Login
Login

Trigintaduonion

{{Short description|Hypercomplex number system}}

{{Infobox number system

| official_name = Trigintaduonions

| symbol = \mathbb T

| type = Hypercomplex algebra

| units = e0, ..., e31

| identity = e0

| properties = {{Plainlist|

}}

}}

In abstract algebra, the trigintaduonions, also known as the {{nowrap|32-ions}}, {{nowrap|32-nions}}, {{nowrap|25-nions}} form a {{nowrap|32-dimensional}} noncommutative and nonassociative algebra over the real numbers.{{cite journal | last1=Saini | first1=Kavita | last2=Raj | first2=Kuldip | title=On generalization for Tribonacci Trigintaduonions | journal=Indian Journal of Pure and Applied Mathematics | publisher=Springer Science and Business Media LLC | volume=52 | issue=2 | year=2021 | issn=0019-5588 | doi=10.1007/s13226-021-00067-y | pages=420–428}}{{cite web | title=Trigintaduonion | website=University of Waterloo | url=https://ece.uwaterloo.ca/~dwharder/Java/doc/ca/uwaterloo/alumni/dwharder/Numbers/Trigintaduonion.html | access-date=2024-10-08}}

Names

The word trigintaduonion is derived from Latin {{linktext|triginta|lang=la}} 'thirty' + {{linktext|duo|lang=la}} 'two' + the suffix -nion, which is used for hypercomplex number systems. Other names include {{nowrap|32-ion}}, {{nowrap|32-nion}}, {{nowrap|25-ion}}, and {{nowrap|25-nion}}.

Definition

Every trigintaduonion is a linear combination of the unit trigintaduonions e_0, e_1, e_2, e_3, ..., e_{31}, which form a basis of the vector space of trigintaduonions. Every trigintaduonion can be represented in the form

:x = x_0 e_0 + x_1 e_1 + x_2 e_2 + \cdots + x_{30} e_{30} + x_{31} e_{31}

with real coefficients {{mvar|xi}}.

The trigintaduonions can be obtained by applying the Cayley–Dickson construction to the sedenions.{{cite web|url=https://mathsci.kaist.ac.kr/~tambour/fichiers/publications/Ensembles_de_nombres.pdf|date=6 September 2011|title=Ensembles de nombres|publisher=Forum Futura-Science|access-date=11 October 2024|language=fr}} Applying the Cayley–Dickson construction to the trigintaduonions yields a 64-dimensional algebra called the 64-ions, 64-nions, sexagintaquatronions, or sexagintaquattuornions.

As a result, the trigintaduonions can also be defined as the following.

An algebra of dimension 4 over the octonions \mathbb{O}:

:\sum_{i=0}^{3} a_i \cdot e_i where a_i \in \mathbb{O} and e_i \notin \mathbb{O}

An algebra of dimension 8 over quaternions \mathbb{H}:

:\sum_{i=0}^{7} a_i \cdot e_i where a_i \in \mathbb{H} and e_i \notin \mathbb{H}

An algebra of dimension 16 over the complex numbers \mathbb{C}:

:\sum_{i=0}^{15} a_i \cdot e_i where a_i \in \mathbb{C} and e_i \notin \mathbb{C}

An algebra of dimension 32 over the real numbers \mathbb{R}:

:\sum_{i=0}^{31} a_i \cdot e_i where a_i \in \mathbb{R} and e_i \notin \mathbb{R}

\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}, \mathbb{S} are all subsets of \mathbb{T}. This relation can be expressed as:

\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \mathbb{T} \subset \cdots

Multiplication

= Properties =

Like octonions and sedenions, multiplication of trigintaduonions is neither commutative nor associative. However, being products of a Cayley–Dickson construction, trigintaduonions have the property of power associativity, which can be stated as that, for any element x of \mathbb{T}, the power x^n is well defined. They are also flexible, and multiplication is distributive over addition.{{cite arXiv | eprint=0907.2047 | last1=Cawagas | first1=Raoul E. | last2=Carrascal | first2=Alexander S. | last3=Bautista | first3=Lincoln A. | last4=Maria | first4=John P. Sta. | last5=Urrutia | first5=Jackie D. | last6=Nobles | first6=Bernadeth | title=The Subalgebra Structure of the Cayley-Dickson Algebra of Dimension 32 (Trigintaduonion) | date=2009 | class=math.RA }} As with the sedenions, the trigintaduonions contain zero divisors and are thus not a division algebra. Furthermore, in contrast to the octonions, both algebras do not even have the property of being alternative.

= Geometric representations =

Whereas octonion unit multiplication patterns can be geometrically represented by PG(2,2) (also known as the Fano plane) and sedenion unit multiplication by PG(3,2), trigintaduonion unit multiplication can be geometrically represented by PG(4,2).

File:Cayley-Salmon configuration g012.png

= Multiplication tables =

The multiplication of the unit trigintaduonions is illustrated in the two tables below. Combined, they form a single 32×32 table with 1024 cells.{{cite journal | last=Weng | first=Zi-Hua | title=Gauge fields and four interactions in the trigintaduonion spaces | journal=Mathematical Methods in the Applied Sciences | publisher=Wiley | date=2024-07-23 | volume=48 | pages=590–604 | issn=0170-4214 | doi=10.1002/mma.10345 | doi-access=free| arxiv=2407.18265 }}

Below is the trigintaduonion multiplication table for e_j, 0 \leq j \leq 15. The top half of this table, for e_i, 0 \leq i \leq 15, corresponds to the multiplication table for the sedenions. The top left quadrant of the table, for e_i, 0 \leq i \leq 7 and e_j, 0 \leq j \leq 7, corresponds to the multiplication table for the octonions.

class="wikitable" style="margin:1em auto; text-align: center;"

