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botanical phyllotaxis. More recently, Nisoli et al. (2009) showed that to be true by constructing a "magnetic cactus" made of magnetic dipoles mounted on bearings stacked along a "stem". They demonstrated that these interacting particles can access novel dynamical phenomena beyond what botany yields: a "dynamical phyllotaxis" family of non local topological
636:
Phyllotaxis has been used as an inspiration for a number of sculptures and architectural designs. Akio Hizume has built and exhibited several bamboo towers based on the
Fibonacci sequence which exhibit phyllotaxis. Saleh Masoumi has proposed a design for an apartment building in which the apartment
604:
diagrammed grids imagined on a cylinder (rhombic lattices). Douady et al. showed that phyllotactic patterns emerge as self-organizing processes in dynamic systems. In 1991, Levitov proposed that lowest energy configurations of repulsive particles in cylindrical geometries reproduce the spirals of
437:
and its second successor. The number of leaves is sometimes called rank, in the case of simple
Fibonacci ratios, because the leaves line up in vertical rows. With larger Fibonacci pairs, the pattern becomes complex and non-repeating. This tends to occur with a basal configuration. Examples can be
550:
between successive leaves is the blind result of this jostling. Since three golden arcs add up to slightly more than enough to wrap a circle, this guarantees that no two leaves ever follow the same radial line from center to edge. The generative spiral is a consequence of the same process that
564:
and George Snow continued these lines of inquiry. Computer modeling and morphological studies have confirmed and refined
Hoffmeister's ideas. Questions remain about the details. Botanists are divided on whether the control of leaf migration depends on chemical
453:
head. This phyllotactic pattern creates an optical effect of criss-crossing spirals. In the botanical literature, these designs are described by the number of counter-clockwise spirals and the number of clockwise spirals. These also turn out to be
244:
phyllotaxis, also called "two-ranked leaf arrangement" is a special case of either opposite or alternate leaf arrangement where the leaves on a stem are arranged in two vertical columns on opposite sides of the stem. Examples include various
1839:
474:. Leaves become initiated in localized areas where auxin concentration is higher. When a leaf is initiated and begins development, auxin begins to flow towards it, thus depleting auxin from area on the
382:
structure where all the leaves are attached at the base of the shoot and the internodes are small or nonexistent. A basal whorl with a large number of leaves spread out in a circle is called a
320:, many species have just two fully developed leaves at a time, the older pair folding back and dying off to make room for the decussately oriented new pair as the plant grows.
478:
close to where the leaf was initiated. This gives rise to a self-propagating system that is ultimately controlled by the ebb and flow of auxin in different regions of the
290:
species, showing its decussate growth in which a single pair of leaves is replaced at a time, leaving just one live active pair of leaves as the old pair withers
624:
Close packing of spheres generates a dodecahedral tessellation with pentaprismic faces. Pentaprismic symmetry is related to the
Fibonacci series and the
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1159:
1534:
1499:
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The rotational angle from leaf to leaf in a repeating spiral can be represented by a fraction of a full rotation around the stem.
577:
rather than
Fibonacci numbers have been observed in a few plants and occasionally, the leaf positioning appears to be random.
1444:
1108:
791:
764:
1307:
Nisoli C, Gabor NM, Lammert PE, Maynard JD, Crespi VH (May 2009). "Static and dynamical phyllotaxis in a magnetic cactus".
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719:
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project in a spiral arrangement around a central axis and none shade the balcony of the apartment directly beneath.
59:
1858:
1527:
458:. In some cases, the numbers appear to be multiples of Fibonacci numbers because the spirals consist of whorls.
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produces the clockwise and counter-clockwise spirals that emerge in densely packed plant structures, such as
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1800:
693:
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17:
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222:), on opposite sides of the stem. An opposite leaf pair can be thought of as a whorl of two leaves.
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466:
The pattern of leaves on a plant is ultimately controlled by the accumulation of the plant hormone
206:
1634:
1463:
Solving the Riddle of
Phyllotaxis: Why the Fibonacci Numbers and the Golden Ratio Occur On Plants
516:
365:
218:
With an opposite leaf arrangement, two leaves arise from the stem at the same level (at the same
494:
330:
52:
961:
Deb, Yamini; Marti, Dominik; Frenz, Martin; Kuhlemeier, Cris; Reinhardt, Didier (2015-06-01).
