- Mathematical Modelling of Natural Phenomena

Transcription

Math. Model. Nat. Phenom.
Vol. 6, No. 2, 2011, pp. 173–186
DOI: 10.1051/mmnp/20116207
The Geometric and Dynamic Essence of Phyllotaxis
P. Atela∗
Department of Mathematics, Smith College, Northampton, MA 01063, USA
Abstract. We present a dynamic geometric model of phyllotaxis based on two postulates, primordia formation and meristem expansion. We find that Fibonacci, Lucas, bijugate and multijugate
are all variations of the same unifying phenomenon and that the difference lies on small changes in
the position of initial primordia. We explore the set of all initial positions and color-code its points
depending on the phyllotactic type of the pattern that arises.
Key words: phyllotaxis, fibonacci, pattern formation, primordia, meristem
AMS subject classification: 00, 51, 92C80
1.
Introduction
a
b
Figure 1: (a) Close view of a pine cone. (b) Palm tree in New Orleans. (c) Mathematical model.
∗
E-mail: [email protected]
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Article published by EDP Sciences and available at http://www.mmnp-journal.org or http://dx.doi.org/10.1051/mmnp/20116207
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P. Atela
Geometric and Dynamic Essence of Phyllotaxis
The aim of this article is to extract the common geometrical and dynamical essence of some
of the models of phyllotaxis (pattern formation) that have been studied in the past and propose a
simple one, based on two main postulates.
We are basing our model mainly on the studies of Douady and Couder [2, 3], who previously
presented a compelling physical experiment involving magnetized drops of ferrofluid into a magnetized dish filled with silicone oil, producing phyllotactic patterns with Fibonacci numbers.
In Botany, a phyllotactic pattern refers to the spiral patterns observed in pine cones, pineapples,
asparagus, flowers, palm trees, artichoke, cauliflower, broccoli, lettuce, some cacti, aloe vera, etc.
(see Figure 1). These patterns typically come as two visible sets of spirals winding in opposite
directions (sometimes a third set is also clearly visible, as in the case of pineapples). These spirals
are called parastichies. The surprising phenomenon that interests us here is that when counting the
parallel parastichies (spirals) in these two sets, most often one gets two consecutive numbers of the
Fibonacci sequence 1, 2, 3, 5, 8, 13, 21, 34, . . .. By far, the Fibonacci pairs are the most common
in nature. In some few cases two consecutive Lucas numbers 3, 4, 7, 11, . . . also appear. And
there are specimens in which one sees consecutive pairs of the sequence 2, 4, 6, 10, 16, . . . (so–
called Bijugate, two times the Fibonacci numbers), and also Multijugate. In these sequences, each
number is the sum of the previous two. (For example, 3 + 5 = 8, 5 + 8 = 13, . . . and also
3 + 4 = 7, 4 + 7 = 11, etc.) We emphasize that, artificially, one can create a pattern that has any
two numbers and it would have an appearance similar to the Fibonacci one in Figure 1c. However,
in plants, when there are visible spirals, the pairs of numbers mentioned above are the ones that
appear.
Here in this introduction we give a schematic overview of what we understand happens as
plants grow, starting at germination, and the geometric effects produced on phyllotactic patterns.
We condense the main features into two postulates, which are stated below.
Figure 2: On the left, primordia in a Norway spruce with the meristem at the tip of the shoot. Meristem’s diameter
is here approx. 500 µm (Rolf Rutishauser, Univ. of Zurich). On the right, view from the top. The arrows point to two
incipient primordia in the active meristem. Notice that they appear “in between” older existing primordia.
As Hofmeister [4] observed, phyllotactic patterns are established at a microscopic scale as
the botanical units called primordia appear at the periphery of the apical meristem, piling up on
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the side of the stem as time passes (see Figures 1 and 2). One observes that, after this initial
appearance, the whole structure that includes and sustains older primordia expands in size, but
the local phyllotactic pattern that has already formed is preserved. In particular, each primordium
maintains its neighbors.
Following van Iterson[10], Snow&Snow[9], Douady&Couder[3], Atela, Golé & Hotton[1] and
others (see also [6], [7], etc.), we think of the meristem as a circular region at the tip of the stem,
which we assume axisymmetric, with primordia forming at its boundary (see Figure 2). It is useful
to model primordia as circles with a given radius, but the important thing is the placement of the
centers of these circles.
