R Graphical Manual

Browse All

Last data update: 2014.03.03

R: Draw a 2D L-System Using Turtle Graphics

drawLsys

R Documentation

Draw a 2D L-System Using Turtle Graphics

Description

This function takes input strings, previously created with Lsys, translates them into 2D turtle graphics instructions, and then plots the results.

Usage

drawLsys(string = NULL, drules = NULL, st = c(5, 50, 0), stepSize = 1,
  ang = 90, which = length(string), shrinkFactor = NULL, ...)

Arguments

`string`	A character vector giving the strings containing the turtle graphics instructions. Created by `Lsys`.
`drules`	A data frame containing columns "symbols" and "action". These contain the input symbols and the corresponding drawing action. See the examples.
`st`	A numeric vector of length 3 giving the screen coordinates where the start of the curve should be placed. The screen is 100 x 100 with the lower left corner as 0,0. The third element is the initial drawing angle in degrees.
`stepSize`	Numeric. The length of the drawing step.
`ang`	Numeric. The angle in degrees when a change in direction is requested.
`which`	Integer. The entries in `string` which should be drawn. Defaults to the last (most complex) entry. If `length(which) > 1` each plot is drawn in its own window.
`shrinkFactor`	A numeric vector of the same length as `string`. As each plot is made, `stepSize` will be divided by the corresponding value in `shrinkFactor`. This allows one to scale down the increasingly large/complex plots to make them occupy a space similar to the less complex plots.
`...`	Additional parameters to be passed to the `grid` drawing routines. Most likely, something of the form `gp = gpar(...)`. See `gpar` and the last example.

Value

None; side effect is a plot.

Warning

Remember that if retAll = TRUE, Lsys returns the initial string plus the results of all iterations. In this case, if you want the 5th iteration, you should specify which = 6 since the initial string is in string[1].

Examples

require('grid')

# Modified Koch curve
rkoch1 <- data.frame(inp = c("F"), out = c("F+F-F-F+F"), stringsAsFactors = FALSE)
k1 <- Lsys(init = "F", rules = rkoch1, n = 3)
dkoch <- data.frame(symbol = c("F", "f", "+", "-", "[", "]"),
action = c("F", "f", "+", "-", "[", "]"), stringsAsFactors = FALSE)
drawLsys(string = k1, stepSize = 3, st = c(10, 50, 0), drules = dkoch)
grid.text("Modified Koch Curve (n = 3)", 0.5, 0.25)

# Classic Koch snowflake
rkoch2 <- data.frame(inp = c("F"), out = c("F-F++F-F"), stringsAsFactors = FALSE)
k2 <- Lsys(init = "F++F++F", rules = rkoch2, n = 4)
drawLsys(string = k2, stepSize = 1, ang = 60, st = c(10, 25, 0), drules = dkoch)
grid.text("Classic Koch Snowflake (n = 4)", 0.5, 0.5)

# Sierpinski Triangle
rSierp <- data.frame(inp = c("A", "B"), out = c("B-A-B", "A+B+A"), stringsAsFactors = FALSE)
s <- Lsys(init = "A", rules = rSierp, n = 6)
dSierp <- data.frame(symbol = c("A", "B", "+", "-", "[", "]"),
action = c("F", "F", "+", "-", "[", "]"), stringsAsFactors = FALSE)
drawLsys(string = s, stepSize = 1, ang = 60, st = c(20, 25, 0), drules = dSierp)
grid.text("Sierpinski Triangle (n = 6)", 0.5, 0.1)

# Islands & Lakes
islands_rules <- data.frame(inp = c("F", "f"), out = c("F+f-FF+F+FF+Ff+FF-f+FF-F-FF-Ff-FFF",
"ffffff"), stringsAsFactors = FALSE)
islands <- Lsys(init = "F+F+F+F", rules = islands_rules, n = 2)
draw_islands <- data.frame(symbol = c("F", "f", "+", "-", "[", "]"),
action = c("F", "f", "+", "-", "[", "]"), stringsAsFactors = FALSE)
drawLsys(string = islands, step = 1, ang = 90, st = c(70, 35, 90),
drules = draw_islands,  gp = gpar(col = "red", fill = "gray"))

# A primitive tree
prim_rules <- data.frame(inp = c("0", "1"),
out = c("1[+0]-0", "11"), stringsAsFactors = FALSE)
primitive_plant <- Lsys(init = "0", rules = prim_rules, n = 7)
draw_prim <- data.frame(symbol = c("0", "1", "+", "-", "[", "]"),
action = c("F", "F", "+", "-", "[", "]"), stringsAsFactors = FALSE)
drawLsys(string = primitive_plant, stepSize = 1, ang = 45, st = c(50, 5, 90),
drules = draw_prim, which = 7)
grid.text("Primitive Tree (n = 6)", 0.5, 0.75)

# A more realistic botanical structure
# Some call this a fractal tree, I think it is more like seaweed
# Try drawing the last iteration (too slow for here, but looks great)
fractal_tree_rules <- data.frame(inp = c("X", "F"),
out = c("F-[[X]+X]+F[+FX]-X", "FF"), stringsAsFactors = FALSE)
fractal_tree <- Lsys(init = "X", rules = fractal_tree_rules, n = 7)
draw_ft <- data.frame(symbol = c("X", "F", "+", "-", "[", "]"),
action = c("f", "F", "+", "-", "[", "]"), stringsAsFactors = FALSE)
drawLsys(string = fractal_tree, stepSize = 2, ang = 25, st = c(50, 5, 90),
drules = draw_ft, which = 5, gp = gpar(col = "chocolate4", fill = "honeydew"))
grid.text("Fractal Seaweed (n = 4)", 0.25, 0.25)