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With an alternate (spiral) pattern, each leaf arises at a different point (node) on the stem.
781:
754:
359:
The whorled arrangement is fairly unusual on plants except for those with particularly short
1185:
Douady S, Couder Y (March 1992). "Phyllotaxis as a physical self-organized growth process".
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In an opposite pattern, if successive leaf pairs are 90 degrees apart, this habit is called
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8:
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Philosophical
Transactions of the Royal Society of London. Series B, Biological Sciences
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360:
102:
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How Long Is the Coast of
Britain? Statistical Self-Similarity and Fractional Dimension
1264:
Levitov LS (January 1991). "Phyllotaxis of flux lattices in layered superconductors".
889:
856:
1833:
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1659:
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1370:"Spiraling solitons: A continuum model for dynamical phyllotaxis of physical systems"
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observed that the spiral phyllotaxis of plants were frequently expressed in both
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36:
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if several leaves arise, or appear to arise, from the same level (at the same
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Alternate distichous leaves will have an angle of 1/2 of a full rotation. In
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857:"Auxin Regulates the Initiation and Radial Position of Plant Lateral Organs"
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End-on view of a plant stem showing consecutive leaves separated by the
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1761:
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1674:
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1096:
The Golden Ratio: The Story of Phi, the World's Most
Astonishing Number
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the angle is 5/13. The numerator and denominator normally consist of a
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98:
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Reinhardt, Didier; Mandel, Therese; Kuhlemeier, Cris (April 2000).
666:
606:
566:
543:
479:
475:
471:
351:
261:
251:
148:
1386:
503:—made observations of the spiral arrangements of plants. In 1754,
323:
If the arrangement is both distichous and decussate, it is called
118:
1718:
1713:
1701:
1679:
1664:
1572:
1228:
Levitov LS (15 March 1991). "Energetic
Approach to Phyllotaxis".
638:
542:, the nascent leaf, forms at the least crowded part of the shoot
418:
410:
310:
286:
1124:
Snow, M.; Snow, R. (1934). "The interpretation of Phyllotaxis".
271:
seedlings, and also in mature plants of related species such as
914:"The shoot apical meristem: the dynamics of a stable structure"
553:
515:
series. Mathematical observations of phyllotaxis followed with
442:
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426:
234:
44:
1751:
811:. Cape Town & London: Darter Bros., Wheldon & Wesley.
720:
712:
614:
467:
402:
398:
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Two different examples of the alternate (spiral) leaf pattern
1020:"The role of auxin transport in plant patterning mechanisms"
963:"Phyllotaxis involves auxin drainage through leaf primordia"
280:
1629:
1567:
1562:
446:
422:
267:
246:
94:
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and his brother Louis connected phyllotaxis ratios to the
130:
1306:
854:
406:
960:
786:. Springer Science & Business Media. pp. 14–.
783:
Illustrated Handbook of Succulent Plants: Aizoaceae A–E
759:. Springer Science & Business Media. pp. 40–.
756:
Illustrated Handbook of Succulent Plants: Crassulaceae
653:, a type of leaf size difference on horizontal shoots
613:
regime of these systems, as well as purely classical
101:. Phyllotactic spirals form a distinctive class of
1426:
1099:(First trade paperback ed.). New York City:
1068:
363:. Examples of trees with whorled phyllotaxis are
1881:
1123:
1528:
534:Insight into the mechanism had to wait until
1184:
911:
596:Physical models of phyllotaxis date back to
27:Arrangement of leaves on the stem of a plant
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1083:
773:
1535:
1521:
1227:
821:
631:
1385:
1320:
1045:
1035:
978:
937:
888:
304:Decussate phyllotaxis also occurs in the
1080:
779:
746:
584:
341:
329:
300:. It is common in members of the family
279:
227:
200:
29:
1476:
1263:
836:
806:
800:
621:in the spectrum of linear excitations.
600:'s experiment of packing hard spheres.
210:has whorls of leaves separated by long
14:
1882:
1542:
1367:
826:. Oxford: Royal Horticultural Society.
815:
449:heads. The most famous example is the
308:. In genera of the Aizoaceae, such as
1516:
1432:
1089:
1017:
752:
560:In modern times, researchers such as
523:'s 1830 and 1830 work, respectively;
1076:. New American Library. p. 121.