(For implementations of Hofmeister’s periodic formation of primordia vs. Snow&Snow’s same
size primordia appearing when there is enough space see, for example, Douady&Couder[3], Atela,
Golé & Hotton[1], also Dumais et al.[5]).
1.1.
Two postulates
We suggest in this article that it is the dynamic combination of two main biological phenomena
that influences phyllotaxis as a plant grows: we call them primordia appearance and meristem
expansion. These form the basis of the dynamic geometric model that we present below.
P1. Primordia appearance: Primordia appear with approximately the same size, “in between”
older primordia at the periphery of the meristem (see Figures 1 and 2).
In this paper we place primordia systematically by placing them following the principle of “in
the first available” spot, the so–called Snow&Snow model (see [3]). This condition can be relaxed
a bit, as the “coin experiment” below shows.
As we mentioned before, after primordia appear and others follow afterwards near them, they
start being away from the active meristem and the whole structure that contains and surrounds
them expands, as one can see in Figures 1 and 2. So, away from the meristem, older primordia
and the diameter of the cross section of the cylindrical shoot continue to expand. However, the
phyllotactic pattern near these primordia had already formed and we assume it does not change
during this growth, all the way to macroscopic level. (There might be some distortions and/or a lot
of tissue growth in between primordia, so that these might seem to grow apart from each other as
is the case of, for example, leaves.)
P2. Meristem expansion: We point out a second type of expansion that we think takes place
in plants and is crucial for the appearance of Fibonacci numbers (and the few other cases, such
as Lucas numbers 3, 4, 7, 11, . . . , bijugate, etc.). When the seed germinates it starts with few
primordia—perhaps one or two—and as new primordia appear, the meristem expands in relation
to primordia size. We emphasize that this meristem expansion is a different phenomenon from
the expansion mentioned in the previous paragraph which the whole structure further away from
the active meristem undergoes. It would be interesting to study in real plants how exactly this
expansion occurs and at what stage or stages of meristem development it does.
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Figure 3: Cutting a cone and unrolling it unto a flat
plane gives us a “wedge” shape.
Figure 4: Rising phyllotaxis. Two coins in the lower
part represent the same single primordium. Some of
the coins on the left boundary have a corresponding
one on the right, representing the same primordium.
2.
The Coin Experiment
We present here a coin experiment modeling plant pattern formation (phyllotaxis) that physically
realizes the two postulates, i.e., as the plant grows, primordia appear during meristem expansion,
followed by, perhaps, periods of meristem’s size stabilization.
Since primordia appear at the outer ring of the circular meristem, we model this expanding ring
process with a conic shape, which we will deal with geometrically by cutting it along its side and
unrolling it flat on a plane, thus getting a “wedge” as Figure 3 shows.
We carried an experiment with coins as primordia. We roughly drew two straight lines forming
a wedge on a piece of paper taped flat on a table, representing an unrolled cone. We then placed two
coins on the lower part on each boundary, both representing the same single primordium because
the two side boundaries of the wedge should be identified. We then started placing coins, piling
them up, by putting new ones “in between” older ones, not too systematically, but usually at the
lower spot possible. As we kept placing coins, we observed rising phyllotaxis with Fibonacci
numbers appearing as we filled the cone. The photograph taken of the result is in Figure 4.†
†
I carried out this simple experiment in April 2006 and reported it with some of the results of the next sections in
the Phyllotaxis session in New Orleans, at the Joint Mathematics Meetings, on Jan 5, 2007.
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3.
The Dynamic Geometric Model
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Figure 5: Example with rising Fibonacci phyllotaxis starting with one primordia. The angle here is arbitrarily 20◦
(the rest of the paper is always 15◦ ). On the left, the voronoi cells.
We report here a systematic numerical study implementing the two postulates of the dynamic
geometric model. This model consists of placing circles of the same size on a circular cone, one
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by one, in an iterative process (one could relax a bit the equal size condition). Each new circle
placed at the lowest possible spot (towards the vertex) (this condition can also be relaxed) without
overlaps with existing ones. Thus, each new circle is tangent to two (sometimes three) previous
circles.
Figure 5 shows a reproduction of the coin experiment using a computer. We used here an
arbitrary angle of 20◦ . Everywhere else in this paper we use 15◦ . The starting circle is placed not
too far apart from itself, so that a second one can be placed tangent to it at the lowest spot. Starting
with only one circle at other heights in this way resulted always in Fibonacci configurations.
