bobo\$222b2o\$177bo7bo4b
o10bo3bo3bo10b2o\$219b2o2bo\$177bo7bo3bo11bo3bobobo27\$160bo3bobobo7bo3bo
bobo3bobobo2\$160bo3bo10bo4bo3bo3bo14bo\$202b2obo\$160bo3bobobo5bo5bobobo
3bobobo8bo2bo\$202bobo\$160bo3bo3bo4bo6bo3bo3bo3bo8bo2b2o2\$160bo3bobobo
3bo7bobobo3bobobo40\$151bobobo3bobobo7bo3bo3bo3bo3bobobo2\$155bo3bo3bo6b
o4bo3bo3bo3bo12bo2b3o\$202bo\$151bobobo3bobobo5bo5bo3bobobo3bobobo8bob4o
\$200bo\$151bo7bo3bo4bo6bo7bo3bo3bo8b5o\$203bo\$151bobobo3bobobo3bo7bo7bo
3bobobo85\$3bo\$2bo73\$obobo3bobobo7bo3bobobo3bobobo3bobobo\$53bo3bobo\$o7b
o10bo8bo3bo3bo3bo12b2ob2obo\$54bo4bo\$obobo3bobobo5bo5bobobo3bobobo3bobo
bo11bo\$53bo4bo\$4bo3bo3bo4bo6bo7bo3bo3bo3bo8bob2ob2o\$53bobo3bo\$obobo3bo
bobo3bo7bobobo3bobobo3bobobo!

```

There's also a 2c/10 orthogonal wickstretcher, but it's easily destroyed by anything that touches it, and is quite rare.

```
#C 2c/10 orthogonal wickstretcher
x = 5, y = 6, rule = B34568/S15678
ob3o\$bobo\$2o2bo\$2o2bo\$bobo\$ob3o!

```

### Polygonal growth

In all of the diagonal wickstretchers, the growing corner sends plows in two directions; these plows hit diagonal edges and stop. But if two such corners are growing perpendicular to each other, their plows can travel until they meet each other; in this way we can obtain quadratic growth. For example, here a 5c/19 corner and an 8c/46 corner fill a quarter-plane:

```
#C Asymptotic shape is a concave quadrilateral, with vertices
#C (0,0), (5t/19, 5t/19), (-13t/77, 40/231), and (-4t/23, 4t/23).
#C Population is asymptotic to 9161/201894 t^2.
x = 31, y = 11, rule = B34568/S15678
bob2o\$2obo\$2o2bo\$2o2bo\$b2o2\$26b3obo\$26bobo\$26bobobo\$26bobo\$26bo!

```

For two corners to be compatible in this way, their speeds s and t must satisfy t >= s/(1+2s) and s >= t/(1+2t); otherwise one corner's plows will catch up with and destroy the other corner. These inequalities are satisfied by all pairs of the speeds mentioned so far, so we can combine any 2, 3, or 4 of them to fill 1/4, 1/2, 3/4, or all of the plane. Here's an artificial example with 4 different corner speeds, which fills 3/4 of the plane:

```
#C Polygonal growth based on diagonal corners of speeds 38c/196 (SW),
#C 8c/46 (SE), 5c/19 (NE), and 10c/56 (NW).  Asymptotic shape is a
#C concave octagon with vertices at (0,0), (-19t/98, -19t/98),
#C (5t/109, -152t/981), (4t/23, -4t/23), (40t/231, -13t/77),
#C (5t/19, 5t/19), (-3t/19, 10t/57), and (-5t/28, 5t/28).
#C Population is asymptotic to 221367875/1848541464 t^2.
x = 87, y = 78, rule = B34568/S15678
9bo\$7bo3bo\$5bob5obobo\$6b10ob2o61b3o\$5b14obobobobobobobobobobobobobobob
obobobobobobobobobobob2obobo2bobobo2bo\$6b79o\$5b82o\$6b79obo\$5b82o\$6b79o
\$5b81o\$6b79o\$6b79o\$5b81o\$6b79o\$6b80o\$8b10ob66o\$8b7ob3o2b65o\$10b5obo2b
66o\$10b6o2bob66o\$12b2o5b65o\$12b2o6b65o\$19b64o\$20b64o\$19b63o\$12b2o6b63o
\$12b2o5b63o\$10b6o2bob63o\$10b5obo2b62o\$8b7ob3o2b61o\$8b10ob62o\$6b76o\$6b
74o\$5b76o\$5b75o\$6b75o\$5b75o\$6b75o\$5b75o\$6b75o\$5b75o\$6b75o\$5b75o\$6b75o\$
5b75o\$6b75o\$5b75o\$6b75o\$5b75o\$6b75o\$4b76o\$5b76o\$4b76o\$5b76o\$4b76o\$5b
76o\$4b76o\$5b77o\$4b77o\$5b77o\$3b78o\$4b79o\$3b79o\$4b79o\$2b80o\$3b81o\$2b81o\$
3b81o\$b83o\$2b82o\$b84o\$2b82o\$86o\$2b82o\$b84o\$2b81obo\$bo2bobobobobobobobo
bobobobobobobobobobo2bobobobobobobobobobobobobobobobobo2bobob2o\$80bo!

```

We can also squeeze a c/4 orthogonal corner between two 5c/19 corners. (It won't fit next to any of the other known corners, since its plows eventually destroy them.) The next pattern has 4 c/4 corners and 4 5c/19 corners; it has the fastest growth of any pattern that I know of.