1821:The Chemical Basis of Morphogenesis
389:
108:
24:
1138:10.1111/j.1469-185X.1934.tb00876.x
1074:The Notebooks of Leonardo da Vinci
25:
1906:
1459:
557:flower disks or pinecone scales.
495:Patterns in nature § History
1643:
912:Traas J, Vernoux T (June 2002).
461:
255:. It also occurs in other plant
174:arrangements of leaves on a stem
156:
147:
129:
117:
1492:
1470:
1453:
1361:
1300:
1221:
1178:
1162:. Smith College. Archived from
1152:
1117:
1062:
1011:
780:Hartmann HE (6 December 2012).
232:Distichous leaf arrangement in
1339:10.1103/PhysRevLett.102.186103
1072:(1971). Taylor, Pamela (ed.).
954:
905:
848:
830:
705:
580:
499:Some early scientists—notably
13:
1:
699:
573:or purely mechanical forces.
1436:The Geometry of Art and Life
1037:10.1371/journal.pbio.0060323
694:Sphere packing in a cylinder
538:proposed a model in 1868. A
7:
1504:World Architecture News.com
1207:10.1103/PhysRevLett.68.2098
753:Eggli U (6 December 2012).
644:
10:
1911:
1404:10.1103/PhysRevE.80.026110
1286:10.1103/PhysRevLett.66.224
1250:10.1209/0295-5075/14/6/006
1018:Smith RS (December 2008).
721:
713:
492:
488:
75:
58:
1851:
1801:D'Arcy Wentworth Thompson
1744:
1652:
1641:
1550:
809:The Flora of South Africa
72: 'leaf' and
34:Crisscrossing spirals of
1368:Nisoli C (August 2009).
842:Introduction to geometry
470:in certain areas of the
207:Veronicastrum virginicum
89: 'arrangement') or
1309:Physical Review Letters
1266:Physical Review Letters
1187:Physical Review Letters
824:Dictionary of Gardening
737:A Greek–English Lexicon
632:In art and architecture
628:of classical geometry.
517:Karl Friedrich Schimper
378:A whorl can occur as a
366:Brabejum stellatifolium
1500:"Open to the elements"
930:10.1098/rstb.2002.1091
822:Chittenden FJ (1951).
593:
511:and counter-clockwise
369:and the related genus
356:
339:
325:secondarily distichous
291:
238:
215:
188:). Leaves may also be
93:is the arrangement of
40:
1635:Widmanstätten pattern
844:. Wiley. p. 169.
728:Liddell, Henry George
588:
405:the angle is 1/3, in
345:
333:
283:
231:
204:
33:
1166:on 27 September 2013
873:10.1105/tpc.12.4.507
425:, it is 3/8, and in
136:Whorled leaf pattern
1869:Mathematics and art
1859:Pattern recognition
1829:Aristid Lindenmayer
1396:2009PhRvE..80b6110N
1331:2009PhRvL.102r6103N
1278:1991PhRvL..66..224L
1242:1991EL.....14..533L
1199:1992PhRvL..68.2098D
1807:On Growth and Form
1707:Logarithmic spiral
1544:Patterns in nature
1380:(2 Pt 2): 026110.
1126:Biological Reviews
980:10.1242/dev.121244
807:Marloth R (1932).
602:Gerrit van Iterson
594:
536:Wilhelm Hofmeister
529:Fibonacci sequence
357:
340:
292:
239:
216:
103:patterns in nature
41:
1877:
1876:
1834:Benoît Mandelbrot
1734:Self-organization
1670:Natural selection
1660:Pattern formation
1446:978-0-486-23542-4
1374:Physical Review E
1193:(13): 2098–2101.
1110:978-0-7679-0816-0
1070:Leonardo da Vinci
973:(11): 1992–2001.
793:978-3-642-56306-5
766:978-3-642-55874-0
689:Three-gap theorem
501:Leonardo da Vinci
456:Fibonacci numbers
274:Kumara plicatilis
259:such as those of
16:(Redirected from
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1890:Plant morphology
1685:Sexual selection
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1433:Ghyka M (1977).
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109:Leaf arrangement
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1103:. p. 110.
