# Structure fractals and para-quaternionic geometry

Julian Ławrynowicz; Massimo Vaccaro

Annales UMCS, Mathematica (2011)

- Volume: 65, Issue: 2, page 63-73
- ISSN: 2083-7402

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topJulian Ławrynowicz, and Massimo Vaccaro. "Structure fractals and para-quaternionic geometry." Annales UMCS, Mathematica 65.2 (2011): 63-73. <http://eudml.org/doc/268160>.

@article{JulianŁawrynowicz2011,

abstract = {It is well known that starting with real structure, the Cayley-Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 27. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteration process p ← p + 2 ← p + 4 ← …, they have constructed 24-dimensional "bipetals" for p = 9 and 27-dimensional "bisepals" for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the "pistil" and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the "stamens". The first named author, M. Nowak-Kępczyk, and S. Marchiafava (2006, 2009a, b) gave an effective, explicit determination of the periods and expressed them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. In contrast to earlier results, the fractal bundle flower structure, in particular petals, sepals, pistils, and stamens are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. Physical concepts of dual and conjugate objects as well as of antiparticles led us to extend the periodicity theorem to structure fractals in para-quaternionic formulation, applying some results in this direction by the second named author. The paper is concluded by outlining some applications.},

author = {Julian Ławrynowicz, Massimo Vaccaro},

journal = {Annales UMCS, Mathematica},

keywords = {Fractal; quaternion; para-quaternion; Clifford structure; para-quaternionic structure; bilinear form; quadratic form; fractal},

language = {eng},

number = {2},

pages = {63-73},

title = {Structure fractals and para-quaternionic geometry},

url = {http://eudml.org/doc/268160},

volume = {65},

year = {2011},

}

TY - JOUR

AU - Julian Ławrynowicz

AU - Massimo Vaccaro

TI - Structure fractals and para-quaternionic geometry

JO - Annales UMCS, Mathematica

PY - 2011

VL - 65

IS - 2

SP - 63

EP - 73

AB - It is well known that starting with real structure, the Cayley-Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 27. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteration process p ← p + 2 ← p + 4 ← …, they have constructed 24-dimensional "bipetals" for p = 9 and 27-dimensional "bisepals" for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the "pistil" and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the "stamens". The first named author, M. Nowak-Kępczyk, and S. Marchiafava (2006, 2009a, b) gave an effective, explicit determination of the periods and expressed them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. In contrast to earlier results, the fractal bundle flower structure, in particular petals, sepals, pistils, and stamens are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. Physical concepts of dual and conjugate objects as well as of antiparticles led us to extend the periodicity theorem to structure fractals in para-quaternionic formulation, applying some results in this direction by the second named author. The paper is concluded by outlining some applications.

LA - eng

KW - Fractal; quaternion; para-quaternion; Clifford structure; para-quaternionic structure; bilinear form; quadratic form; fractal

UR - http://eudml.org/doc/268160

ER -

## References

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- Ławrynowicz, J., Marchiafava, S. and Nowak-Kępczyk, M., Quaternionic background of the periodicity of petal and sepal structures in some fractals of the flower type, Proceedings of the 5th ISAAC Congress, Catania, July 25-30, 2005: More Progresses in Analysis, ed. by H. Begehr and F. Nicolosi, World Scientific, Singapore, 2009, pp. 987-996. Zbl1181.81072
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- Ławrynowicz, J., Ogata, T. and Suzuki, O., Differential and integral calculus for Schauder basis on a fractal set (I) (Schauder basis 80 years after), Lvov Mathematical School in the Period 1915-45 as Seen Today, ed. by B. Bojarski, J. Ławrynowicz and Ya. G. Prytuła, Banach Center Publications 87, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 2009, pp. 115-140. Zbl1218.37062
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- Lounesto, P., Clifford Algebra and Spinors, London Math. Soc. Lecture Notes 239, Cambridge Univ. Press, Cambridge, 1997, 2nd ed. (vol. 286), ibid. 2001.
- Vaccaro, M., Subspaces of a para-quaternionic Hermitian vector space, 2010, ArXiv:1011.2947v1 [math. D6]; Internat. J. of Geom. Methods in Modern Phys. 8 (2011), in print.
- Vaccaro, M., Basics of linear para-quaternionic geometry I. Hermitian para-type structure on a real vector space, Bull. Soc. Sci. Lett. Łódź Sér. Rech. Déform. 61 (2011), no. 1, 23-36. Zbl1293.53039
- Vaccaro, M., Basics of linear para-quaternionic geometry II. Decomposition of a generic subspace of a para-quaternionic hermitian vector space, Bull. Soc. Sci. Lett. Łódź Sér. Rech. Déform. 61 (2011), no. 2, 17-34. Zbl1275.53033

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