Talk:Sedenion

message: I wonder  about the following feature of the sedenions as claimed in the article:
They have multiplicative inverses and at the same time zero divisors.
In a matrix algebra, these two features cannot coexist, since a divisor of
zero necessarily has determinant zero and thus it is not invertible.
I"d like to see an example of two sedenions with inverses and a product of zero.
--J"org Knappen
:What the article says is:The sedenions have a multiplicative identity element 1 and multiplicative inverses, but they are not a division algebra. This is because they have zero divisors.Take a look at zero divisors which demonstrates the principle with matrices. A given sedenion might be a zero divisor when paired with one particular other, but nothing stops either having an inverse as far as I can see. HTH --Phil | Talk 11:56, Jan 11, 2005 (UTC)
::Seen that -- nice demostration of what is possible in infinitely many dimensions. My argument above is for finite matrices (what"s the determinant of an infinite matrix, anyway?) I just like to see an example for sedenions, too --J"org Knappen
:::J"org - Matrices form an associative algebra. Sedenions are non-associative. You are not able to map a sedenion algebra on matrices (as opposed to e.g. octonions or quaternions). For examples of calculations with sedenions see e.g. the two references to articles from K. Carmody. Please note, however, that the sedenions in there are of a different type as the ones displayed in the multiplication table (I'll try to have this corrected). --65.185.222.50 13:19, 12 September 2005 (UTC)
replies: text:  Within the current article it says   (e_3+e_{10})(e_6-e_{15}) = 0 , and the inverse of any s can be calculated by s^*/|s|^2, so (e_3+e_{10})^{-1} = -1/2 (e_3+e_{10}),  (e_6-e_{15})^{-1}=-1/2 (e_6-e_{15}). Generally, the inverses of (nontrivial) zero divisors are again zero divisors. Summsumm2 (talk) 15:19, 5 July 2019 (UTC)
thread_title:  Multiplicative inverses '''and''' zero divisors 

message: when were they discovered? are there larger algebras with the same properties? 83.79.181.211 19:05, 29 September 2005 (UTC)
replies: text: About the term "sedenion" and the discovery; as far as I know, the cited articles are the earliest publications I could find that go deeper into arithmetic laws of the sedenions, and thereby solidify the term (so it's quite recent). Surely, as part of the Cayley-Dickson construction, their existence was discovered earlier within that program. Any earlier known uses of the term "sedenion"? Thanks, Jens Koeplinger 12:40, 15 September 2006 (UTC)
thread_title: History

message: Substituted at 02:35, 5 May 2016 (UTC)
thread_title: Assessment comment

message: For a double simetry in a sedenion, resulting in a Quaternion,
resultant wave function in the Quantum mechanics, is a matrix that results in the Elementary particle of the standard model (without Higgs).
Please, add the simplification.
--188.171.58.12 (talk) 22:17, 26 December 2019 (UTC)
thread_title:  Double simetry in Sedenion 

message: Could the multiplication table be fixed such that it shows -e correctly for all of the elements? In my browser (Chrome) it shows the negative sign above the e for e values with two-digit subscripts, and it took me a moment to figure out what this was. TricksterWolf (talk) 18:55, 29 February 2020 (UTC)
thread_title:  Multiplication Table 

message: There are actually 168 primitive zero divisors. The Quaternion triplets mentioned are those created by the Cayley-Dickson doubling process with sequential new basis elements but is not unique, others exist. Please consult my paper: https://vixra.org/abs/2010.0086?ref=11771562 where all 168 primitive zero divisors for the standard Sedenion form mentioned are itemized, the 15 Octonion subalgebra candidates are itemized by Sedenion basis element subsets, an algebraic proof all 15 cannot be oriented as proper Octonion algebras, a method to determine if a set of seven oriented Quaternion triplets otherwise suitable for an Octonion Algebra actually describes one, a method to find all sets of 5 Octonion subalgebra candidates that cannot all be properly oriented and how this shows the maximal number of Octonion Algebra candidates that can be oriented as proper Octonion is eight; any seven sharing a common basis element plus any one of the other eight.
The definition variation for all Cayley-Dickson algebras through Sedenions is discussed in detail in my paper describing all valid Quaternion triplet orientations. A doubling process by variation is provided. There is much information provided that could improve the wiki for Octonion Algebra.
EightOnions (talk) 23:43, 14 January 2021 (UTC)
thread_title:  Incorrect number of primitive zero divisors 

message: In Richard D. Schafer (1954) "On the algebras formed by the Cayley–Dickson process", American Journal of Mathematics 76: 435–46 doi:10.2307/2372583 it is demonstrated that all Cayley - Dickson algebras are flexible alternative. I feel that only saying 'sedenions are not an alternative algebra', thought it is true, does not describe the essence of sedenions.  — Preceding unsigned comment added by Crodrigue1 (talk • contribs) 01:44, 30 January 2021 (UTC)
thread_title:  Flexible alternativity 

message: I think there are 2 mistakes in the Sedenion Zero divisor list:
"(e1+e13)×(e7+e-14) = 0" is False, but "(e1+e13)×(e2+e-14) == 0" is True.
"(e6+e10)×(e7+e-13) = 0" is False, but "(e6+e12)×(e7+e-13) == 0" is True.
Correction would then be:
 {e1,e13,e7,e-14} -> changed to -> {e1,e13,e2,e-14}
 {e6,e10,e7,e-13} -> changed to -> {e6,e12,e7,e-13}
I am not math certified. I wrote a program to locate zero divisors and found these 2 differences. Maybe someone could verify. Peawormsworth (talk) 19:36, 6 August 2022 (UTC)
thread_title:  Zero Divisor correction 

message: Names have been given to various further algebras, but this appears inconsistent. The 32 unit algebra is referred to in one section as the "pathions" but elsewhere as the "trigintaduonions" (this name presumably courtesy of Raoul E. Cawagas). Both of these are inconsistent with the established naming scheme given by the names of the quaternions, octonions and sedenions. The name of the sedenions shows that the 32 unit algebra following it should be the duotriginions, as the tens prefix (den-) is preceeded by the ones prefix (se-), and since the "ta" of "triginta" would be omitted to fit the established nomenclaturalogical pattern ("...-nions"). This would give:
Real numbers (1)
Complex numbers (2)
Quaternions (4)
Octonions (8)
Sedenions (16)
Duotriginions (32)
Quatersexaginions (64) *
Cenoctoviginions (128)
Duocensequinquaginions (256) *
and as the article says, ad infinitum.
*: The established prefixes "se" and "quater" might be considered only due to ease of pronunciation, and "sex" and "quattuor" (the correct latin prefixes) may in fact be more appropriate. HYPERIAPATH (talk) 22:03, 25 May 2024 (UTC)
thread_title:  Names of further algebras