Results


R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(LindenmayeR)
Loading required package: stringr
Loading required package: grid
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/LindenmayeR/drawLsys.Rd_%03d_medium.png", width=480, height=480)
> ### Name: drawLsys
> ### Title: Draw a 2D L-System Using Turtle Graphics
> ### Aliases: drawLsys
> ### Keywords: plot
> 
> ### ** Examples
> 
> require('grid')
> 
> # Modified Koch curve
> rkoch1 <- data.frame(inp = c("F"), out = c("F+F-F-F+F"), stringsAsFactors = FALSE)
> k1 <- Lsys(init = "F", rules = rkoch1, n = 3)

Cycle 0 string has length 1
Cycle 0: F 
[[1]]
     start end
[1,]     1   1

[[1]]
  start end    insert
1     1   1 F+F-F-F+F


Cycle 1 string has length 9
Cycle 1: F+F-F-F+F
[[1]]
     start end
[1,]     1   1
[2,]     3   3
[3,]     5   5
[4,]     7   7
[5,]     9   9

[[1]]
  start end    insert
1     1   1 F+F-F-F+F
2     3   3 F+F-F-F+F
3     5   5 F+F-F-F+F
4     7   7 F+F-F-F+F
5     9   9 F+F-F-F+F


Cycle 2 string has length 49
Cycle 2: F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F
[[1]]
      start end
 [1,]     1   1
 [2,]     3   3
 [3,]     5   5
 [4,]     7   7
 [5,]     9   9
 [6,]    11  11
 [7,]    13  13
 [8,]    15  15
 [9,]    17  17
[10,]    19  19
[11,]    21  21
[12,]    23  23
[13,]    25  25
[14,]    27  27
[15,]    29  29
[16,]    31  31
[17,]    33  33
[18,]    35  35
[19,]    37  37
[20,]    39  39
[21,]    41  41
[22,]    43  43
[23,]    45  45
[24,]    47  47
[25,]    49  49

[[1]]
   start end    insert
1      1   1 F+F-F-F+F
2      3   3 F+F-F-F+F
3      5   5 F+F-F-F+F
4      7   7 F+F-F-F+F
5      9   9 F+F-F-F+F
6     11  11 F+F-F-F+F
7     13  13 F+F-F-F+F
8     15  15 F+F-F-F+F
9     17  17 F+F-F-F+F
10    19  19 F+F-F-F+F
11    21  21 F+F-F-F+F
12    23  23 F+F-F-F+F
13    25  25 F+F-F-F+F
14    27  27 F+F-F-F+F
15    29  29 F+F-F-F+F
16    31  31 F+F-F-F+F
17    33  33 F+F-F-F+F
18    35  35 F+F-F-F+F
19    37  37 F+F-F-F+F
20    39  39 F+F-F-F+F
21    41  41 F+F-F-F+F
22    43  43 F+F-F-F+F
23    45  45 F+F-F-F+F
24    47  47 F+F-F-F+F
25    49  49 F+F-F-F+F


Cycle 3 string has length 249
Cycle 3: F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F+F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F-F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F-F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F+F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F
> dkoch <- data.frame(symbol = c("F", "f", "+", "-", "[", "]"),
+ action = c("F", "f", "+", "-", "[", "]"), stringsAsFactors = FALSE)
> drawLsys(string = k1, stepSize = 3, st = c(10, 50, 0), drules = dkoch)
> grid.text("Modified Koch Curve (n = 3)", 0.5, 0.25)
> 
> # Classic Koch snowflake
> rkoch2 <- data.frame(inp = c("F"), out = c("F-F++F-F"), stringsAsFactors = FALSE)
> k2 <- Lsys(init = "F++F++F", rules = rkoch2, n = 4)

Cycle 0 string has length 7
Cycle 0: F++F++F 
[[1]]
     start end
[1,]     1   1
[2,]     4   4
[3,]     7   7

[[1]]
  start end   insert
1     1   1 F-F++F-F
2     4   4 F-F++F-F
3     7   7 F-F++F-F


Cycle 1 string has length 28
Cycle 1: F-F++F-F++F-F++F-F++F-F++F-F
[[1]]
      start end
 [1,]     1   1
 [2,]     3   3
 [3,]     6   6
 [4,]     8   8
 [5,]    11  11
 [6,]    13  13
 [7,]    16  16
 [8,]    18  18
 [9,]    21  21
[10,]    23  23
[11,]    26  26
[12,]    28  28

[[1]]
   start end   insert
1      1   1 F-F++F-F
2      3   3 F-F++F-F
3      6   6 F-F++F-F
4      8   8 F-F++F-F
5     11  11 F-F++F-F
6     13  13 F-F++F-F
7     16  16 F-F++F-F
8     18  18 F-F++F-F
9     21  21 F-F++F-F
10    23  23 F-F++F-F
11    26  26 F-F++F-F
12    28  28 F-F++F-F


Cycle 2 string has length 112
Cycle 2: F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F
[[1]]
      start end
 [1,]     1   1
 [2,]     3   3
 [3,]     6   6
 [4,]     8   8
 [5,]    10  10
 [6,]    12  12
 [7,]    15  15
 [8,]    17  17
 [9,]    20  20
[10,]    22  22
[11,]    25  25
[12,]    27  27
[13,]    29  29
[14,]    31  31
[15,]    34  34
[16,]    36  36
[17,]    39  39
[18,]    41  41
[19,]    44  44
[20,]    46  46
[21,]    48  48
[22,]    50  50
[23,]    53  53
[24,]    55  55
[25,]    58  58
[26,]    60  60
[27,]    63  63
[28,]    65  65
[29,]    67  67
[30,]    69  69
[31,]    72  72
[32,]    74  74
[33,]    77  77
[34,]    79  79
[35,]    82  82
[36,]    84  84
[37,]    86  86
[38,]    88  88
[39,]    91  91
[40,]    93  93
[41,]    96  96
[42,]    98  98
[43,]   101 101
[44,]   103 103
[45,]   105 105
[46,]   107 107
[47,]   110 110
[48,]   112 112

