Periodicity Theorem for Structure Fractals in Quaternionic Formulation

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 iteraton 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 present paper aims at an effective, explicit determination of the periods and expressing them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. The proof of the Periodicity Theorem is given in the case where the index of the generator of the algebra in question exceeds the order of the initial algebra. In contrast to earlier results, the fractal bundle flower structure, in particular sepals, bisepals, perianths, and calyces are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. The same concerns canonical two-layer pairing of prianth sepals and the relationship of fractal bundles of algebraic structure with the Hurwitz problem.

On the Ternary Approach to Clifford Structures and Ising Lattices

“…We conclude with a study of a composition algebra and self-dual perfect JNW-systems. Continuing our previous papers [10,11,13] we prove in this case it is one of quaternion and octonion algebras, and this needs a study of the product table of sedenion algebra. The idea of applying the JNW-systems, considered in Section 3 and thereafter, to binary and ternary alloys may be to some extent illustrated for instance by the first step of fractal construction related to an AB 3 binary alloy of an fcc lattice and (111) surface orientation [2, Fig.…”

“…which give rise to two kinds of fractals: of flower type and of branch type, and to the problem of their duality [8,10]. The objects of type (3.1) appear as a result of calculations in [16] and [20], but they inspire in us a relationship with the two types of fractals mentioned.…”

Structure fractals and para-quaternionic geometry

“…The proof is analogous to that of the corresponding theorem given in [4][5][6]. However, in order to understand two important differences:…”

“…6 in [4], but we have to take into account that the number of basic squares was 4 p+q−2 = 4 q , and now it amounts at 4 p+q−1 = 4 q+1 . The complete configuration related to ( The data for the initial lifting s = 11 corresponding to q = 3 can easily be checked out from Fig.…”

“…The difference in formula (1) here and in [4] is due to the replacement of matrix units 1, iσ 1 , iσ 2 , iσ 3 of the usual quaternions by the units 1, i = iσ 2 , j = σ 1 , k = σ 3 of para-quaternions, so that our i, j, and k mean j, (1/i)i, and (1/i)k in [4], respectively. This is due to our definition of the real Clifford algebraH of para-quaternions as generated by 1 and imaginary units i, j, k satisfying…”