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in a plant
Figure 6: On the left, the dynamic geometric model in its initial stages. The initial configuration is that of circles 1
and 2. More circles are then placed, one at a time, in the lowest possible spot avoiding overlaps. Notice that at the edge
of the meristem (at the top), the phyllotactic pattern is Fibonacci (3,5), as it is indicated using circle 40 as reference. It
evolves to Figure 9a. On the right, the configuration viewed “around the stem”. In a plant the lower part, away from
the meristem, would have also expanded in size, preserving the pattern, just as one sees in Figure 2.
From now on, we will deal with configurations that arise starting with exactly two different
circles. The initial position of these two first circles determines an outcome. An example is shown
in Figure 6, where circles numbered 1 and 2 determine the configuration that arises. After a few
iterations the configuration evolves in this case into rising Fibonacci phyllotaxis. In the figure
on the right we see what the configuration looks like when seen wrapped around the stem. In a
plant the lower part, away from the meristem, would have also expanded considerably in size, but
preserving the pattern, just as one sees in Figure 2.
Different initial positions result in different outcomes, i.e., sometimes Lucas numbers, bijugate
configurations and even multijugate configurations (see Figure 7). We will see more about this in
the next section.
We emphasize that the two postulates make any of these phyllotactic patterns to progressively
evolve as a “fibonacci–like” sequence, with each number being the sum of the previous two. That
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Figure 7: Different initial positions result in different configurations. Circle 1 is fixed, circle 2 is varied. (a) is the
Fibonacci configuration of Figure 6. (b) Lucas, with (3,4) at the top. (c) Bijugate with (4,6) at the top. (d) Multijugate (3,6).
is, in each case, the phyllotactic type of the pattern follows a sequence of pairs of numbers
(m, n) → (m + n, n) → (m + n, m + 2n) → . . .
We will see how these transitions occur in the next section.
Primordia formation with the meristem’s diameter stabilized at a certain size is modeled by
extending the “top” of the cone with a cylinder. When the meristem stops expanding and primordia
keep appearing, the phyllotactic type of the pattern remains constant even if the structure away
from the meristem expands to macroscopic level. An example is shown in Figure 8. If, on the
contrary, we continue with meristem expansion, we observe that the pattern transitions further to
(8, 5), (8, 13), (21, 13), (21, 34), . . . as we add more and more circles.
4.
Varying Initial Positions. Parameter Space
We fix the angle of the wedge arbitrarily at 15◦ (other angles give similar results). As mentioned
before, different initial positions of the first two circles result in different outcome configurations.
We found mainly four types of outcomes:
Fibonacci: 1, 2, 3, 5, 8, 13, 21, . . . ,
Lucas: 1, 3, 4, 7, 11, 18, 29, . . .
Bijugate: 2, 4, 6, 10, 16, 26, 42, . . .
Multijugate (Trijugate): 3, 6, 9, 15, 24, 39, . . .
The bijugate numbers are two times the Fibonacci numbers and the trijugate are three times these.
We note that in very few unstable isolated boundary cases we got, for example 5, 7, 12, . . ..
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meristem
(5,3)
(5,3)
This region would
have expanded by now
(considerably) in a plant
Figure 8: Experiment implementing the meristem expansion postulate and then, after a while, the stabilization of its
size. While meristem expands, we have rising phyllotaxis as we can see in the lower part: (1, 1) → (2, 1) → (2, 3) →
(5, 3). When meristem stops expanding, we can see a constant phyllotactic pattern (5,3) or (5,3,8). On the right, we
chose a different way to render primordia than in Figure 6 making the bumps a little larger but still at the exact location
of the circles. Again, in a plant, the region below the meristem would have grown and expanded greatly but preserving
the phyllotactic pattern.
Figures 9 and 10 show the four examples of Figure 7 with 312 circles. We chose these as
arbitrary representatives of the four types we find. We took the radius of the circles as the unit of
measure. In all four examples, circle 1 is in the same place, at a distance 24.4 from the vertex.
Circle 2 is in different locations. Figure 11 shows the top layer of these four configurations after
1200 circles.