[[1]]
   start end   insert
1      1   1 F-F++F-F
2      3   3 F-F++F-F
3      6   6 F-F++F-F
4      8   8 F-F++F-F
5     10  10 F-F++F-F
6     12  12 F-F++F-F
7     15  15 F-F++F-F
8     17  17 F-F++F-F
9     20  20 F-F++F-F
10    22  22 F-F++F-F
11    25  25 F-F++F-F
12    27  27 F-F++F-F
13    29  29 F-F++F-F
14    31  31 F-F++F-F
15    34  34 F-F++F-F
16    36  36 F-F++F-F
17    39  39 F-F++F-F
18    41  41 F-F++F-F
19    44  44 F-F++F-F
20    46  46 F-F++F-F
21    48  48 F-F++F-F
22    50  50 F-F++F-F
23    53  53 F-F++F-F
24    55  55 F-F++F-F
25    58  58 F-F++F-F
26    60  60 F-F++F-F
27    63  63 F-F++F-F
28    65  65 F-F++F-F
29    67  67 F-F++F-F
30    69  69 F-F++F-F
31    72  72 F-F++F-F
32    74  74 F-F++F-F
33    77  77 F-F++F-F
34    79  79 F-F++F-F
35    82  82 F-F++F-F
36    84  84 F-F++F-F
37    86  86 F-F++F-F
38    88  88 F-F++F-F
39    91  91 F-F++F-F
40    93  93 F-F++F-F
41    96  96 F-F++F-F
42    98  98 F-F++F-F
43   101 101 F-F++F-F
44   103 103 F-F++F-F
45   105 105 F-F++F-F
46   107 107 F-F++F-F
47   110 110 F-F++F-F
48   112 112 F-F++F-F


Cycle 3 string has length 448
Cycle 3: F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F
[[1]]
       start end
  [1,]     1   1
  [2,]     3   3
  [3,]     6   6
  [4,]     8   8
  [5,]    10  10
  [6,]    12  12
  [7,]    15  15
  [8,]    17  17
  [9,]    20  20
 [10,]    22  22
 [11,]    25  25
 [12,]    27  27
 [13,]    29  29
 [14,]    31  31
 [15,]    34  34
 [16,]    36  36
 [17,]    38  38
 [18,]    40  40
 [19,]    43  43
 [20,]    45  45
 [21,]    47  47
 [22,]    49  49
 [23,]    52  52
 [24,]    54  54
 [25,]    57  57
 [26,]    59  59
 [27,]    62  62
 [28,]    64  64
 [29,]    66  66
 [30,]    68  68
 [31,]    71  71
 [32,]    73  73
 [33,]    76  76
 [34,]    78  78
 [35,]    81  81
 [36,]    83  83
 [37,]    85  85
 [38,]    87  87
 [39,]    90  90
 [40,]    92  92
 [41,]    95  95
 [42,]    97  97
 [43,]   100 100
 [44,]   102 102
 [45,]   104 104
 [46,]   106 106
 [47,]   109 109
 [48,]   111 111
 [49,]   113 113
 [50,]   115 115
 [51,]   118 118
 [52,]   120 120
 [53,]   122 122
 [54,]   124 124
 [55,]   127 127
 [56,]   129 129
 [57,]   132 132
 [58,]   134 134
 [59,]   137 137
 [60,]   139 139
 [61,]   141 141
 [62,]   143 143
 [63,]   146 146
 [64,]   148 148
 [65,]   151 151
 [66,]   153 153
 [67,]   156 156
 [68,]   158 158
 [69,]   160 160
 [70,]   162 162
 [71,]   165 165
 [72,]   167 167
 [73,]   170 170
 [74,]   172 172
 [75,]   175 175
 [76,]   177 177
 [77,]   179 179
 [78,]   181 181
 [79,]   184 184
 [80,]   186 186
 [81,]   188 188
 [82,]   190 190
 [83,]   193 193
 [84,]   195 195
 [85,]   197 197
 [86,]   199 199
 [87,]   202 202
 [88,]   204 204
 [89,]   207 207
 [90,]   209 209
 [91,]   212 212
 [92,]   214 214
 [93,]   216 216
 [94,]   218 218
 [95,]   221 221
 [96,]   223 223
 [97,]   226 226
 [98,]   228 228
 [99,]   231 231
[100,]   233 233
[101,]   235 235
[102,]   237 237
[103,]   240 240
[104,]   242 242
[105,]   245 245
[106,]   247 247
[107,]   250 250
[108,]   252 252
[109,]   254 254
[110,]   256 256
[111,]   259 259
[112,]   261 261
[113,]   263 263
[114,]   265 265
[115,]   268 268
[116,]   270 270
[117,]   272 272
[118,]   274 274
[119,]   277 277
[120,]   279 279
[121,]   282 282
[122,]   284 284
[123,]   287 287
[124,]   289 289
[125,]   291 291
[126,]   293 293
[127,]   296 296
[128,]   298 298
[129,]   301 301
[130,]   303 303
[131,]   306 306
[132,]   308 308
[133,]   310 310
[134,]   312 312
[135,]   315 315
[136,]   317 317
[137,]   320 320
[138,]   322 322
[139,]   325 325
[140,]   327 327
[141,]   329 329
[142,]   331 331
[143,]   334 334
[144,]   336 336
[145,]   338 338
[146,]   340 340
[147,]   343 343
[148,]   345 345
[149,]   347 347
[150,]   349 349
[151,]   352 352
[152,]   354 354
[153,]   357 357
[154,]   359 359
[155,]   362 362
[156,]   364 364
[157,]   366 366
[158,]   368 368
[159,]   371 371
[160,]   373 373
[161,]   376 376
[162,]   378 378
[163,]   381 381
[164,]   383 383
[165,]   385 385
[166,]   387 387
[167,]   390 390
[168,]   392 392
[169,]   395 395
[170,]   397 397
[171,]   400 400
[172,]   402 402
[173,]   404 404
[174,]   406 406
[175,]   409 409
[176,]   411 411
[177,]   413 413
[178,]   415 415
[179,]   418 418
[180,]   420 420
[181,]   422 422
[182,]   424 424
[183,]   427 427
[184,]   429 429
[185,]   432 432
[186,]   434 434
[187,]   437 437
[188,]   439 439
[189,]   441 441
[190,]   443 443
[191,]   446 446
[192,]   448 448