```
#C Asymptotic shape is a concave 16-gon whose vertices in gen t are at
#C (t/4, 0), (5t/22, t/11), (5t/19, 5t/19), and their reflections.
#C Population in gen t is asymptotic to 49/209 t^2.
x = 31, y = 31, rule = B34568/S15678
22b3o\$6bo8bo4bobobobo\$8bo4bobobo2b8o\$2b7o2bob15o\$b27o\$2b26o\$28obo\$ob
26o\$29o\$2b25o\$b26o\$3b25o\$3b24o\$2b27o\$3b25o\$b29o\$3b25o\$2b27o\$4b24o\$3b
25o\$4b26o\$4b25o\$2b29o\$3b26obo\$bob28o\$3b26o\$3b27o\$3b15obo2b7o\$3b8o2bobo
bo4bo\$4bobobobo4bo8bo\$6b3o!

```

### The c/8 corner

There's one other common form of diagonal growth, which doesn't seem to exist as a wickstretcher; its speed is c/8. It also produces plows, but with a different leading edge that can't turn a corner or hit a diagonal line cleanly. Here it's shown filling the first quadrant:

```
#C c/8 corner grows northeast.  Asymptotic shape is a quadrilateral with
#C vertices (0,0), (t/9, 0), (t/8, t/8), and (0,t/9).  Population is
#C asymptotic to t^2/72.  (This is the slowest form of quadratic growth
#C that I know of.)
x = 5, y = 5, rule = B34568/S15678
ob3o\$2o\$2obo\$2o\$b3o!

```

The most common form of infinite growth has 4 of the c/8 corners:

```
#C Asymptotic shape is concave octagon whose vertices in gen t are at
#C (t/9, 0), (t/8, t/8), and their rotations.  Population in gen t is
#C asymptotic to 1/18 t^2.
x = 5, y = 5, rule = B34568/S15678
obobo\$o2b2o\$2o2bo\$2bobo\$5o!

```

In this example all 4 corners start their c/8 growth within 300 gens. But often it takes much longer. E.g. here one corner starts growing in gen 630 and the others about gens 9000, 22000, and 61000:

```
x = 6, y = 5, rule = B34568/S15678
o2b2o\$ob3o\$o4bo\$2bo2bo\$o2bobo!

```

If the inequalities stated earlier applied to the c/8 corner, it would only be compatible with perpendicular corners with speeds between c/10 and c/6, slower than the others mentioned so far. But when the modified plows from the c/8 meet the normal ones from faster corners, the collisions produce flaws which slow down the normal plows. So in fact the c/8 corner is compatible with the 8c/46, 10c/56, 12c/66, and 14c/76 corners. In the last 3 cases, the collision point between the plows stays a bounded distance from the c/8 corner, so the asymptotic shapes are quadrilaterals rather than pentagons. This seems like an odd coincidence, and I don't know if there's a good explanation for it. (It's conceivable that the slope of the top edge adjacent to the 2nc/(10n+6) corner could depend on initial conditions, and that some such corner is compatible with the c/8 corner for some n > 7, but I haven't seen any evidence of that.)

Here are examples of these pairs of corners:

```
#C Perpendicular corners with speeds c/8 and 8c/46.
#C Asymptotic shape is concave pentagon with vertices (0,0), (4t/23, 4t/23),
#C (-7t/65, 8t/65), (-t/8, t/8), and (-t/9, 0).  (It takes about 17000 gens
#C for this to settle down.)
x = 31, y = 15, rule = B34568/S15678
b2o\$4obobobobobobobobobo2bobo\$ob25ob2o\$30o\$b28o\$31o\$b28o\$30o\$b28o\$30o\$
b27o\$29o\$b27o\$28o\$2obobobobobobobobobobobobo!

```

Here's a natural example of the above; it takes about 17000 gens for the southern edge to settle down.