!colspan="2" rowspan="2"| e_ie_j

!colspan="16" |e_j

e_0

! e_1

! e_2

! e_3

! e_4

! e_5

! e_6

! e_7

! e_8

! e_9

! e_{10}

! e_{11}

! e_{12}

! e_{13}

! e_{14}

! e_{15}

rowspan="32" | e_i

! width="30pt" | e_0

| width="30pt" | e_0

| width="30pt" | e_1

| width="30pt" | e_2

| width="30pt" | e_3

| width="30pt" | e_4

| width="30pt" | e_5

| width="30pt" | e_6

| width="30pt" | e_7

| width="30pt" | e_8

| width="30pt" | e_9

| width="30pt" | e_{10}

| width="30pt" | e_{11}

| width="30pt" | e_{12}

| width="30pt" | e_{13}

| width="30pt" | e_{14}

| width="30pt" | e_{15}

e_{1}

| e_{1}

| -e_{0}

| e_{3}

| -e_{2}

| e_{5}

| -e_{4}

| -e_{7}

| e_{6}

| e_{9}

| -e_{8}

| -e_{11}

| e_{10}

| -e_{13}

| e_{12}

| e_{15}

| -e_{14}

e_{2}

| e_{2}

| -e_{3}

| -e_{0}

| e_{1}

| e_{6}

| e_{7}

| -e_{4}

| -e_{5}

| e_{10}

| e_{11}

| -e_{8}

| -e_{9}

| -e_{14}

| -e_{15}

| e_{12}

| e_{13}

e_{3}

| e_{3}

| e_{2}

| -e_{1}

| -e_{0}

| e_{7}

| -e_{6}

| e_{5}

| -e_{4}

| e_{11}

| -e_{10}

| e_{9}

| -e_{8}

| -e_{15}

| e_{14}

| -e_{13}

| e_{12}

e_{4}

| e_{4}

| -e_{5}

| -e_{6}

| -e_{7}

| -e_{0}

| e_{1}

| e_{2}

| e_{3}

| e_{12}

| e_{13}

| e_{14}

| e_{15}

| -e_{8}

| -e_{9}

| -e_{10}

| -e_{11}

e_{5}

| e_{5}

| e_{4}

| -e_{7}

| e_{6}

| -e_{1}

| -e_{0}

| -e_{3}

| e_{2}

| e_{13}

| -e_{12}

| e_{15}

| -e_{14}

| e_{9}

| -e_{8}

| e_{11}

| -e_{10}

e_{6}

| e_{6}

| e_{7}

| e_{4}

| -e_{5}

| -e_{2}

| e_{3}

| -e_{0}

| -e_{1}

| e_{14}

| -e_{15}

| -e_{12}

| e_{13}

| e_{10}

| -e_{11}

| -e_{8}

| e_{9}

e_{7}

| e_{7}

| -e_{6}

| e_{5}

| e_{4}

| -e_{3}

| -e_{2}

| e_{1}

| -e_{0}

| e_{15}

| e_{14}

| -e_{13}

| -e_{12}

| e_{11}

| e_{10}

| -e_{9}

| -e_{8}

e_{8}

| e_{8}

| -e_{9}

| -e_{10}

| -e_{11}

| -e_{12}

| -e_{13}

| -e_{14}

| -e_{15}

| -e_{0}

| e_{1}

| e_{2}

| e_{3}

| e_{4}

| e_{5}

| e_{6}

| e_{7}

e_{9}

| e_{9}

| e_{8}

| -e_{11}

| e_{10}

| -e_{13}

| e_{12}

| e_{15}

| -e_{14}

| -e_{1}

| -e_{0}

| -e_{3}

| e_{2}

| -e_{5}

| e_{4}

| e_{7}

| -e_{6}

e_{10}

| e_{10}

| e_{11}

| e_{8}

| -e_{9}

| -e_{14}

| -e_{15}

| e_{12}

| e_{13}

| -e_{2}

| e_{3}

| -e_{0}

| -e_{1}

| -e_{6}

| -e_{7}

| e_{4}

| e_{5}

e_{11}

| e_{11}

| -e_{10}

| e_{9}

| e_{8}

| -e_{15}

| e_{14}

| -e_{13}

| e_{12}

| -e_{3}

| -e_{2}

| e_{1}

| -e_{0}

| -e_{7}

| e_{6}

| -e_{5}

| e_{4}

e_{12}

| e_{12}

| e_{13}

| e_{14}

| e_{15}

| e_{8}

| -e_{9}

| -e_{10}

| -e_{11}

| -e_{4}

| e_{5}

| e_{6}

| e_{7}

| -e_{0}

| -e_{1}

| -e_{2}

| -e_{3}

e_{13}

| e_{13}

| -e_{12}

| e_{15}

| -e_{14}

| e_{9}

| e_{8}

| e_{11}

| -e_{10}

| -e_{5}

| -e_{4}

| e_{7}

| -e_{6}

| e_{1}

| -e_{0}

| e_{3}

| -e_{2}

e_{14}

| e_{14}

| -e_{15}

| -e_{12}

| e_{13}

| e_{10}

| -e_{11}

| e_{8}

| e_{9}

| -e_{6}

| -e_{7}

| -e_{4}

| e_{5}

| e_{2}

| -e_{3}

| -e_{0}

| e_{1}

e_{15}

| e_{15}

| e_{14}

| -e_{13}

| -e_{12}

| e_{11}

| e_{10}

| -e_{9}

| e_{8}

| -e_{7}

| e_{6}