[[1]]
    start end   insert
1       1   1 F-F++F-F
2       3   3 F-F++F-F
3       6   6 F-F++F-F
4       8   8 F-F++F-F
5      10  10 F-F++F-F
6      12  12 F-F++F-F
7      15  15 F-F++F-F
8      17  17 F-F++F-F
9      20  20 F-F++F-F
10     22  22 F-F++F-F
11     25  25 F-F++F-F
12     27  27 F-F++F-F
13     29  29 F-F++F-F
14     31  31 F-F++F-F
15     34  34 F-F++F-F
16     36  36 F-F++F-F
17     38  38 F-F++F-F
18     40  40 F-F++F-F
19     43  43 F-F++F-F
20     45  45 F-F++F-F
21     47  47 F-F++F-F
22     49  49 F-F++F-F
23     52  52 F-F++F-F
24     54  54 F-F++F-F
25     57  57 F-F++F-F
26     59  59 F-F++F-F
27     62  62 F-F++F-F
28     64  64 F-F++F-F
29     66  66 F-F++F-F
30     68  68 F-F++F-F
31     71  71 F-F++F-F
32     73  73 F-F++F-F
33     76  76 F-F++F-F
34     78  78 F-F++F-F
35     81  81 F-F++F-F
36     83  83 F-F++F-F
37     85  85 F-F++F-F
38     87  87 F-F++F-F
39     90  90 F-F++F-F
40     92  92 F-F++F-F
41     95  95 F-F++F-F
42     97  97 F-F++F-F
43    100 100 F-F++F-F
44    102 102 F-F++F-F
45    104 104 F-F++F-F
46    106 106 F-F++F-F
47    109 109 F-F++F-F
48    111 111 F-F++F-F
49    113 113 F-F++F-F
50    115 115 F-F++F-F
51    118 118 F-F++F-F
52    120 120 F-F++F-F
53    122 122 F-F++F-F
54    124 124 F-F++F-F
55    127 127 F-F++F-F
56    129 129 F-F++F-F
57    132 132 F-F++F-F
58    134 134 F-F++F-F
59    137 137 F-F++F-F
60    139 139 F-F++F-F
61    141 141 F-F++F-F
62    143 143 F-F++F-F
63    146 146 F-F++F-F
64    148 148 F-F++F-F
65    151 151 F-F++F-F
66    153 153 F-F++F-F
67    156 156 F-F++F-F
68    158 158 F-F++F-F
69    160 160 F-F++F-F
70    162 162 F-F++F-F
71    165 165 F-F++F-F
72    167 167 F-F++F-F
73    170 170 F-F++F-F
74    172 172 F-F++F-F
75    175 175 F-F++F-F
76    177 177 F-F++F-F
77    179 179 F-F++F-F
78    181 181 F-F++F-F
79    184 184 F-F++F-F
80    186 186 F-F++F-F
81    188 188 F-F++F-F
82    190 190 F-F++F-F
83    193 193 F-F++F-F
84    195 195 F-F++F-F
85    197 197 F-F++F-F
86    199 199 F-F++F-F
87    202 202 F-F++F-F
88    204 204 F-F++F-F
89    207 207 F-F++F-F
90    209 209 F-F++F-F
91    212 212 F-F++F-F
92    214 214 F-F++F-F
93    216 216 F-F++F-F
94    218 218 F-F++F-F
95    221 221 F-F++F-F
96    223 223 F-F++F-F
97    226 226 F-F++F-F
98    228 228 F-F++F-F
99    231 231 F-F++F-F
100   233 233 F-F++F-F
101   235 235 F-F++F-F
102   237 237 F-F++F-F
103   240 240 F-F++F-F
104   242 242 F-F++F-F
105   245 245 F-F++F-F
106   247 247 F-F++F-F
107   250 250 F-F++F-F
108   252 252 F-F++F-F
109   254 254 F-F++F-F
110   256 256 F-F++F-F
111   259 259 F-F++F-F
112   261 261 F-F++F-F
113   263 263 F-F++F-F
114   265 265 F-F++F-F
115   268 268 F-F++F-F
116   270 270 F-F++F-F
117   272 272 F-F++F-F
118   274 274 F-F++F-F
119   277 277 F-F++F-F
120   279 279 F-F++F-F
121   282 282 F-F++F-F
122   284 284 F-F++F-F
123   287 287 F-F++F-F
124   289 289 F-F++F-F
125   291 291 F-F++F-F
126   293 293 F-F++F-F
127   296 296 F-F++F-F
128   298 298 F-F++F-F
129   301 301 F-F++F-F
130   303 303 F-F++F-F
131   306 306 F-F++F-F
132   308 308 F-F++F-F
133   310 310 F-F++F-F
134   312 312 F-F++F-F
135   315 315 F-F++F-F
136   317 317 F-F++F-F
137   320 320 F-F++F-F
138   322 322 F-F++F-F
139   325 325 F-F++F-F
140   327 327 F-F++F-F
141   329 329 F-F++F-F
142   331 331 F-F++F-F
143   334 334 F-F++F-F
144   336 336 F-F++F-F
145   338 338 F-F++F-F
146   340 340 F-F++F-F
147   343 343 F-F++F-F
148   345 345 F-F++F-F
149   347 347 F-F++F-F
150   349 349 F-F++F-F
151   352 352 F-F++F-F
152   354 354 F-F++F-F
153   357 357 F-F++F-F
154   359 359 F-F++F-F
155   362 362 F-F++F-F
156   364 364 F-F++F-F
157   366 366 F-F++F-F
158   368 368 F-F++F-F
159   371 371 F-F++F-F
160   373 373 F-F++F-F
161   376 376 F-F++F-F
162   378 378 F-F++F-F
163   381 381 F-F++F-F
164   383 383 F-F++F-F
165   385 385 F-F++F-F
166   387 387 F-F++F-F
167   390 390 F-F++F-F
168   392 392 F-F++F-F
169   395 395 F-F++F-F
170   397 397 F-F++F-F
171   400 400 F-F++F-F
172   402 402 F-F++F-F
173   404 404 F-F++F-F
174   406 406 F-F++F-F
175   409 409 F-F++F-F
176   411 411 F-F++F-F
177   413 413 F-F++F-F
178   415 415 F-F++F-F
179   418 418 F-F++F-F
180   420 420 F-F++F-F
181   422 422 F-F++F-F
182   424 424 F-F++F-F
183   427 427 F-F++F-F
184   429 429 F-F++F-F
185   432 432 F-F++F-F
186   434 434 F-F++F-F
187   437 437 F-F++F-F
188   439 439 F-F++F-F
189   441 441 F-F++F-F
190   443 443 F-F++F-F
191   446 446 F-F++F-F
192   448 448 F-F++F-F