```
x = 7, y = 5, rule = B34568/S15678
bo3bo\$b2obobo\$o3b3o\$o2b4o\$bobob2o!

#C Perpendicular corners with speeds c/8 and 10c/56.
#C Asymptotic shape is quadrilateral with vertices (0,0), (5t/28, 5t/28),
#C (-t/8, t/8), and (-t/9, 0).
x = 35, y = 17, rule = B34568/S15678
b2o\$4obobobobobobobobobobo2bobobo\$ob29ob2o\$34o\$b32o\$35o\$b32o\$34o\$b32o\$
34o\$b31o\$33o\$b31o\$33o\$b31o\$32o\$bobobobobobobobobobobobobobobo!

#C Perpendicular corners with speeds c/8 and 12c/66.
#C Asymptotic shape is quadrilateral with vertices (0,0), (2t/11, 2t/11),
#C (-t/8, t/8), and (-t/9, 0).
x = 39, y = 19, rule = B34568/S15678
b2o\$4obobobobobobobobobobobo2bobobobo\$ob33ob2o\$38o\$b36o\$39o\$b36o\$38o\$b
36o\$38o\$b35o\$37o\$b35o\$37o\$b35o\$37o\$b35o\$36o\$2obobobobobobobobobobobobo
bobobobo!

#C Perpendicular corners with speeds c/8 and 14c/76.
#C Asymptotic shape is quadrilateral with vertices (0,0), (7t/38, 7t/38),
#C (-t/8, t/8), and (-t/9, 0).
x = 43, y = 21, rule = B34568/S15678
b2o\$4obobobobobobobobobobobobo2bobobobobo\$ob37ob2o\$42o\$b40o\$43o\$b40o\$
42o\$b40o\$42o\$b39o\$41o\$b39o\$41o\$b39o\$41o\$b39o\$41o\$b39o\$40o\$bobobobobobo
bobobobobobobobobobobobobo!

```

If we try the same sort of thing with a 16c/86 corner, the c/8 corner eventually gets destroyed, no matter how wide the initial pattern is.

```
#C Perpendicular corners with speeds c/8 and 16c/86.
#C The c/8 corner gets destroyed about gen 11200 and this becomes a
#C period 2 oscillator in gen 37347.
x = 195, y = 23, rule = B34568/S15678
b2o\$4obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo
bobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo
bobobobobobobobobobobobobobobobobobobobo2bobobobobobo\$ob189ob2o\$194o\$b
192o\$195o\$b192o\$194o\$b192o\$194o\$b191o\$193o\$b191o\$193o\$b191o\$193o\$b191o
\$193o\$b191o\$193o\$b191o\$192o\$2obobobobobobobobobobobobobobobobobobobobo
bobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo
bobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo
bobobobo!

```

### Some stranger patterns

So far all of the patterns that I've mentioned have asymptotic shapes that are polygons, and each edge is spatially periodic. I don't know if nonpolygonal asymptotic shapes can occur, but there are some patterns whose edges aren't periodic, and others which don't have an asymptotic shape. The ones that I know of are based on the flaw-pulling interaction of two c/4 corners.

First, here's a pattern whose asymptotic shape is a hexagon, but two of its edges are not periodic. The top edge (with slope 1/6) is made up of ever-lengthening straight pieces of slopes 0 and 1/4, and the left edge is similarly jagged.

```
#C Asymptotic shape is convex hexagon with vertices at (t/5, -t/5),
#C (t/4, 0), (t/5, t/5), (-t/7, t/7), (-t/5, -t/5), (0, -t/4). The
#C top and left edges (with slopes 1/6 and 6) are not spatially
#C periodic.
x = 6, y = 6, rule = B34568/S15678
2b3o\$2bo\$2o2b2o\$o2bo\$obo2bo\$2bobo!

```

Most patterns in this rule run quite efficiently in hashlife. However, whenever an edge with slope 1/6 forms, the program slows things down enormously and uses a lot of memory. Several times my computer has crashed when using Golly to run such things. (I understand that that's because I have such an out-of-date computer; on newer ones Golly will slow down but shouldn't crash.)

This pattern has no asymptotic shape:

```
#C No asymptotic shape
#C Pattern is approximately square, symmetric under 90 degree rotation,
#C and usually has 2 hills with slopes 1/4 and -1/4 along the north
#C edge.  But at certain times there's only one hill, sometimes at the
#C west end of the edge, sometimes at the east end.  Specifically,
#C define a tribonacci sequence by a(0)=75, a(1)=130, a(2)=255, and
#C a(n)=a(n-1)+a(n-2)+a(n-3) for n>=3.  Then there's only one hill, at
#C the west end, at times  a(2n)+6n+4  (i.e. 79, 265, 861, 2887, ...);
#C there's only one hill, at the east end, at times  a(2n+1)+6n+2
#C (i.e. 132, 468, 1574, 5290, ...).
x = 10, y = 10, rule = B34568/S15678
3b2obo\$b3obob2o\$bo2b3obo\$obobo3b2o\$b2ob4obo\$ob4ob2o\$2o3bobobo\$bob3o2bo
\$b2obob3o\$3bob2o!

```

The pattern below is similar, but only the top and bottom have the shifting hills.

```
#C No asymptotic shape
x = 7, y = 7, rule = B34568/S15678
bobo\$2b2obo\$obo2b2o\$3ob3o\$2o2bobo\$bob2o\$3bobo!

```

This one also has shifting hills on the top and bottom, but not until about gen 48000:

```x = 7, y = 7, rule = B34568/S15678
bobobo\$o4bo\$2bo3bo\$o5bo\$o3bo\$bo4bo\$bobobo!

```

I haven't seen this occur naturally, but it's also possible to have shifting hills on just one edge:

```x = 38, y = 27, rule = B34568/S15678
23bo\$21bo3bo\$4bobobobobo2bobobob5obobo\$4b26obo\$3b31o\$b34o\$2b33o\$b35o\$
2b33o\$37o\$ob34o\$37o\$2b34o\$b37o\$2b34obo\$b37o\$3b33o\$2b35o\$3b33o\$3b34o\$4b
31o\$6bob28o\$8bobob21obo\$12bobob13obobo\$16bobob5obobo\$20bo3bo\$22bo!

```

### Growth spurts (5c/19 plus c/8)

If a pattern has growing corners of speeds 5c/19 or 2nc/(10n+6) in opposite directions, it will probably settle down to a wickstretcher, possibly a 2-ended one. For example:

```
#C About gen 4200 this becomes a 2-ended wickstretcher growing at 5c/19
#C southwest and 8c/46 northeast.
x = 6, y = 5, rule = B34568/S15678
2obobo\$2bo2bo\$2b2o\$2bob2o\$2o2bo!

```

But if one corner has speed c/8, then things can get more interesting, as in this pattern:

```x = 5, y = 5, rule = B34568/S15678
5o\$2ob2o\$2obo\$2o2bo\$4o!

```

Starting around gen 800, this grows at 5c/19 southwest, and at c/8 in the other 3 diagonal directions. The plows from the 5c/19 corner reach the adjacent corners around gen 2200, stopping them. After that, the southwest part grows as a wide 5c/19 wickstretcher.

In gen 2672 the half of the pattern above the SW-NE diagonal has 3 edges, with slopes 24/5, 1, and 1/9. But then the modified plows from the northeast c/8 corner reach the edge of slope 1, and 3 new edges appear, with slopes 2, 1/8, and -4/19. During gens 2672 to 5197 the top half of the pattern grows, changing shape several times; the table below shows the slopes of the edges, from bottom to top, during the process. (The generations are approximate, since it's hard to define exactly when an edge appears or disappears.)

```Generations  Slopes
-----------  -------------------------------------
2672         24/5    1                         1/9
2672-2890    24/5    1    2    1/8    -4/19    1/9
2890-3216    24/5         2    1/8    -4/19    1/9
3216-3295    24/5              1/8    -4/19    1/9
3295-3470    24/5    1         1/8    -4/19    1/9
3470         24/5    1                -4/19    1/9
3470-4003    24/5    1    0           -4/19    1/9
4003         24/5    1    0                    1/9
4003-4551    24/5    1    0    6/41            1/9
4551-5197    24/5    1         6/41            1/9
5197         24/5    1                         1/9

```

The process repeats, with growth spurts in the upper half starting in gens 5197, 14496, 49049, and 177181. Meanwhile, similar growth spurts occur in the bottom half of the pattern, starting at gens 2770, 5240, 14430, and 48317.

If all growth spurts were of the same type, such patterns would grow forever, with no asymptotic shape, and would expand by a factor asymptotic to 415/112 = 3.705... during each spurt. But variations sometimes occur. For this pattern the last spurt in the lower half is different: Gen 88710 corresponds to gen 3470 above, but instead of a vertical edge starting to form immediately, it's delayed until gen 115060:

```Generations    Slopes
-----------    -----------------------------------
88710         5/24    1                -19/4    9
88710-96510   5/24    1    1/2    8    -19/4    9
96510-108200  5/24         1/2    8    -19/4    9
108200         5/24                8    -19/4    9
108200-115060  5/24    1           8    -19/4    9
115060         5/24    1                -19/4    9

```

Because of the delay, the slope -19/4 edge catches up with the c/8 corner, stopping it about gen 153850. The pattern takes until gen 355901 to settle down, becoming a 5c/19 diagonal wickstretcher with width about 130000.

I've seen many examples of patterns like this, combining 5c/19 and c/8 corners. In each case, after some number of growth spurts, the c/8 corner gets stopped and the pattern becomes a 5c/19 wickstretcher. Since most spurts take about 3.7 times as long as the preceding ones, it doesn't take very many of them to make a pattern last for a long time. Here's the longest-lasting one that I've seen:

```
#C Becomes a 5c/19 diagonal wickstretcher about gen 5.88205*10^15, with
#C width about 2.2*10^15.  This goes through 22 growth spurts on each
#C side of the diagonal, before the last one on the bottom stops the
#C c/8 corner about gen 2.5*10^15.
x = 6, y = 6, rule = B34568/S15678
b4o\$o2bobo\$obo\$o3b2o\$ob4o\$b3o!

```

I don't know if all such patterns always behave this way, or if there are some in which infinitely many growth spurts occur. With enough work, we could probably model this as a "3n+1" type of problem, and find patterns that last much longer than the ones I've seen.

Here are some more examples:

```
#C Becomes a 5c/19 diagonal wickstretcher in gen 654988, with
x = 5, y = 5, rule = B34568/S15678
4bo\$5o\$2o2bo\$4bo\$ob3o!

#C Becomes a 5c/19 diagonal wickstretcher in gen 15379926, with
x = 5, y = 5, rule = B34568/S15678
o3bo\$ob2o\$o2bo\$5o\$b2obo!

#C Becomes a 5c/19 diagonal wickstretcher in gen 242490606, with
x = 5, y = 5, rule = B34568/S15678
b3o\$5o\$bobo\$2bo\$2obo!

#C Becomes a 5c/19 diagonal wickstretcher about gen 3.60238*10^10,
x = 5, y = 5, rule = B34568/S15678
5o\$4o\$o\$bobo\$2ob2o!

#C Becomes a 5c/19 diagonal wickstretcher about gen 8.1838*10^12,
x = 5, y = 5, rule = B34568/S15678
b3o\$4o\$4bo\$bo2bo\$2obo!

#C Becomes a 5c/19 diagonal wickstretcher about gen 3.44*10^13,
x = 5, y = 5, rule = B34568/S15678
5o\$2bobo\$obobo\$o2b2o\$3bo!

```

### Growth spurts (2nc/(10n+6) plus c/8)

Although they're much less common, there are also some patterns which combine a 2nc/(10n+6) corner and a c/8 corner. (Sometimes there are two c/8 corners if n <= 7). These also undergo growth spurts like those above, although variant types seem to be more common. Some of the variants produce c/4 corners which eventually stop both of the diagonal corners. So instead of turning into a wide wickstretcher, these often stop growing completely. Here are some examples:

```
#C Grows at c/8 northeast and 8c/46 southwest for a while, but
#C becomes a period 2 oscillator in gen 13512, with populations
#C 9353701 and 9353234.
x = 6, y = 6, rule = B34568/S15678
2bobo\$2ob3o\$obob2o\$obo\$2bo\$ob4o!

#C Grows at c/8 northeast and 8c/46 southwest for a while, but
#C becomes a period 2 oscillator in gen 1622613, with population
x = 7, y = 5, rule = B34568/S15678
b2obobo\$bo3bo\$obobobo\$2b5o\$b2o3bo!

#C Grows at c/8 northeast and southeast and 8c/46 southwest for about
#C 3.5*10^11 gens.  Starting about gen 2.4*10^12 it grows at c/8 in all
#C 4 diagonal directions.  (This has an edge of slope -6 during gens
#C 1.2*10^10 to 1.6*10^10, which slows down hashlife.)
x = 6, y = 6, rule = B34568/S15678
2bo\$o2b3o\$3ob2o\$o4bo\$2bobo\$bo!

#C Grows at c/8 northeast and southeast and 8c/46 southwest for about
#C 8*10^12 gens.  Starting about gen 7*10^13 it grows at c/8 in all
#C 4 diagonal directions.
x = 5, y = 8, rule = B34568/S15678
obo\$2bobo\$obobo\$b2o\$bo2bo\$ob2o\$o2b2o\$2b3o!