| -e_{5}

| -e_{4}

| e_{3}

| e_{2}

| -e_{1}

| -e_{0}

e_{16}

| e_{16}

| -e_{17}

| -e_{18}

| -e_{19}

| -e_{20}

| -e_{21}

| -e_{22}

| -e_{23}

| -e_{24}

| -e_{25}

| -e_{26}

| -e_{27}

| -e_{28}

| -e_{29}

| -e_{30}

| -e_{31}

e_{17}

| e_{17}

| e_{16}

| -e_{19}

| e_{18}

| -e_{21}

| e_{20}

| e_{23}

| -e_{22}

| -e_{25}

| e_{24}

| e_{27}

| -e_{26}

| e_{29}

| -e_{28}

| -e_{31}

| e_{30}

e_{18}

| e_{18}

| e_{19}

| e_{16}

| -e_{17}

| -e_{22}

| -e_{23}

| e_{20}

| e_{21}

| -e_{26}

| -e_{27}

| e_{24}

| e_{25}

| e_{30}

| e_{31}

| -e_{28}

| -e_{29}

e_{19}

| e_{19}

| -e_{18}

| e_{17}

| e_{16}

| -e_{23}

| e_{22}

| -e_{21}

| e_{20}

| -e_{27}

| e_{26}

| -e_{25}

| e_{24}

| e_{31}

| -e_{30}

| e_{29}

| -e_{28}

e_{20}

| e_{20}

| e_{21}

| e_{22}

| e_{23}

| e_{16}

| -e_{17}

| -e_{18}

| -e_{19}

| -e_{28}

| -e_{29}

| -e_{30}

| -e_{31}

| e_{24}

| e_{25}

| e_{26}

| e_{27}

e_{21}

| e_{21}

| -e_{20}

| e_{23}

| -e_{22}

| e_{17}

| e_{16}

| e_{19}

| -e_{18}

| -e_{29}

| e_{28}

| -e_{31}

| e_{30}

| -e_{25}

| e_{24}

| -e_{27}

| e_{26}

e_{22}

| e_{22}

| -e_{23}

| -e_{20}

| e_{21}

| e_{18}

| -e_{19}

| e_{16}

| e_{17}

| -e_{30}

| e_{31}

| e_{28}

| -e_{29}

| -e_{26}

| e_{27}

| e_{24}

| -e_{25}

e_{23}

| e_{23}

| e_{22}

| -e_{21}

| -e_{20}

| e_{19}

| e_{18}

| -e_{17}

| e_{16}

| -e_{31}

| -e_{30}

| e_{29}

| e_{28}

| -e_{27}

| -e_{26}

| e_{25}

| e_{24}

e_{24}

| e_{24}

| e_{25}

| e_{26}

| e_{27}

| e_{28}

| e_{29}

| e_{30}

| e_{31}

| e_{16}

| -e_{17}

| -e_{18}

| -e_{19}

| -e_{20}

| -e_{21}

| -e_{22}

| -e_{23}

e_{25}

| e_{25}

| -e_{24}

| e_{27}

| -e_{26}

| e_{29}

| -e_{28}

| -e_{31}

| e_{30}

| e_{17}

| e_{16}

| e_{19}

| -e_{18}

| e_{21}

| -e_{20}

| -e_{23}

| e_{22}

e_{26}

| e_{26}

| -e_{27}

| -e_{24}

| e_{25}

| e_{30}

| e_{31}

| -e_{28}

| -e_{29}

| e_{18}

| -e_{19}

| e_{16}

| e_{17}

| e_{22}

| e_{23}

| -e_{20}

| -e_{21}

e_{27}

| e_{27}

| e_{26}

| -e_{25}

| -e_{24}

| e_{31}

| -e_{30}

| e_{29}

| -e_{28}

| e_{19}

| e_{18}

| -e_{17}

| e_{16}

| e_{23}

| -e_{22}

| e_{21}

| -e_{20}

e_{28}

| e_{28}

| -e_{29}

| -e_{30}

| -e_{31}

| -e_{24}

| e_{25}

| e_{26}

| e_{27}

| e_{20}

| -e_{21}

| -e_{22}

| -e_{23}

| e_{16}

| e_{17}

| e_{18}

| e_{19}

e_{29}

| e_{29}

| e_{28}

| -e_{31}

| e_{30}

| -e_{25}

| -e_{24}

| -e_{27}

| e_{26}

| e_{21}

| e_{20}

| -e_{23}

| e_{22}

| -e_{17}

| e_{16}

| -e_{19}

| e_{18}

e_{30}

| e_{30}

| e_{31}

| e_{28}

| -e_{29}

| -e_{26}

| e_{27}

| -e_{24}

| -e_{25}

| e_{22}

| e_{23}

| e_{20}

| -e_{21}

| -e_{18}

| e_{19}

| e_{16}

| -e_{17}

e_{31}

| e_{31}

| -e_{30}

| e_{29}

| e_{28}

| -e_{27}

| -e_{26}

| e_{25}

| -e_{24}

| e_{23}

| -e_{22}

| e_{21}

| e_{20}

| -e_{19}

| -e_{18}

| e_{17}

| e_{16}

Below is the trigintaduonion multiplication table for e_j, 16 \leq j \leq 31.

class="wikitable" style="margin:1em auto; text-align: center;"