Cycle 4 string has length 1792
Cycle 4: F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F-F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F
> drawLsys(string = k2, stepSize = 1, ang = 60, st = c(10, 25, 0), drules = dkoch)
> grid.text("Classic Koch Snowflake (n = 4)", 0.5, 0.5)
> 
> # Sierpinski Triangle
> rSierp <- data.frame(inp = c("A", "B"), out = c("B-A-B", "A+B+A"), stringsAsFactors = FALSE)
> s <- Lsys(init = "A", rules = rSierp, n = 6)

Cycle 0 string has length 1
Cycle 0: A 
[[1]]
     start end
[1,]     1   1

[[1]]
  start end insert
1     1   1  B-A-B


Cycle 1 string has length 5
Cycle 1: B-A-B
[[1]]
     start end
[1,]     3   3

[[2]]
     start end
[1,]     1   1
[2,]     5   5

[[1]]
  start end insert
1     3   3  B-A-B

[[2]]
  start end insert
1     1   1  A+B+A
2     5   5  A+B+A


Cycle 2 string has length 17
Cycle 2: A+B+A-B-A-B-A+B+A
[[1]]
     start end
[1,]     1   1
[2,]     5   5
[3,]     9   9
[4,]    13  13
[5,]    17  17

[[2]]
     start end
[1,]     3   3
[2,]     7   7
[3,]    11  11
[4,]    15  15

[[1]]
  start end insert
1     1   1  B-A-B
2     5   5  B-A-B
3     9   9  B-A-B
4    13  13  B-A-B
5    17  17  B-A-B

[[2]]
  start end insert
1     3   3  A+B+A
2     7   7  A+B+A
3    11  11  A+B+A
4    15  15  A+B+A


Cycle 3 string has length 53
Cycle 3: B-A-B+A+B+A+B-A-B-A+B+A-B-A-B-A+B+A-B-A-B+A+B+A+B-A-B
[[1]]
      start end
 [1,]     3   3
 [2,]     7   7
 [3,]    11  11
 [4,]    15  15
 [5,]    19  19
 [6,]    23  23
 [7,]    27  27
 [8,]    31  31
 [9,]    35  35
[10,]    39  39
[11,]    43  43
[12,]    47  47
[13,]    51  51

[[2]]
      start end
 [1,]     1   1
 [2,]     5   5
 [3,]     9   9
 [4,]    13  13
 [5,]    17  17
 [6,]    21  21
 [7,]    25  25
 [8,]    29  29
 [9,]    33  33
[10,]    37  37
[11,]    41  41
[12,]    45  45
[13,]    49  49
[14,]    53  53

[[1]]
   start end insert
1      3   3  B-A-B
2      7   7  B-A-B
3     11  11  B-A-B
4     15  15  B-A-B
5     19  19  B-A-B
6     23  23  B-A-B
7     27  27  B-A-B
8     31  31  B-A-B
9     35  35  B-A-B
10    39  39  B-A-B
11    43  43  B-A-B
12    47  47  B-A-B
13    51  51  B-A-B

[[2]]
   start end insert
1      1   1  A+B+A
2      5   5  A+B+A
3      9   9  A+B+A
4     13  13  A+B+A
5     17  17  A+B+A
6     21  21  A+B+A
7     25  25  A+B+A
8     29  29  A+B+A
9     33  33  A+B+A
10    37  37  A+B+A
11    41  41  A+B+A
12    45  45  A+B+A
13    49  49  A+B+A
14    53  53  A+B+A