#C Grows at c/8 northeast and 10c/56 southwest for a while, but
#C becomes a period 2 oscillator in gen 609320 with population
x = 7, y = 5, rule = B34568/S15678
o2bob2o\$o2b4o\$4b3o\$2b4o\$3ob2o!

#C Grows at c/8 northeast and southeast and 10c/56 southwest for a
#C while, but starts growing at c/8 in all 4 diagonal directions about
#C gen 4.6*10^13.
x = 6, y = 6, rule = B34568/S15678
3b3o\$2o2bo2\$3ob2o\$3b3o\$bo3bo!

#C Grows at c/8 northeast and southeast and 10c/56 southwest for about
#C 6*10^14 gens, but starts growing at c/8 in all 4 diagonal directions
x = 5, y = 8, rule = B34568/S15678
2o2bo\$bo2bo\$3o\$o\$2o2bo\$4o\$2b2o\$o2b2o!

#C THIS TAKES A LONG TIME TO SETTLE DOWN
#C Grows at c/8 northeast and southeast and 10c/56 southwest for about
#C 1.4*10^21 gens, but starts growing at c/8 in all 4 diagonal directions
x = 5, y = 6, rule = B34568/S15678
2bo\$obobo\$o3bo\$b2obo\$b3o\$b2o!

#C Grows at c/8 northeast and 12c/66 southwest for a while, but
#C becomes a period 2 oscillator in gen 3741593, with population
x = 6, y = 6, rule = B34568/S15678
ob2o\$b4o\$bob3o\$2bo2bo\$o3b2o\$b5o!

#C Grows at c/8 northeast and 12c/66 southwest for about 3.3*10^15 gens,
#C but becomes a period 2 oscillator about gen 1.19983*10^16, with
x = 7, y = 5, rule = B34568/S15678
5o\$2b4o\$obobo\$obo2bo\$2b3obo!

#C Grows at c/8 northeast and 14c/76 southwest for a while, but
#C becomes a period 2 oscillator about gen 5.93079*10^10, with
x = 6, y = 6, rule = B34568/S15678
2b4o\$ob2o\$5o\$o2bo\$2o3bo\$3ob2o!

#C Grows at c/8 northeast and 14c/76 southwest for a while, but
#C becomes a wide 14c/76 wickstretcher in gen 9488900.
x = 7, y = 5, rule = B34568/S15678
b2o3bo\$3o3bo\$3b4o\$obo3bo\$bob2obo!

#C Grows at c/8 northeast and 16c/86 southwest for a while, but
#C becomes a period 2 oscillator in gen 11018300, with population
x = 6, y = 5, rule = B34568/S15678
b4o\$o2b3o\$3o2bo\$o3b2o\$b3o!

#C Grows at c/8 northeast and 16c/86 southwest for about 3*10^10 gens,
#C but becomes a period 2 oscillator about gen 9.84602*10^10, with
x = 7, y = 5, rule = B34568/S15678
b2ob3o\$bo3bo\$bo3bo\$ob3o\$2b5o!

#C Grows at c/8 northeast and 24c/126 southwest for about 2.1*10^10
#C gens, but becomes a period 2 oscillator about gen 9.25*10^10 with
x = 8, y = 5, rule = B34568/S15678
3b3o\$b2ob3o\$2b3o2bo\$obo2bobo\$bob5o!

#C Grows at c/8 northeast and 26c/136 southwest for about 4.8*10^7
#C gens, but becomes a period 2 oscillator in gen 227251018, with
x = 7, y = 5, rule = B34568/S15678
2obo\$4b3o\$4b2o\$ob4o\$bobo2bo!

#C Grows at c/8 northeast and 32c/166 southwest for about 2.8*10^6
#C gens, but becomes a period 2 oscillator in gen 28207881, with
x = 7, y = 5, rule = B34568/S15678
obobobo\$o2b2o\$bobob2o\$o2b3o\$4obo!

#C Grows at c/8 northeast and 42c/216 southwest for a while, but
#C becomes a period 2 oscillator in gen 2353516 with population
x = 7, y = 5, rule = B34568/S15678
bob4o\$2ob3o\$2obo2bo\$3obo\$o3bobo!

```

Finally, here are some patterns of this type which I've been unable to run to completion. Each of these eventually produces an edge of slope -6 or -1/6, which causes Golly to slow down and use a lot of memory. Maybe someone with more memory will be able to run these to completion.