!colspan="2" rowspan="2"| e_ie_j

!colspan="16" |e_j

e_{16}

! e_{17}

! e_{18}

! e_{19}

! e_{20}

! e_{21}

! e_{22}

! e_{23}

! e_{24}

! e_{25}

! e_{26}

! e_{27}

! e_{28}

! e_{29}

! e_{30}

! e_{31}

rowspan="32" | e_i

! width="30pt" | e_0

| width="30pt" | e_{16}

| width="30pt" | e_{17}

| width="30pt" | e_{18}

| width="30pt" | e_{19}

| width="30pt" | e_{20}

| width="30pt" | e_{21}

| width="30pt" | e_{22}

| width="30pt" | e_{23}

| width="30pt" | e_{24}

| width="30pt" | e_{25}

| width="30pt" | e_{26}

| width="30pt" | e_{27}

| width="30pt" | e_{28}

| width="30pt" | e_{29}

| width="30pt" | e_{30}

| width="30pt" | e_{31}

e_{1}

| e_{17}

| -e_{16}

| -e_{19}

| e_{18}

| -e_{21}

| e_{20}

| e_{23}

| -e_{22}

| -e_{25}

| e_{24}

| e_{27}

| -e_{26}

| e_{29}

| -e_{28}

| -e_{31}

| e_{30}

e_{2}

| e_{18}

| e_{19}

| -e_{16}

| -e_{17}

| -e_{22}

| -e_{23}

| e_{20}

| e_{21}

| -e_{26}

| -e_{27}

| e_{24}

| e_{25}

| e_{30}

| e_{31}

| -e_{28}

| -e_{29}

e_{3}

| e_{19}

| -e_{18}

| e_{17}

| -e_{16}

| -e_{23}

| e_{22}

| -e_{21}

| e_{20}

| -e_{27}

| e_{26}

| -e_{25}

| e_{24}

| e_{31}

| -e_{30}

| e_{29}

| -e_{28}

e_{4}

| e_{20}

| e_{21}

| e_{22}

| e_{23}

| -e_{16}

| -e_{17}

| -e_{18}

| -e_{19}

| -e_{28}

| -e_{29}

| -e_{30}

| -e_{31}

| e_{24}

| e_{25}

| e_{26}

| e_{27}

e_{5}

| e_{21}

| -e_{20}

| e_{23}

| -e_{22}

| e_{17}

| -e_{16}

| e_{19}

| -e_{18}

| -e_{29}

| e_{28}

| -e_{31}

| e_{30}

| -e_{25}

| e_{24}

| -e_{27}

| e_{26}

e_{6}

| e_{22}

| -e_{23}

| -e_{20}

| e_{21}

| e_{18}

| -e_{19}

| -e_{16}

| e_{17}

| -e_{30}

| e_{31}

| e_{28}

| -e_{29}

| -e_{26}

| e_{27}

| e_{24}

| -e_{25}

e_{7}

| e_{23}

| e_{22}

| -e_{21}

| -e_{20}

| e_{19}

| e_{18}

| -e_{17}

| -e_{16}

| -e_{31}

| -e_{30}

| e_{29}

| e_{28}

| -e_{27}

| -e_{26}

| e_{25}

| e_{24}

e_{8}

| e_{24}

| e_{25}

| e_{26}

| e_{27}

| e_{28}

| e_{29}

| e_{30}

| e_{31}

| -e_{16}

| -e_{17}

| -e_{18}

| -e_{19}

| -e_{20}

| -e_{21}

| -e_{22}

| -e_{23}

e_{9}

| e_{25}

| -e_{24}