Notice how the patterns evolve. Take the example in Figure 9(a). A region with two clear sets
of parastichies and phyllotactic pattern (m, n) = (3, 5) has each circle being tangent to exactly
two below it (e.g., circle 56 has 53 and 51). The pattern evolves with a third set of parastichies
becoming more and more distinct until we reach a brief transition region, where we have a few
triple tangencies (a circle tangent to three circles below it, e.g., circle 83 has 79, 75 and 78 below
it) and three sets of parastichies with numbers m, n and m + n. A bit later, after the transition, the
pattern becomes (m + n, n) = (8, 5).
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(8,13)
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285
277
268
260
252
244
237
230
221
212
(8,5)
243
235
265
257
250
273
306
298
290
282
(11,7)
311
303
295
286
278
270
263
308
299
291
283
276
312
304
296
289
281
261
309
7
3
5
Figure 9: Here and in the next figure, the same four configurations of Figure 7 but with 312 circles. In (a) circle 56
and others have only two tangent circles below it, and circle 83, which is in a transition region, has three.
181
P. Atela
(10,6)
310
299
289
294
284
274
268
258
248
262
242
222
205
195
209
179
169
159
213
203
183
173
163
153
133
123
114
104
233
187
177
167
157
98
111
74
70
59
55
45
43
39
41
37
34
32
27
24
25
18
Bijugate
c
10
1
23
20
16
12
8
4
9
5
2
253
229
200
186
186
205
191
177
176
183
168
162
139
138
129
188
149
143
129
140
133
119
105 102
96
(6,3)
21
85
81
89
77
42
36
29
20
19
Multijugate
(trijugate)
d
6
37
33
30
25
16
15
12
46
41
28
26
24
21
18
13
7
5
2
59
52
36
31
17
11
4
1
39
34
67
57
50
88
75
63
45
42
9
8
60
48
23
19
78
113
98
84
71
66
54
29
27
22
14
56
51
35
32
69
44
40
80
122
107
91
87
73
62
58
47
38
83
76
65
53
90
147
131
115
101
94
171
156
141
125
109
104
165
150
135
118
112
97
92
72
61
43
100
79
68
49
106
86
74
55
51
110
127
121
117
144
136
130
126
120
95
82
139
134
103
99
64
11
3
108
93
15
7
123
70
67
128
145
178
173
158
153
194
187
182
167
161
154
148
142
137
116 114
111
94
146
132
124
151
169
163
157
176
203
196
192
185
179
172
166
160
155
181
175
170
189
234
218
213
206
201
195
228
222
216
210
204
198
190
184
180
164
159
199
193
212
207
242
237
231
225
219
256
251
244
240
232
227
221
215
208
202
174
152
149
217
211
197
223
254
248
271
265
259
287
280
274
303
297
289
283
268
263
257
312
306
299
293
278
272
241
236
230
307
301
288
282
250
246
239
233
226
220
214
243
235
296
267
260
255
249
291
276
270
264
258
252
245
238
224
266
261
273
285
279
310
305
298
294
286
281
26
17
13
269
209
212
275
57
48
31
28
(6,9)
83
63
38
33
30
22
14
40
35
238
247
73
54
44
264
284
308
302
295
290
120
89
69
50
46
262
300
110
79
60
280
311
304
100
95
66
56
126
106
75
155
135
277
166
145
116
85
72
52
47
122
112
81
62
58
53
49
87
152
132
91
172
292
202
192
182
162
142
101
97
68
64
138
107
78
168
148
128
188
305
228
218
208
198
178
158
118
113
93
80
154
124
84
184
164
134
204
309
254
244
234
224
214
194
174
144
103
99
90
76
65
119
109
96
82
61
115
170
130
125
190
160
140
210
240
230
220
200
180
150
146
86
71
165
105
102
92
88
77
121
185
156
216
196
245
235
225
206
175
136
131
127
108
171
141
137
117
181
161
221
201
241
295
270
260
250
311
285
276
266
256
301
291
281
271
261
251
231
211
191
151
147
237
217
197
247
227
207
257
297
287
277
267
307
303
293
283
273
263
253
243
223
193
143
(4,6)
239
219
189
259
300
290
279
269
249
229
199
286
309
306
296
275
255
236
215
292
265
246
226
302
282
272
252
232
298
288
278
312
308
304
10
6
3
Figure 10: Here and in the previous figure, the same four configurations of Figure 7 but with 312 circles.