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662:Fermat's spiral
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583:
525:Auguste Bravais
521:Alexander Braun
519:and his friend
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349:phyllotaxis of
184:(also known as
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1506:. 11 Dec 2012.
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1272:(2): 224–227.
1230:Europhys. Lett
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1101:Broadway Books
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861:The Plant Cell
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626:golden section
609:emerge in the
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413:it is 2/5, in
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1135:
1131:
1127:
1120:
1112:
1106:
1102:
1098:
1097:
1092:
1086:
1084:
1075:
1071:
1065:
1057:
1053:
1048:
1043:
1038:
1033:
1029:
1025:
1021:
1014:
1006:
1002:
998:
994:
990:
986:
981:
976:
972:
968:
964:
957:
949:
945:
940:
935:
931:
927:
923:
919:
915:
908:
900:
896:
891:
886:
882:
878:
874:
870:
866:
862:
858:
851:
843:
839:
833:
825:
818:
810:
803:
795:
789:
785:
784:
776:
768:
762:
758:
757:
749:
743:
739:
738:
733:
732:Scott, Robert
729:
725:
717:
708:
704:
695:
692:
690:
687:
685:
682:
680:
677:
675:
674:
670:
668:
665:
663:
660:
658:
655:
652:
649:
648:
642:
640:
629:
627:
622:
620:
616:
612:
608:
603:
599:
592:
587:
578:
576:
575:Lucas numbers
572:
568:
563:
558:
556:
555:
549:
545:
541:
537:
532:
530:
526:
522:
518:
514:
510:
506:
502:
496:
486:
484:
481:
477:
473:
469:
462:Determination
459:
457:
452:
448:
444:
441:
436:
432:
428:
424:
420:
416:
412:
408:
404:
400:
395:
387:
385:
381:
376:
374:
373:
368:
367:
362:
355:
353:
348:
344:
337:
332:
328:
326:
321:
319:
318:
313:
312:
307:
303:
299:
298:
289:
288:
282:
278:
276:
275:
270:
269:
264:
263:
258:
254:
253:
248:
243:
237:
236:
230:
226:
223:
221:
213:
209:
208:
203:
199:
198:) on a stem.
197:
193:
192:
187:
183:
179:
175:
159:
150:
132:
124:Opposite leaf
120:
106:
104:
100:
96:
92:
88:
85:
78:
74:
71:
68:
61:
57:
54:
53:Ancient Greek
50:
46:
39:
38:
32:
19:
1838:
1819:
1805:
1771:
1697:Chaos theory
1620:Tessellation
1582:
1503:
1494:
1482:. Retrieved
1472:
1462:
1455:
1435:
1428:
1377:
1373:
1363:
1312:
1308:
1302:
1269:
1265:
1236:(6): 533–9.
1233:
1229:
1223:
1190:
1186:
1180:
1170:24 September
1168:. Retrieved
1164:the original
1154:
1129:
1125:
1119:
1095:
1091:Livio, Mario
1073:
1064:
1030:(12): e323.
1027:
1024:PLOS Biology
1023:
1013:
970:
966:
956:
921:
917:
907:
864:
860:
850:
841:
832:
823:
817:
808:
802:
782:
775:
755:
748:
735:
707:
671:
635:
623:
595:
591:golden angle
559:
552:
548:golden angle
533:
513:golden ratio
498:
480:meristematic
465:
396:
393:
377:
370:
364:
358:
350:
338:leaf pattern
324:
322:
315:
309:
302:Crassulaceae
295:
293:
285:
272:
266:
260:
250:
241:
240:
233:
224:
217:
205:
189:
185:
181:
177:
171:
90:
83:
80:
73:
66:
63:
56:
48:
42:
35:
1815:Alan Turing
1773:Liber Abaci
1692:Mathematics
1598:in crystals
1588:Soap bubble
1583:Phyllotaxis
1484:18 November
1479:"Star Cage"
967:Development
684:Plastochron
657:Decussation
651:Anisophylly
581:Mathematics
49:phyllotaxis
1884:Categories
1762:Empedocles
1757:Pythagoras
1675:Camouflage
1613:in biology
1608:in flowers
1578:Parastichy
838:Coxeter HS
700:References
679:Parastichy
569:among the
540:primordium
493:See also:
483:topography
415:sunflowers
361:internodes
317:Conophytum
242:Distichous
212:internodes
172:The basic
99:plant stem
91:phyllotaxy
51:(from
18:Distichous
1864:Emergence
1767:Fibonacci
1460:Adler I.