| e_{27}

| -e_{26}

| e_{29}

| -e_{28}

| -e_{31}

| e_{30}

| e_{17}

| -e_{16}

| e_{19}

| -e_{18}

| e_{21}

| -e_{20}

| -e_{23}

| e_{22}

e_{10}

| e_{26}

| -e_{27}

| -e_{24}

| e_{25}

| e_{30}

| e_{31}

| -e_{28}

| -e_{29}

| e_{18}

| -e_{19}

| -e_{16}

| e_{17}

| e_{22}

| e_{23}

| -e_{20}

| -e_{21}

e_{11}

| e_{27}

| e_{26}

| -e_{25}

| -e_{24}

| e_{31}

| -e_{30}

| e_{29}

| -e_{28}

| e_{19}

| e_{18}

| -e_{17}

| -e_{16}

| e_{23}

| -e_{22}

| e_{21}

| -e_{20}

e_{12}

| e_{28}

| -e_{29}

| -e_{30}

| -e_{31}

| -e_{24}

| e_{25}

| e_{26}

| e_{27}

| e_{20}

| -e_{21}

| -e_{22}

| -e_{23}

| -e_{16}

| e_{17}

| e_{18}

| e_{19}

e_{13}

| e_{29}

| e_{28}

| -e_{31}

| e_{30}

| -e_{25}

| -e_{24}

| -e_{27}

| e_{26}

| e_{21}

| e_{20}

| -e_{23}

| e_{22}

| -e_{17}

| -e_{16}

| -e_{19}

| e_{18}

e_{14}

| e_{30}

| e_{31}

| e_{28}

| -e_{29}

| -e_{26}

| e_{27}

| -e_{24}

| -e_{25}

| e_{22}

| e_{23}

| e_{20}

| -e_{21}

| -e_{18}

| e_{19}

| -e_{16}

| -e_{17}

e_{15}

| e_{31}

| -e_{30}

| e_{29}

| e_{28}

| -e_{27}

| -e_{26}

| e_{25}

| -e_{24}

| e_{23}

| -e_{22}

| e_{21}

| e_{20}

| -e_{19}

| -e_{18}

| e_{17}

| -e_{16}

e_{16}

| -e_{0}

| e_{1}

| e_{2}

| e_{3}

| e_{4}

| e_{5}

| e_{6}

| e_{7}

| e_{8}

| e_{9}

| e_{10}

| e_{11}

| e_{12}

| e_{13}

| e_{14}

| e_{15}

e_{17}

| -e_{1}

| -e_{0}

| -e_{3}

| e_{2}

| -e_{5}

| e_{4}

| e_{7}

| -e_{6}

| -e_{9}

| e_{8}

| e_{11}

| -e_{10}

| e_{13}

| -e_{12}

| -e_{15}

| e_{14}

e_{18}

| -e_{2}

| e_{3}

| -e_{0}

| -e_{1}

| -e_{6}

| -e_{7}

| e_{4}

| e_{5}

| -e_{10}

| -e_{11}

| e_{8}

| e_{9}

| e_{14}

| e_{15}

| -e_{12}

| -e_{13}

e_{19}

| -e_{3}

| -e_{2}

| e_{1}

| -e_{0}

| -e_{7}

| e_{6}

| -e_{5}

| e_{4}

| -e_{11}

| e_{10}

| -e_{9}

| e_{8}

| e_{15}

| -e_{14}

| e_{13}

| -e_{12}

e_{20}

| -e_{4}

| e_{5}

| e_{6}

| e_{7}

| -e_{0}

| -e_{1}

| -e_{2}

| -e_{3}

| -e_{12}

| -e_{13}

| -e_{14}

| -e_{15}

| e_{8}

| e_{9}

| e_{10}

| e_{11}

e_{21}

| -e_{5}

| -e_{4}

| e_{7}

| -e_{6}

| e_{1}

