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Discrete lattices on the single bearing spiral: from geometry to botany. (English) Zbl 1504.92026

Darvas, György (ed.), Complex symmetries. Cham: Birkhäuser. 105-122 (2021).
Summary: This work is based on the “bearing spiral hypothesis”. It refers primarily to organic objects with helical or spiral structures and shapes. It assumes the existence of a single bearing (generative) spiral (or helix) that sets the trajectory of the elements’ multiplication. The elements, regularly located on the bearing spiral, form discrete lattices that can be studied using elementary geometry. The key point of the hypothesis is the interpretation of the “mode of existence” of the bearing spiral. Since nothing has been identified resembling this spiral “in physical expression” in plant anatomy, it is necessary to initially agree that the bearing spiral can be a spatial expression of a cyclic process.
For the entire collection see [Zbl 1485.52001].

MSC:

92C15 Developmental biology, pattern formation
92C80 Plant biology
Full Text: DOI

References:

[1] Adler, D. Barabe, R. V. Jean (1997) A History of the Study of Phyllotaxis. Oxford Journals Life Sciences, Annals of Botany; 80, 3, 231-244. https://doi.org/10.1006/anbo.1997.0422 · doi:10.1006/anbo.1997.0422
[2] Bravais, L., Bravais, A. (1839) Essai sur la disposition generate des feuilles rectiseriees, Ann. Sci. Nat. Bot. 12: 5-14, 65-77.
[3] Cook T.A. (1914) The curves of Life, Being an Account of Spiral Formation and Their Application to Growth in Nature, To Science and Art, London: Constable and Co Ltd, 479 p.
[4] Coxeter H.S.M., S.L. Greitzer (1969) Geometry revised, John Wiley & Sons, Inc. 469 p. · Zbl 0181.48101
[5] Douady S., Couder Y. (1996) Phyllotaxis as a Dynamical Self Organizing Process. Parts I, II, III. J. Theor. Biol., 178, 255-312. https://doi.org/10.1006/jtbi.1996.0026 · doi:10.1006/jtbi.1996.0026
[6] Church A.H. (1904) On the Relation of Phyllotaxis to Mechanical Laws, London: Williams and Norgate, 353 p. https://doi.org/10.5962/bhl.title.57125 · doi:10.5962/bhl.title.57125
[7] Darvas, G. (2007) Symmetry, Basel: Birkhäuser, xi +508 p. · Zbl 1142.00002
[8] Gola E.M, Banasiak A. (2016) Diversity of phyllotaxis in land plants in reference to the shoot apical meristem structure, Acta Soc Bot Pol., 85 (4), 3529. https://doi.org/10.5586/asbp.3529 · doi:10.5586/asbp.3529
[9] Gurevich, D. (2020) Discrete lattices on the single bearing spiral: From botany to music, Symmetry: Culture and Science, 31, 4, 449-463. https://doi.org/10.26830/symmetry_2020_4_449 · Zbl 1499.92005 · doi:10.26830/symmetry_2020_4_449
[10] Hardy G.H., Wright E.M. (1938) An Introduction to the Theory of Numbers, 4th Ed, Oxford, 1960. · Zbl 0020.29201
[11] Harnum, J. (2001) Basic Music Theory. How to Read, Write and Understand Written Music, Sol-Ut Press, 333 p.
[12] Jean, R.V. (1994) Phyllotaxis: A Systemic Study in Plant Morphogenesis, Cambridge University Press, 386 р. (digitally printed version 2009). https://doi.org/10.1017/CBO9780511666933 · Zbl 0818.92001 · doi:10.1017/CBO9780511666933
[13] Kunz, M., F. Rothen (1989), Phyllotaxis or the properties of spiral lattices. Part III: An algebraic model of morphogenesis, J. Phys. France 2, 2131-2172. https://doi.org/10.1051/jp1:1992273 · doi:10.1051/jp1:1992273
[14] Mauseth, J.D. (1998) Botany. An Introduction to plant biology, Jones and bartlett Pub., 868 p.
[15] Naylor M. (2002) Golden, √2, and π flowers: a spiral story, Mathematic Magazine 75, 163-172. https://doi.org/10.1080/0025570X.2002.11953126 · doi:10.1080/0025570X.2002.11953126
[16] Okabe T., Ishida A., and Yoshimura J. (2019) The unified rule of phyllotaxis explaining both spiral and non-spiral arrangements. J R Soc Interface. Feb; 16 (151). https://doi.org/10.1098/rsif.2018.0850 · doi:10.1098/rsif.2018.0850
[17] Petoukhov S.B. (1981) Biomechanics, bionics and symmetry, Moscow: Nauka, 239 p. (in Russian). · Zbl 0513.92001
[18] Ridley J. N. (1986) Ideal phyllotaxis on general surfaces of revolution, Math. Biosc., 79, 1, May, 1-24. https://doi.org/10.1016/0025-5564(86)90013-1 · Zbl 0601.92002 · doi:10.1016/0025-5564(86)90013-1
[19] Ridley J. N. (1982) Packing efficiency in sunflower Heads, Math. Biosc., 58, 1, February, 129-139. https://doi.org/10.1016/0025-5564(82)90056-6 · Zbl 0487.92001 · doi:10.1016/0025-5564(82)90056-6
[20] Rivier N., Occelli R., Pantaloni J. and Lissowski A. (1984) Structure of Bénard convection cells, phyllotaxis and crystallography in cylindrical symmetry, J. Physique, 45, 49-63. https://doi.org/10.1051/jphys:0198400450104900 · doi:10.1051/jphys:0198400450104900
[21] Rothen F., Koch A.-J. (1989) Phyllotaxis, or the properties of spiral lattices. Part I. Shape invariance under compression, J. Phys. France, 50 633-657. https://doi.org/10.1051/jphys:01989005006063300 · doi:10.1051/jphys:01989005006063300
[22] Sadoc, J.-F., N. Rivier and J. Charvolin. (2012) Phyllotaxis: a non-conventional crystalline solution to packing efficiency in situations with radial symmetry, Acta Cryst., A68, 470-483. https://doi.org/10.1107/S0108767312018910 · doi:10.1107/S0108767312018910
[23] Schimper, C. F. (1836) Geometrische Anordnung der um eine Axe periferischen Blattgebilde, Verhandl. Schez. Naturf. Ges. 21, 113-17.
[24] Thompson, D’Arcy W. (1942) On Growth and Form, Cambridge University Press (A new edition, 1945), 1116 p. · Zbl 0063.07372
[25] Vogel H. (1979) A Better Way to Construct the Sunflower Head, Mathematical Biosciences 44, 119-189. https://doi.org/10.1016/0025-5564(79)90080-4 · doi:10.1016/0025-5564(79)90080-4
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