Cycle 4 string has length 161
Cycle 4: A+B+A-B-A-B-A+B+A+B-A-B+A+B+A+B-A-B+A+B+A-B-A-B-A+B+A-B-A-B+A+B+A+B-A-B-A+B+A-B-A-B-A+B+A-B-A-B+A+B+A+B-A-B-A+B+A-B-A-B-A+B+A+B-A-B+A+B+A+B-A-B+A+B+A-B-A-B-A+B+A
[[1]]
      start end
 [1,]     1   1
 [2,]     5   5
 [3,]     9   9
 [4,]    13  13
 [5,]    17  17
 [6,]    21  21
 [7,]    25  25
 [8,]    29  29
 [9,]    33  33
[10,]    37  37
[11,]    41  41
[12,]    45  45
[13,]    49  49
[14,]    53  53
[15,]    57  57
[16,]    61  61
[17,]    65  65
[18,]    69  69
[19,]    73  73
[20,]    77  77
[21,]    81  81
[22,]    85  85
[23,]    89  89
[24,]    93  93
[25,]    97  97
[26,]   101 101
[27,]   105 105
[28,]   109 109
[29,]   113 113
[30,]   117 117
[31,]   121 121
[32,]   125 125
[33,]   129 129
[34,]   133 133
[35,]   137 137
[36,]   141 141
[37,]   145 145
[38,]   149 149
[39,]   153 153
[40,]   157 157
[41,]   161 161

[[2]]
      start end
 [1,]     3   3
 [2,]     7   7
 [3,]    11  11
 [4,]    15  15
 [5,]    19  19
 [6,]    23  23
 [7,]    27  27
 [8,]    31  31
 [9,]    35  35
[10,]    39  39
[11,]    43  43
[12,]    47  47
[13,]    51  51
[14,]    55  55
[15,]    59  59
[16,]    63  63
[17,]    67  67
[18,]    71  71
[19,]    75  75
[20,]    79  79
[21,]    83  83
[22,]    87  87
[23,]    91  91
[24,]    95  95
[25,]    99  99
[26,]   103 103
[27,]   107 107
[28,]   111 111
[29,]   115 115
[30,]   119 119
[31,]   123 123
[32,]   127 127
[33,]   131 131
[34,]   135 135
[35,]   139 139
[36,]   143 143
[37,]   147 147
[38,]   151 151
[39,]   155 155
[40,]   159 159

[[1]]
   start end insert
1      1   1  B-A-B
2      5   5  B-A-B
3      9   9  B-A-B
4     13  13  B-A-B
5     17  17  B-A-B
6     21  21  B-A-B
7     25  25  B-A-B
8     29  29  B-A-B
9     33  33  B-A-B
10    37  37  B-A-B
11    41  41  B-A-B
12    45  45  B-A-B
13    49  49  B-A-B
14    53  53  B-A-B
15    57  57  B-A-B
16    61  61  B-A-B
17    65  65  B-A-B
18    69  69  B-A-B
19    73  73  B-A-B
20    77  77  B-A-B
21    81  81  B-A-B
22    85  85  B-A-B
23    89  89  B-A-B
24    93  93  B-A-B
25    97  97  B-A-B
26   101 101  B-A-B
27   105 105  B-A-B
28   109 109  B-A-B
29   113 113  B-A-B
30   117 117  B-A-B
31   121 121  B-A-B
32   125 125  B-A-B
33   129 129  B-A-B
34   133 133  B-A-B
35   137 137  B-A-B
36   141 141  B-A-B
37   145 145  B-A-B
38   149 149  B-A-B
39   153 153  B-A-B
40   157 157  B-A-B
41   161 161  B-A-B

[[2]]
   start end insert
1      3   3  A+B+A
2      7   7  A+B+A
3     11  11  A+B+A
4     15  15  A+B+A
5     19  19  A+B+A
6     23  23  A+B+A
7     27  27  A+B+A
8     31  31  A+B+A
9     35  35  A+B+A
10    39  39  A+B+A
11    43  43  A+B+A
12    47  47  A+B+A
13    51  51  A+B+A
14    55  55  A+B+A
15    59  59  A+B+A
16    63  63  A+B+A
17    67  67  A+B+A
18    71  71  A+B+A
19    75  75  A+B+A
20    79  79  A+B+A
21    83  83  A+B+A
22    87  87  A+B+A
23    91  91  A+B+A
24    95  95  A+B+A
25    99  99  A+B+A
26   103 103  A+B+A
27   107 107  A+B+A
28   111 111  A+B+A
29   115 115  A+B+A
30   119 119  A+B+A
31   123 123  A+B+A
32   127 127  A+B+A
33   131 131  A+B+A
34   135 135  A+B+A
35   139 139  A+B+A
36   143 143  A+B+A
37   147 147  A+B+A
38   151 151  A+B+A
39   155 155  A+B+A
40   159 159  A+B+A