| -e_{0}

| e_{3}

| -e_{2}

| -e_{13}

| e_{12}

| -e_{15}

| e_{14}

| -e_{9}

| e_{8}

| -e_{11}

| e_{10}

e_{22}

| -e_{6}

| -e_{7}

| -e_{4}

| e_{5}

| e_{2}

| -e_{3}

| -e_{0}

| e_{1}

| -e_{14}

| e_{15}

| e_{12}

| -e_{13}

| -e_{10}

| e_{11}

| e_{8}

| -e_{9}

e_{23}

| -e_{7}

| e_{6}

| -e_{5}

| -e_{4}

| e_{3}

| e_{2}

| -e_{1}

| -e_{0}

| -e_{15}

| -e_{14}

| e_{13}

| e_{12}

| -e_{11}

| -e_{10}

| e_{9}

| e_{8}

e_{24}

| -e_{8}

| e_{9}

| e_{10}

| e_{11}

| e_{12}

| e_{13}

| e_{14}

| e_{15}

| -e_{0}

| -e_{1}

| -e_{2}

| -e_{3}

| -e_{4}

| -e_{5}

| -e_{6}

| -e_{7}

e_{25}

| -e_{9}

| -e_{8}

| e_{11}

| -e_{10}

| e_{13}

| -e_{12}

| -e_{15}

| e_{14}

| e_{1}

| -e_{0}

| e_{3}

| -e_{2}

| e_{5}

| -e_{4}

| -e_{7}

| e_{6}

e_{26}

| -e_{10}

| -e_{11}

| -e_{8}

| e_{9}

| e_{14}

| e_{15}

| -e_{12}

| -e_{13}

| e_{2}

| -e_{3}

| -e_{0}

| e_{1}

| e_{6}

| e_{7}

| -e_{4}

| -e_{5}

e_{27}

| -e_{11}

| e_{10}

| -e_{9}

| -e_{8}

| e_{15}

| -e_{14}

| e_{13}

| -e_{12}

| e_{3}

| e_{2}

| -e_{1}

| -e_{0}

| e_{7}

| -e_{6}

| e_{5}

| -e_{4}

e_{28}

| -e_{12}

| -e_{13}

| -e_{14}

| -e_{15}

| -e_{8}

| e_{9}

| e_{10}

| e_{11}

| e_{4}

| -e_{5}

| -e_{6}

| -e_{7}

| -e_{0}

| e_{1}

| e_{2}

| e_{3}

e_{29}

| -e_{13}

| e_{12}

| -e_{15}

| e_{14}

| -e_{9}

| -e_{8}

| -e_{11}

| e_{10}

| e_{5}

| e_{4}

| -e_{7}

| e_{6}

| -e_{1}

| -e_{0}

| -e_{3}

| e_{2}

e_{30}

| -e_{14}

| e_{15}

| e_{12}

| -e_{13}

| -e_{10}

| e_{11}

| -e_{8}

| -e_{9}

| e_{6}

| e_{7}

| e_{4}

| -e_{5}

| -e_{2}

| e_{3}

| -e_{0}

| -e_{1}

e_{31}

| -e_{15}

| -e_{14}

| e_{13}

| e_{12}

| -e_{11}

| -e_{10}

| e_{9}

| -e_{8}

| e_{7}

| -e_{6}

| e_{5}

| e_{4}

| -e_{3}

| -e_{2}

| e_{1}

| -e_{0}

= Triples =

There are 155 distinguished triples (or triads) of imaginary trigintaduonion units in the trigintaduonion multiplication table, which are listed below. In comparison, the octonions have 7 such triples, the sedenions have 35, while the sexagintaquatronions have 651.{{cite OEIS|A171477}}