```
#C UNKNOWN
#C Grows at c/8 northeast and 8c/46 southwest for at least 1.8*10^13
#C gens.  This has an edge of slope -6 during gens 1.8*10^10 to
#C 2.4*10^10.  An edge of slope -1/6 forms around gen 1.8*10^13, and
#C I haven't run it much beyond that.
x = 8, y = 5, rule = B34568/S15678
2bob4o\$3bob2o\$bobo\$3obo2bo\$2bob3o!

#C UNKNOWN
#C Starting about gen 61000, this grows at c/8 northeast and
#C southeast and 8c/46 southwest for at least 3.6*10^12 gens.
#C This has an edge of slope -6 during gens 1.1*10^11 to 1.3*10^11.
#C Another one forms about gen 3.6*10^12.
x = 5, y = 8, rule = B34568/S15678
bo2bo\$b2obo\$2bobo\$2bo\$o2b2o\$b2o\$o3bo\$o2bo!

#C UNKNOWN
#C Grows at c/8 northeast and southeast and 8c/46 southwest for at
#C least 7.4*10^15 gens.  This has an edge of slope -6 during gens
#C 2.4*10^10 to 3.2*10^10.  Another one forms about gen 7.4*10^15.
x = 7, y = 5, rule = B34568/S15678
2obob2o\$2ob2obo\$3ob3o\$bo2bobo\$4o2bo!

>#C UNKNOWN
#C Grows at c/8 northeast and 10c/56 southwest for at least
#C 1.9*10^13 gens, at which point an edge of slope -1/6 forms.
x = 8, y = 5, rule = B34568/S15678
o2bobobo\$2o3b3o\$2bo4bo\$b2o3bo\$3o3b2o!

#C UNKNOWN
#C Grows at c/8 northeast and 12c/66 southwest for at least
#C 6.4*10^11 gens.  This has an edge of slope -6 during gens
#C 1.1*10^9 to 1.6*10^9, another during gens 1.9*10^10 to
#C 2.6*10^10, and a third during gens 5.6*10^10 to 7.1*10^10.
#C An edge of slope -1/6 forms about gen 6.4*10^11.
x = 7, y = 5, rule = B34568/S15678
bobob2o\$bobo\$3obobo\$b6o\$bobo!

#C UNKNOWN
#C Grows at c/8 northeast and 18c/96 southwest for at least 9*10^11 gens.
#C This has an edge of slope -1/6 during gens 10^9 to 1.2*10^9, and
#C one of slope -6 from gen 7.3*10^11 to at least 9*10^11.
x = 5, y = 6, rule = B34568/S15678
b2o\$obo\$obobo\$ob2o\$3bo\$4o!

```

### The 25c/150 corner

After writing the rest of this, I ran across another diagonal corner, with speed 25c/150:

```
#C UNKNOWN
#C Grows at 25c/150 to the northeast for at least 3.4*10^9 gens.
x = 7, y = 5, rule = B34568/S15678
bob3o\$2bob3o\$b2obobo\$bobo\$4obo!

```

I don't know exactly what this pattern does. Like the c/8 corner, some of the plows that the corner produces are unusual, and it doesn't seem to work as a wickstretcher. At times up to 3.4*10^9, the pattern's shape is usually a pentagon:

```
___ A
B  ___---   |
|         |
|         |
|        |
C |        |
\      |
\_____|
D      E

```

At A it grows at 25c/150 to the northeast, sending plows west and south which form the edges AB and AE (with slopes 1/7 and 7). There are messy reactions at B and E, which sometimes send plows south and west, moving the edges BC and DE west and south. Usually when the plows reach C and D, they just extend the edge CD without moving it sideways. But on rare occasions a modified plow causes a messy reaction at C or D, moving the edge southwest.

Growth to the south and west seems to be sublinear; at gen 3.4*10^9 these edges have only moved about 528000 and 594000 units, respectively. If the pattern continues in this way, then the asymptotic shape is the quadrilateral with vertices (0,0), (t/7, 0), (t/6, t/6), and (0,t/7). But it's possible that some more complex reaction will occur somewhere which will change things completely. E.g. if a 5c/19 corner were to form at B or E, its plows would eventually reach corner A, stopping the 25c/150 growth.

Putting two 25c/150 corners together seems to give more orderly growth. But I haven't determined exactly when southward growth occurs, and I can't guarantee that this will run forever either:

```
#C UNKNOWN
#C 25c/150 corners grow northwest and northeast for at least
#C 9.3*10^8 gens.
x = 9, y = 26, rule = B34568/S15678
4bo\$3b3o\$b7o\$bob3obo\$b7o\$3b3o\$b7o\$2b5o\$b7o\$2b5o\$9o\$ob5obo\$9o\$2b5o\$b7o\$
2b5o\$b7o\$2b5o\$b7o\$2b5o\$b7o\$2b5o\$b7o\$2b5o\$b7o\$3bobo!

```

Putting four of these corners together, or putting one of them between two of the other known corners (slower than 5c/19), eliminates the messy corner reactions. E.g. here one grows between a 40c/206 corner and a c/8 corner:

```
#C Grows at 40c/206 northwest, 25c/150 northeast, and c/8 southeast.
#C Asymptotic shape is the concave octagon with vertices (0,0),
#C (0, -t/9), (t/8, -t/8), (25t/206, -19t/206), (325t/1994, 281t/1994),
#C (t/6, t/6), (21t/329, 50t/329), and (-20t/103, 20t/103). Population
#C is asymptotic to  133714003/2432544408 t^2.
#C (The vertex at (325t/1994, 281t/1994) is hard to see; the exterior
#C angle there is less than 2 degrees.)
x = 87, y = 55, rule = B34568/S15678
4bo\$obobobobo\$b8obobo\$b12obobo56b3o\$17obobobobobo42bobobobobobob3o\$b
26obobobobobobobobob2obobobobobobobobobobobob11obobo\$85o\$b85o\$85o\$b85o
\$85o\$b85o\$85o\$b85o\$85o\$b85o\$85o\$b85o\$85o\$b85o\$85o\$b85o\$85o\$b85o\$85o\$b
85o\$85o\$b85o\$85o\$b85o\$85o\$b85o\$85o\$b85o\$85o\$b85o\$85o\$b85o\$85o\$b85o\$85o
\$b85o\$b84o\$86o\$b84o\$86o\$b84o\$11ob74o\$b7ob3o2b71o\$8obo2b74o\$2b7o2bob72o
\$2bobobo5b74o\$13b74o\$12b72ob2o\$12b2obobobobobobobobobobobobobobobobobo
bobobobobobobobobobobobobobobobobob3o!

```

We can also combine this corner with others growing antiparallel to it, to get more examples of growth spurts. Unfortunately such patterns run very slowly in hashlife, and I haven't been able to track any of them to completion. E.g.:

```
#C UNKNOWN
#C Grows at 25c/150 northeast and 5c/19 southwest for at least
#C 1.5*10^10 gens.
x = 16, y = 18, rule = B34568/S15678
10bob3o\$11bob3o\$10b2obobo\$10bobo\$9b4obo9\$5o\$o\$4o2\$obo!

```

### Growth spurts (c/4 plus c/8)

Here are some more patterns with growth spurts, combining a c/4 corner growing south and a c/8 corner growing northeast. Because of long-lasting edges of slope 6, these run slowly in hashlife.

```
#C Grows at c/8 northeast and c/4 south, with growth spurts on the
#C northwest side, until about gen 1.5*10^6.  Becomes a period 2
#C oscillator in gen 8895153, with population about 2.64204*10^12.
x = 5, y = 6, rule = B34568/S15678
3bo\$3o\$o2b2o\$obo\$ob2o\$b2o!

#C Starting around gen 22000, this grows at c/8 northeast and c/4
#C south, with growth spurts on the northwest side, until about
#C gen 2*10^9.  By gen 1.4*10^10, it becomes a period 2 oscillator
x = 7, y = 5, rule = B34568/S15678
o3b2o\$b3o\$2o\$bobo2bo\$4o2bo!

```

The next pattern is similar; I haven't tracked it to completion.

```
#C UNKNOWN
#C Grows at c/8 northeast and c/4 south, with growth spurts on the
#C northwest side for at least 5*10^9 gens.
x = 5, y = 8, rule = B34568/S15678
2ob2o\$bo2bo\$b4o\$obo\$2ob2o\$2ob2o\$obobo\$2bobo!

```

### Questions

What do the patterns labelled "UNKNOWN" do? Perhaps someone using Golly on a bigger computer than mine can finish running some of them.

Is there a simple reason why perpendicular corners with speeds c/8 and either 10c/56, 12c/66, or 14c/76 have asymptotic shapes which are quadrilaterals instead of pentagons?

Can a pattern undergo infinitely many growth spurts?

Can solids decay? I.e. if a large square is full of live cells at some time, is it possible for it to be empty at some later time?

What other rules have linear, space-filling patterns? In addition to B3456/S15678 with optional B8 and S0 and B37/S345678 with optional B8 and S012, there's at least one other:

```
#C Population growth is linear, but every cell in the plane eventually
#C becomes and remains alive. This works by a complicated interaction of
#C several plows of two types.
x = 5, y = 5, rule = B345678/S015678
b2obo\$o2bo\$bo\$bobo\$5o!

```