Cycle 5 string has length 485
Cycle 5: B-A-B+A+B+A+B-A-B-A+B+A-B-A-B-A+B+A-B-A-B+A+B+A+B-A-B+A+B+A-B-A-B-A+B+A+B-A-B+A+B+A+B-A-B+A+B+A-B-A-B-A+B+A+B-A-B+A+B+A+B-A-B-A+B+A-B-A-B-A+B+A-B-A-B+A+B+A+B-A-B-A+B+A-B-A-B-A+B+A+B-A-B+A+B+A+B-A-B+A+B+A-B-A-B-A+B+A-B-A-B+A+B+A+B-A-B-A+B+A-B-A-B-A+B+A-B-A-B+A+B+A+B-A-B-A+B+A-B-A-B-A+B+A+B-A-B+A+B+A+B-A-B+A+B+A-B-A-B-A+B+A-B-A-B+A+B+A+B-A-B-A+B+A-B-A-B-A+B+A-B-A-B+A+B+A+B-A-B+A+B+A-B-A-B-A+B+A+B-A-B+A+B+A+B-A-B+A+B+A-B-A-B-A+B+A+B-A-B+A+B+A+B-A-B-A+B+A-B-A-B-A+B+A-B-A-B+A+B+A+B-A-B
[[1]]
       start end
  [1,]     3   3
  [2,]     7   7
  [3,]    11  11
  [4,]    15  15
  [5,]    19  19
  [6,]    23  23
  [7,]    27  27
  [8,]    31  31
  [9,]    35  35
 [10,]    39  39
 [11,]    43  43
 [12,]    47  47
 [13,]    51  51
 [14,]    55  55
 [15,]    59  59
 [16,]    63  63
 [17,]    67  67
 [18,]    71  71
 [19,]    75  75
 [20,]    79  79
 [21,]    83  83
 [22,]    87  87
 [23,]    91  91
 [24,]    95  95
 [25,]    99  99
 [26,]   103 103
 [27,]   107 107
 [28,]   111 111
 [29,]   115 115
 [30,]   119 119
 [31,]   123 123
 [32,]   127 127
 [33,]   131 131
 [34,]   135 135
 [35,]   139 139
 [36,]   143 143
 [37,]   147 147
 [38,]   151 151
 [39,]   155 155
 [40,]   159 159
 [41,]   163 163
 [42,]   167 167
 [43,]   171 171
 [44,]   175 175
 [45,]   179 179
 [46,]   183 183
 [47,]   187 187
 [48,]   191 191
 [49,]   195 195
 [50,]   199 199
 [51,]   203 203
 [52,]   207 207
 [53,]   211 211
 [54,]   215 215
 [55,]   219 219
 [56,]   223 223
 [57,]   227 227
 [58,]   231 231
 [59,]   235 235
 [60,]   239 239
 [61,]   243 243
 [62,]   247 247
 [63,]   251 251
 [64,]   255 255
 [65,]   259 259
 [66,]   263 263
 [67,]   267 267
 [68,]   271 271
 [69,]   275 275
 [70,]   279 279
 [71,]   283 283
 [72,]   287 287
 [73,]   291 291
 [74,]   295 295
 [75,]   299 299
 [76,]   303 303
 [77,]   307 307
 [78,]   311 311
 [79,]   315 315
 [80,]   319 319
 [81,]   323 323
 [82,]   327 327
 [83,]   331 331
 [84,]   335 335
 [85,]   339 339
 [86,]   343 343
 [87,]   347 347
 [88,]   351 351
 [89,]   355 355
 [90,]   359 359
 [91,]   363 363
 [92,]   367 367
 [93,]   371 371
 [94,]   375 375
 [95,]   379 379
 [96,]   383 383
 [97,]   387 387
 [98,]   391 391
 [99,]   395 395
[100,]   399 399
[101,]   403 403
[102,]   407 407
[103,]   411 411
[104,]   415 415
[105,]   419 419
[106,]   423 423
[107,]   427 427
[108,]   431 431
[109,]   435 435
[110,]   439 439
[111,]   443 443
[112,]   447 447
[113,]   451 451
[114,]   455 455
[115,]   459 459
[116,]   463 463
[117,]   467 467
[118,]   471 471
[119,]   475 475
[120,]   479 479
[121,]   483 483

[[2]]
       start end
  [1,]     1   1
  [2,]     5   5
  [3,]     9   9
  [4,]    13  13
  [5,]    17  17
  [6,]    21  21
  [7,]    25  25
  [8,]    29  29
  [9,]    33  33
 [10,]    37  37
 [11,]    41  41
 [12,]    45  45
 [13,]    49  49
 [14,]    53  53
 [15,]    57  57
 [16,]    61  61
 [17,]    65  65
 [18,]    69  69
 [19,]    73  73
 [20,]    77  77
 [21,]    81  81
 [22,]    85  85
 [23,]    89  89
 [24,]    93  93
 [25,]    97  97
 [26,]   101 101
 [27,]   105 105
 [28,]   109 109
 [29,]   113 113
 [30,]   117 117
 [31,]   121 121
 [32,]   125 125
 [33,]   129 129
 [34,]   133 133
 [35,]   137 137
 [36,]   141 141
 [37,]   145 145
 [38,]   149 149
 [39,]   153 153
 [40,]   157 157
 [41,]   161 161
 [42,]   165 165
 [43,]   169 169
 [44,]   173 173
 [45,]   177 177
 [46,]   181 181
 [47,]   185 185
 [48,]   189 189
 [49,]   193 193
 [50,]   197 197
 [51,]   201 201
 [52,]   205 205
 [53,]   209 209
 [54,]   213 213
 [55,]   217 217
 [56,]   221 221
 [57,]   225 225
 [58,]   229 229
 [59,]   233 233
 [60,]   237 237
 [61,]   241 241
 [62,]   245 245
 [63,]   249 249
 [64,]   253 253
 [65,]   257 257
 [66,]   261 261
 [67,]   265 265
 [68,]   269 269
 [69,]   273 273
 [70,]   277 277
 [71,]   281 281
 [72,]   285 285
 [73,]   289 289
 [74,]   293 293
 [75,]   297 297
 [76,]   301 301
 [77,]   305 305
 [78,]   309 309
 [79,]   313 313
 [80,]   317 317
 [81,]   321 321
 [82,]   325 325
 [83,]   329 329
 [84,]   333 333
 [85,]   337 337
 [86,]   341 341
 [87,]   345 345
 [88,]   349 349
 [89,]   353 353
 [90,]   357 357
 [91,]   361 361
 [92,]   365 365
 [93,]   369 369
 [94,]   373 373
 [95,]   377 377
 [96,]   381 381
 [97,]   385 385
 [98,]   389 389
 [99,]   393 393
[100,]   397 397
[101,]   401 401
[102,]   405 405
[103,]   409 409
[104,]   413 413
[105,]   417 417
[106,]   421 421
[107,]   425 425
[108,]   429 429
[109,]   433 433
[110,]   437 437
[111,]   441 441
[112,]   445 445
[113,]   449 449
[114,]   453 453
[115,]   457 457
[116,]   461 461
[117,]   465 465
[118,]   469 469
[119,]   473 473
[120,]   477 477
[121,]   481 481
[122,]   485 485

