# Who pioneered the study of the sedenions?

I found lots of background information about the discovery of both imaginary and complex numbers, and enough information about the first two types of hypercomplex numbers; quaternions and octonions (also known as Cayley Numbers). However, I haven't found who were the mathematicians that pioneered the study of the sedenions.

It is known that this can be achieved through the Cayley-Dickson doubling process. This question is not about the usefulness of the sedenions, or the operations that are allowed on them, but who pioneered their study and when.

• J.-E. Pin contributed this: Here is an early reference. The term sedenion is not used in this paper, but the Cayley-Dickson process is used to produce algebras with dimension a power of 2 (including 16). R. D. Schafer, On the algebras formed by the Cayley-Dickson process, Amer. J. Math. 76 (1954), 435-446. Source: math.stackexchange.com/questions/3526551/… – Mr. J. Larios Feb 3 '20 at 3:35
• @Danu: He does actually mention 'hyper-complex numbers'; I was merely pointing out - if he didn't happen to know - that a large part of that study of those system come under Clifford algebras. – Mozibur Ullah Mar 16 '20 at 18:59

There are (at least) two different types of numbers called "sedenions". The first ones were introduced by Muses in 1980, who called them $$16$$-ions, and renamed into "sedenions" by Carmody in 1988. However, Sorgsepp and Lohmus beat him to the name in Binary and Ternary Sedenions (1981), which they used more in line with "quaternions" and "octonions" (sedecim is Latin for sixteen), i.e. for the output of the Cayley-Dickson construction:

"The possibility of an extension of the 8-dimensional Cayley algebra into a 16-dimensional sedenion algebra is investigated. For retaining some desirable properties, the binary sedenion algebra obtained by the Cayley-Dickson process is modified into an algebra with a ternary product. Both for binary and ternary sedenion algebras, the generalized Cauchy-Riemann and Laplace equations are written out. The finite projective geometries formed by hypercomplex sedenion units are briefly discussed."

As Imaeda and Imaeda, who studied the Cayley-Dickson sedenions further, explain in Sedenions: algebra and analysis (2000):

"Some parts of our work paralells that of other authors [3-5]; but unlike the sedenions discussed in [3,4] our sedenions are non-alternative, non-modular and posess real norm. In [5] alternativity is restored by introducing ternary products. Should Hurwitz theorem be one possible indication of non-conformity of sedenions to current mathematical methods in physics? It is clear from Muses [6] and Gursey [7], and other authors that some new algebraic and analytical structures, such that might be offered by a $$16$$-dimensional algebra, may hold a key to better formulation of physical theories. Furthermore, its relation to the Lie group $$G_2$$ remains compelling under any circumstances."

The "other authors" are Carmody ([3-4]) and Sorgsepp-Lohmus ([5]). In Carmody's Circular and hyperbolic quaternions, octonions, and sedenions — Further results (1997) we find a prefatory note by Muses, written at Carmody's request, which reads in particular:

"From the prevalence of counterquaternions in quantum physics, I feel sure that counteroctonions and sedonions will inevitably be entrained as well on the frontiers of physics. By 1980 I had discovered/invented them and called them 16-ions [3, p. 76] and perhaps that designation is least opaque. In any case, their 16 elementary units ($$i_0$$ through $$i_7$$ and $$ε_0$$ through $$ε_7$$) plus the $$8$$ from $$w$$-arithmetic ($$w_0$$ through $$w_7$$) form a $$24$$-dimensional vector space. This is a watershed in hypernumber arithmetic and algebra because the equation $$x^0 - 1 = 0$$ has elementary hypernumber solutions only if $$x=\pm i_n$$, $$x=\pm ε_n$$ or $$x=\pm w_n$$."

The reference [3] is to Muses's Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries (1980) that details his previous announcements in 1976 and 1978. In the abstract we read:

"It turns out that more than three kinds of $$i$$-type hypemumbers and more than three hinds of the $$ε$$-type are needed to ensure the necessary nilpotent and noncommutative algebra required in unified field theory. It is also shown that “more than three” here means “at least seven”; and it turns out that a l6-dimensional arithmetic is needed for such computations. The following paper contextualizes, characterizes, and specifies that arithmetic at the apex of a hierarchy susceptible of clear geometric definition. And the hypemumbers needed in quantized unified field theory are specified."

Carmody's paper quoted above is a follow-up to his Circular and hyperbolic quaternions, octonions, and sedenions (1988) that names them "sedenions" with the following explanation:

"Muses has exhibited the fact that a 16-dimensional system comprising the elements $$1$$, $$i_1$$ to $$i_7$$, $$ε_1$$ to $$ε_7$$, and a sixteenth element is possible. This system is called the sedenions after the Latin sedecim, sixteen."

• The inclusion of Muses (and Carmody) is not convenient because Muses introduced something called Hypernumbers, which is very different from Hypercomplex Numbers. When I introduced the question I wasn't referring to the rare work of Muses, but the better known Complex Numbers, Hamilton Quaternions, Octonions (Cayley Numbers), the doubling process from Cayley-Dickson, etc... Even the talk page from Wikipedia shows that those two topics should not be mixed into the same article. – Mr. J. Larios Feb 7 '20 at 4:52

On Bibliography of Quaternions and Allied Mathematics by Alexander Macfarlane I found this:

On page 72; James Byrnie Shaw 1896 Sedenions (title). American Assoc. Proc., 45, 26.

I couldn't find this reference, but the same author wrote this book:

Synopsis of Linear Associative Algebra: A Report on its Natural Development and Results Reached up to the Present Time. 1907.

Early on Sedenions were also known as "quadriquaternions".

Section XIX: Sylvester Algebras. Page 93. Covers "Nonions" (9-ions), and "Sedenions" (16-ions). Here the Sedenions are attributed to James Joseph Sylvester.

On page 76; James Joseph Sylvester 1883-4 On quaternions, nonions, sedenions, etc. Johns Hopkins Univ. Circ., 3. Nos. 7 and 9. 4, No. 28.

This second reference can be found among The Collected Mathematical Papers of James Joseph Sylvester, [Volume IV(1882—1897)]:

Sylvester, James Joseph (1973) [1904], Baker, Henry Frederick (ed.), The collected mathematical papers of James Joseph Sylvester, IV, New York: AMS Chelsea Publishing, ISBN 978-0-8218-4238-6

Then, as far as I understood, James Joseph Sylvester appears on the literature as the proponent of two Algebras; the one from the Nonions and the Sedenions. I will leave as an open question for the community to confirm, correct, or debunk that J.J. Sylvester "discovered" or "was the pioneer" of their study in 1883.

• I think you are right about Sylvester. Full text of his 1883 note is viewable on archive. It does not include the word "sedenions" except in the title, but they are among the systems defined there. Looks like Sorgsepp and Lohmus were unaware of Sylvester or Shaw, they do not cite them either in 1981 or in 1998. – Conifold Feb 7 '20 at 5:53