182
P. Atela
21
13
1196
1184
1162
a
1194
1173
1152
1131
1109
1182
1161
1140
1118
1190
1169
1148
1127
1106
1179
1158
1135
1115
1198
1188
1167
1146
1125
1154
1133
1112
1103
1176
29
1190
1179
b
1161
1150
1173
1155
1143
1132
1124
1113
1102
1184
1176
1166
1106
1117
1151
1140
1129
1111
1122
1104
1115
1184
c
1168
1142
1116
1189
1178
1163
1152
1126
1136
1198
1119
1109
1181
1171
1145
1192
1166
1155
1129
1103
1176
1140
1114
1124
1108
24
1200
d
1161
1193
1187
1173
1136
1112
1179
1163
1149
1195
1125
1101
1168
1155
1139
1115
1183
1144
1131
1107
1171
1157
1120
1109
1104
1177
1115
1124
1138
1114
1125
1114
1103
1192
1181
1118
1107
1196
1185
1167
1156
1145
1134
1125
1174
1163
1152
1142
1131
1121
1111
1160
1145
1130
1106
1121
1105
1138
1127
1120
1109
1137
1182
1167
1104
1119
1198
1134
1111
1181
1165
1150
1103
1118
1135
1110
1180
1166
1151
1126
1102
1102
1141
1117
1188
1172
1156
1132
1108
1148
1123
1164
1132
1122
1112
1158
1148
1138
1128
1118
1110
1196
1189
1175
1158
1142
1127
1101
1174
1164
1154
1144
1200
1190
1180
1170
1161
1135
1127
1117
1107
1190
1174
1159
1143
1128
1113
1113
1143
1133
1123
1153
1196
1188
1179
1169
1159
1149
1139
1197
1191
1152
1156
1195
1185
1175
1165
1131
1121
1191
1183
1147
1137
1176
1199
1173
1162
1151
1184
1169
1153
1110
1199
1170
1160
1149
1139
1128
1116
1105
1187
1177
1199
1192
1162
1147
1133
1130
1120
1123
1112
1101
1141
15
1186
1120
1113
1105
1188
1178
1168
1157
1146
1135
1197
1186
1175
1164
1153
1141
1130
1119
1167
1157
1146
1134
1104
1193
1182
1171
1159
1147
1200
1189
1193
1182
1172
1160
1150
1102
1141
1134
1126
1119
1111
10
1197
1186
1177
1137
1108
26
16
1194
1126
1194
1165
1154
1144
1133
1183
1172
1162
1117
1110
1147
1139
1132
1124
1155
11
1191
1180
1169
1158
1148
1136
1198
1187
1108
1137
1130
1123
1116
1175
1168
1160
1153
1145
1189
1181
1174
1164
1157
1149
1142
1138
1129
1121
1114
1107
1151
1143
1136
1128
1101
18
1195
1150
1144
1193
1186
1178
1170
1163
1156
1199
1191
1183
1177
1171
1166
1159
1197
1192
1185
1180
1172
1165
1122
1200
1195
1187
1106
1194
1178
1140
1116
1170
1154
1129
1185
1146
1122
1105
Figure 11: The configurations of Figures 7, 9 and 10 after 1200 circles. Only the top portion is shown. (a) Fibonacci
(21,13). (b) Lucas (11,18) in transition with the third set of spirals with number 29, but circles non–tangential.
(c) Bijugate in transition (10,16, 26). (d) Multijugate (15, 24).
183
P. Atela
4.1.
Parameter Space. The Lense–Spaces
We study now systematically the set of initial positions of two circles—the “parameter space.” As
we mentioned before, the position of these two initial circles 1 and 2 determines the outcome. We
take the radius of the circles as the unit of measure. We ask that the centers of circles 1 and 2 be
less than two diameters apart (distance = 4) on either side (recall that we are on a cone) so that
circle 3 (and perhaps also circle 4) can be placed tangent to both. Fixing the “height” parameter h
of circle 1, which is its distance to the cone’s vertex, we have then a two–dimensional lense–shaped
cross section of parameter space as the region of possibilities where circle 2 may be placed (see
Figure 12). We refer to these as “lense spaces.”
30.64
24.4
19.15
Figure 12: Parameter space. On the left, the two extremes in height h for placing circle 1, h = 30.64 and h = 19.15.