[[1]]
    start end insert
1       3   3  B-A-B
2       7   7  B-A-B
3      11  11  B-A-B
4      15  15  B-A-B
5      19  19  B-A-B
6      23  23  B-A-B
7      27  27  B-A-B
8      31  31  B-A-B
9      35  35  B-A-B
10     39  39  B-A-B
11     43  43  B-A-B
12     47  47  B-A-B
13     51  51  B-A-B
14     55  55  B-A-B
15     59  59  B-A-B
16     63  63  B-A-B
17     67  67  B-A-B
18     71  71  B-A-B
19     75  75  B-A-B
20     79  79  B-A-B
21     83  83  B-A-B
22     87  87  B-A-B
23     91  91  B-A-B
24     95  95  B-A-B
25     99  99  B-A-B
26    103 103  B-A-B
27    107 107  B-A-B
28    111 111  B-A-B
29    115 115  B-A-B
30    119 119  B-A-B
31    123 123  B-A-B
32    127 127  B-A-B
33    131 131  B-A-B
34    135 135  B-A-B
35    139 139  B-A-B
36    143 143  B-A-B
37    147 147  B-A-B
38    151 151  B-A-B
39    155 155  B-A-B
40    159 159  B-A-B
41    163 163  B-A-B
42    167 167  B-A-B
43    171 171  B-A-B
44    175 175  B-A-B
45    179 179  B-A-B
46    183 183  B-A-B
47    187 187  B-A-B
48    191 191  B-A-B
49    195 195  B-A-B
50    199 199  B-A-B
51    203 203  B-A-B
52    207 207  B-A-B
53    211 211  B-A-B
54    215 215  B-A-B
55    219 219  B-A-B
56    223 223  B-A-B
57    227 227  B-A-B
58    231 231  B-A-B
59    235 235  B-A-B
60    239 239  B-A-B
61    243 243  B-A-B
62    247 247  B-A-B
63    251 251  B-A-B
64    255 255  B-A-B
65    259 259  B-A-B
66    263 263  B-A-B
67    267 267  B-A-B
68    271 271  B-A-B
69    275 275  B-A-B
70    279 279  B-A-B
71    283 283  B-A-B
72    287 287  B-A-B
73    291 291  B-A-B
74    295 295  B-A-B
75    299 299  B-A-B
76    303 303  B-A-B
77    307 307  B-A-B
78    311 311  B-A-B
79    315 315  B-A-B
80    319 319  B-A-B
81    323 323  B-A-B
82    327 327  B-A-B
83    331 331  B-A-B
84    335 335  B-A-B
85    339 339  B-A-B
86    343 343  B-A-B
87    347 347  B-A-B
88    351 351  B-A-B
89    355 355  B-A-B
90    359 359  B-A-B
91    363 363  B-A-B
92    367 367  B-A-B
93    371 371  B-A-B
94    375 375  B-A-B
95    379 379  B-A-B
96    383 383  B-A-B
97    387 387  B-A-B
98    391 391  B-A-B
99    395 395  B-A-B
100   399 399  B-A-B
101   403 403  B-A-B
102   407 407  B-A-B
103   411 411  B-A-B
104   415 415  B-A-B
105   419 419  B-A-B
106   423 423  B-A-B
107   427 427  B-A-B
108   431 431  B-A-B
109   435 435  B-A-B
110   439 439  B-A-B
111   443 443  B-A-B
112   447 447  B-A-B
113   451 451  B-A-B
114   455 455  B-A-B
115   459 459  B-A-B
116   463 463  B-A-B
117   467 467  B-A-B
118   471 471  B-A-B
119   475 475  B-A-B
120   479 479  B-A-B
121   483 483  B-A-B

[[2]]
    start end insert
1       1   1  A+B+A
2       5   5  A+B+A
3       9   9  A+B+A
4      13  13  A+B+A
5      17  17  A+B+A
6      21  21  A+B+A
7      25  25  A+B+A
8      29  29  A+B+A
9      33  33  A+B+A
10     37  37  A+B+A
11     41  41  A+B+A
12     45  45  A+B+A
13     49  49  A+B+A
14     53  53  A+B+A
15     57  57  A+B+A
16     61  61  A+B+A
17     65  65  A+B+A
18     69  69  A+B+A
19     73  73  A+B+A
20     77  77  A+B+A
21     81  81  A+B+A
22     85  85  A+B+A
23     89  89  A+B+A
24     93  93  A+B+A
25     97  97  A+B+A
26    101 101  A+B+A
27    105 105  A+B+A
28    109 109  A+B+A
29    113 113  A+B+A
30    117 117  A+B+A
31    121 121  A+B+A
32    125 125  A+B+A
33    129 129  A+B+A
34    133 133  A+B+A
35    137 137  A+B+A
36    141 141  A+B+A
37    145 145  A+B+A
38    149 149  A+B+A
39    153 153  A+B+A
40    157 157  A+B+A
41    161 161  A+B+A
42    165 165  A+B+A
43    169 169  A+B+A
44