1439:. Dover.
1387:0907.2576
1258:250864634
1160:"History"
1093:(2003) .
989:1477-9129
881:1040-4651
639:balconies
611:nonlinear
571:primordia
567:gradients
562:Mary Snow
531:in 1837.
509:clockwise
451:sunflower
440:composite
438:found in
372:Macadamia
354:rupestris
347:Decussate
336:decussate
306:Aizoaceae
297:decussate
182:alternate
1593:Symmetry
1551:Patterns
1420:27552596
1412:19792203
1347:19518890
1294:10043542
1215:10045303
1146:86184933
1056:19090623
1005:13800404
997:25953346
948:12079669
899:10760240
840:(1961).
667:L-system
645:See also
607:solitons
544:meristem
476:meristem
472:meristem
352:Crassula
262:Gasteria
252:Boophone
249:such as
178:opposite
1852:Related
1719:Crystal
1714:Physics
1702:Fractal
1680:Mimicry
1665:Biology
1573:Meander
1392:Bibcode
1355:4596630
1327:Bibcode
1274:Bibcode
1238:Bibcode
1195:Bibcode
1047:2602727
939:1692983
740:at the
489:History
443:flowers
411:apricot
384:rosette
311:Lithops
287:Lithops
191:whorled
67:phúllon
1895:Leaves
1745:People
1653:Causes
1443:
1418:
1410:
1353:
1345:
1292:
1256:
1213:
1144:
1107:
1054:
1044:
1003:
995:
987:
946:
936:
897:
890:139849
887:
879:
790:
763:
714:φύλλον
619:maxons
615:rotons
554:Protea
546:. The
431:almond
427:willow
421:, and
419:poplar
257:habits
235:Clivia
186:spiral
95:leaves
60:φύλλον
45:botany
1752:Plato
1558:Crack
1416:S2CID
1382:arXiv
1351:S2CID
1317:arXiv
1254:S2CID
1142:S2CID
1001:S2CID
722:τάξις
468:auxin
403:hazel
399:beech
380:basal
97:on a
84:táxis
77:τάξις
55:
1630:Wave
1568:Foam
1563:Dune
1486:2012
1441:ISBN
1408:PMID
1343:PMID
1290:PMID
1211:PMID
1172:2013
1105:ISBN
1052:PMID
993:PMID
985:ISSN
944:PMID
895:PMID
877:ISSN
788:ISBN
761:ISBN
617:and
598:Airy
447:seed
445:and
429:and
423:pear
409:and
401:and
314:and
268:Aloe
220:node
196:node
180:and
176:are
1400:doi
1335:doi
1313:102
1282:doi
1246:doi
1203:doi
1134:doi
1042:PMC
1032:doi
975:doi
971:142
934:PMC
926:doi
922:357
885:PMC
869:doi
407:oak
265:or
43:In
1886::
1502:.
1414:.
1406:.
1398:.
1390:.
1378:80
1376:.
1372:.
1349:.
1341:.
1333:.
1325:.
1311:.
1288:.
1280:.
1270:66
1268:.
1252:.
1244:.
1234:14
1232:.
1209:.
1201:.
1191:68
1189:.
1140:.
1128:.
1082:^
1050:.
1040:.
1026:.
1022:.
999:.
991:.
983:.
969:.
965:.
942:.
932:.
920:.
916:.
893:.
883:.
875:.
865:12
863:.
859:.
734:;
730:;
726:.
718:,
485:.
417:,
386:.
375:.
334:A
327:.
284:A
277:.
105:.
47:,
1536:e
1529:t
1522:v
1488:.
1466:.
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1422:.
1402::
1394::
1384::
1357:.
1337::
1329::
1319::
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1284::
1276::
1260:.
1248::
1240::
1217:.
1205::
1197::
1174:.
1148:.
1136::
1130:9
1113:.
1058:.
1034::
1028:6
1007:.
977::
950:.
928::
901:.
871::
796:.
769:.
214:.
87:)
81:(
70:)
64:(
20:)
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