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The purpose of this paper is to identify all eight of the basic Cayley–Dickson doubling products. A Cayley–Dickson algebra \({\mathbb{A}_{N+1}}\) of dimension \({2^{N+1}}\) consists of all ordered pairs of elements of a Cayley–Dickson algebra \({\mathbb{A}_{N}}\) of dimension \({2^N}\) where the product \({(a, b)(c, d)}\) of elements of \({\mathbb{...
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The purpose of this paper is to identify all of the basic Cayley-Dickson doubling products. A Cayley-Dickson algebra A N+1 of dimension 2 N+1 consists of all ordered pairs of elements of a Cayley-Dickson algebra A N of dimension 2 N where the product (a, b)(c, d) of elements of A N+1 is defined in terms of a pair of second degree binomials (f (a, b...
A detailed analysis of the quaternion generic Jacobi-Fourier moments (QGJFMs) for color image description is presented. In order to reach numerical stability, a recursive approach is used during the computation of the generic Jacobi radial polynomials. Moreover, a search criterion is performed to establish the best values for the parameters α and β...
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... (a, b)(c, d) = (ac − bd, ad + bc) (1) To produce the quaternions by a doubling product on the complex numbers requires that one take conjugation into consideration in such a way that, for real numbers the product reduces to the one above. There are eight (and only eight) distinct Cayley-Dickson doubling products [4] which accomplish this. For each of the eight, the conjugate of an ordered pair (a, b) is defined recursively by ...
... Only two of these eight, P 3 and P ⊤ 3 have been investigated. The eight algebras resulting from these products are isomorphic [4] and all have the same elements and the same unit basis vectors e 0 , e 1 , e 2 , · · · , e n , · · · . The basis vectors will be defined below. ...
... In [4] the products P 0 , P ⊤ 0 ,P 1 , P ⊤ 1 ,P 2 , and P ⊤ 2 were derived. Further investigation has shown that for the product P 2 (and its corresponding transpose) there is a simple closed form formula for ω. ...
Although the Cayley-Dickson algebras are twisted group algebras, little attention has been paid to the nature of the Cayley-Dickson twist. One reason is that the twist appears to be highly chaotic and there are other interesting things about the algebras to focus attention upon. However, if one uses a doubling product for the algebras different from yet equivalent to the ones commonly used and if one uses a numbering of the basis vectors different from the standard basis a quite beautiful and highly periodic twist emerges. This leads easily to a simple closed form formula for the product of any two basis vectors of a Cayley-Dickson algebra.
... The unit basis vectors {e k } of Cayley-Dickson algebras may be represented as a twisted group with e 0 as the group identity. For each of the eight Cayley-Dickson doubling products [4] there is a twisting map ω(p, q) : N 2 0 → {±1} (where N 0 represents the non-negative integers) with the property that e p e q = ω(p, q)e p⊕q where ⊕ is a group operation on N 0 consisting of the 'bit-wise exclusive or' of the binary representations of non-negative integers. ...
... There exist eight basic Cayley-Dickson doubling products [4] which are listed in Table 1. A search of the literature on Cayley-Dickson algebras reveals the use of only the two doubling products P 3 and P ⊤ 3 . ...
... Using the general tree to calculate e p e q A version of the general quaternion tree is developed in [4]. The general tree depicted in Figures 3 and 4 suffices for Cayley-Dickson algebras of any dimension for products P 0 through P 3 . ...
Regarding the Cayley-Dickson algebras as twisted group algebras, this paper reveals some basic periodic properties of these twists.