The dotted regions around circle 1 denote a distance of less than 4, where circle 2 can be placed. (Some locations
circle 2 overlaps with circle 1.) Thus, the intersection of the two dotted circles form a lense–shaped region where
circle 2 can be placed without leaving too wide a space in either side with circle 1. On the right, h = 24.4 with the
four positions of circle 2 of Figures 7, 9 and 10 (see also Figure 13).
The four configurations in Figures 7, 9 and 10 have circle 1 at same location h = 24.4. So, these
four configurations are represented by four corresponding points in the lense h = 24.4. These four
points are the centers of circle 2 (one for for each case) and they are plotted in Figures 12 and 13.
In Figure 13 we scanned this lense–space (with ∆ = .007 in both directions) and color-coded
each point according to the phyllotactic type of the outcome after 300 iterations: (f) Fibonacci (the
large white spaces), (l) Lucas, (b) Bijugate, and (m) Multijugate (trijugate).
Figure 14 shows the results of scanning the lense spaces h = 19.16 and h = 22.16, the large
white regions within the lenses correspond to Fibonacci cases. For h = 19.16 there are only
Fibonacci and bijugate configurations.
We scanned the lense spaces at intervals ∆h = 0.2 using ∆ = 0.007 in both directions.
Figure 15 shows four other scanned lense spaces, corresponding to h = 23.6, 24.2, 24.8 and 25.4.
184
P. Atela
b
f
l
l
b
24.4
m
Figure 13: Lense space with circle 1 at h =
l
m
24.4. Each point inside the lense represents an
initial position for circle 2. (For this value of h,
circle 2 does not ever overlap circle 1.) We
scanned the lense space with ∆ = .007 and
colored each point according to the phyllotactic type of the outcome observed. In white, and
marked with (f), fibonacci configurations. (l) lucas, (b) bijugate, and (m) multijugate (in fact,
trijugate). The four black points correspond to
the four configurations in Figures 7, 9 and 10.
b
Figure 14: Two lense spaces, corresponding to h = 19.16 on the left, and h = 22.16 on the right . Circle 1 is placed
on the edge of the wedge as we did in Figure 12. The dotted circles here denote regions where circle 2 would overlap
with circle 1.
185
P. Atela
27
26
Fibonacci
27
27
27
26
26
26
25.4
Bijugate
25
Lucas
24.2
24
23.6
23
22
- 1.5
Trijugate
1.0
- 1.5
25
25
24
24
24
23
23
23
22
22
22
25
24.8
1.0
- 1.5
1.0
- 1.5
1.0
Figure 15: Four lense spaces, corresponding to h = 23.6, 24.2, 24.8 and 25.4. For these values of h, circle 2 does
not overlap with circle 1 when placed within the lense.
References
[1] P. Atela, C. Gole, S. Hotton. A dynamical system for plant pattern formation: a rigorous
analysis. J. Nonlinear Sci., 12 (2002), No. 9, 641–676.
[2] S. Douady, Y. Couder. Phyllotaxis as a physical self organized growth process. Phys. Rev.
Lett., (1992), No. 68, 2098–2101.
[3] S. Douady, Y. Couder. Phyllotaxis as a Self Organizing Iterative Process, Parts I, II and III.
J. Theor. Biol., (1996), No. 178, 255–312.
[4] W. Hofmeister. Allgemeine Morphologie der Gewachse, in Handbuch der Physiologishcen
Botanik. Engelmann, Leipzig, (1868), No. 1, 405–664.
[5] S. Hotton, V. Johnson, J. Wilbarger, K. Zwieniecki, P. Atela, C. Gole, J. Dumais. The possible and the actual in phyllotaxis: Bridging the gap between empirical observations and
iterative models. Journal of Plant Growth Regulation, (2006), No. 25, 313–323.
[6] R. V. Jean. Phyllotaxis, a Systemic Study in Plant Morphogenesis. Cambridge University
Press, 1994.
[7] R.V. Jean, D. Barab, editors. Symmetry in Plants. World Scientific, Singapore, 1994.
[8] L.S. Levitov. Energetic Approach to Phyllotaxis. Europhys. Lett., 14 (1991), No. 6, 533–
539.
[9] M. Snow, R. Snow. Minimum Areas and Leaf Determination. Proc. Roy. Soc., (1952),
No. B139, 545–566.
[10] G. van Iterson. Mathematische und Microscopisch-Anatamische Studien über Blattstellungen, nebst Betrshungen über den Schalenbau der miliolinen. G.-Fischer-Verlag, Jena, 1907.
186

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