  • 45 triples of type {α, α, β}: {3, 13, 14}, {3, 21, 22}, {3, 25, 26}, {5, 11, 14}, {5, 19, 22}, {5, 25, 28}, {6, 11, 13}, {6, 19, 21}, {6, 26, 28}, {7, 9, 14}, {7, 10, 13}, {7, 11, 12}, {7, 17, 22}, {7, 18, 21}, {7, 19, 20}, {7, 25, 30}, {7, 26, 29}, {7, 27, 28}, {9, 19, 26}, {9, 21, 28}, {10, 19, 25}, {10, 22, 28}, {11, 17, 26}, {11, 18, 25}, {11, 19, 24}, {11, 21, 30}, {11, 22, 29}, {11, 23, 28}, {12, 21, 25}, {12, 22, 26}, {13, 17, 28}, {13, 19, 30}, {13, 20, 25}, {13, 21, 24}, {13, 22, 27}, {13, 23, 26}, {14, 18, 28}, {14, 19, 29}, {14, 20, 26}, {14, 21, 27}, {14, 22, 24}, {14, 23, 25}, {15, 19, 28}, {15, 21, 26}, {15, 22, 25}
  • 20 triples of type {β, β, β}: {3, 5, 6}, {3, 9, 10}, {3, 17, 18}, {3, 29, 30}, {5, 9, 12}, {5, 17, 20}, {5, 27, 30}, {6, 10, 12}, {6, 18, 20}, {6, 27, 29}, {9, 17, 24}, {9, 23, 30}, {10, 18, 24}, {10, 23, 29}, {12, 20, 24}, {12, 23, 27}, {15, 17, 30}, {15, 18, 29}, {15, 20, 27}, {15, 23, 24}
  • 15 triples of type {β, β, β}: {3, 12, 15}, {3, 20, 23}, {3, 24, 27}, {5, 10, 15}, {5, 18, 23}, {5, 24, 29}, {6, 9, 15}, {6, 17, 23}, {6, 24, 30}, {9, 18, 27}, {9, 20, 29}, {10, 17, 27}, {10, 20, 30}, {12, 17, 29}, {12, 18, 30}
  • 60 triples of type {α, β, γ}: {1, 6, 7}, {1, 10, 11}, {1, 12, 13}, {1, 14, 15}, {1, 18, 19}, {1, 20, 21}, {1, 22, 23}, {1, 24, 25}, {1, 26, 27}, {1, 28, 29}, {2, 5, 7}, {2, 9, 11}, {2, 12, 14}, {2, 13, 15}, {2, 17, 19}, {2, 20, 22}, {2, 21, 23}, {2, 24, 26}, {2, 25, 27}, {2, 28, 30}, {3, 4, 7}, {3, 8, 11}, {3, 16, 19}, {3, 28, 31}, {4, 9, 13}, {4, 10, 14}, {4, 11, 15}, {4, 17, 21}, {4, 18, 22}, {4, 19, 23}, {4, 24, 28}, {4, 25, 29}, {4, 26, 30}, {5, 8, 13}, {5, 16, 21}, {5, 26, 31}, {6, 8, 14}, {6, 16, 22}, {6, 25, 31}, {7, 8, 15}, {7, 16, 23}, {7, 24, 31}, {8, 17, 25}, {8, 18, 26}, {8, 19, 27}, {8, 20, 28}, {8, 21, 29}, {8, 22, 30}, {9, 16, 25}, {9, 22, 31}, {10, 16, 26}, {10, 21, 31}, {11, 16, 27}, {11, 20, 31}, {12, 16, 28}, {12, 19, 31}, {13, 16, 29}, {13, 18, 31}, {14, 16, 30}, {14, 17, 31}
  • 15 triples of type {β, γ, γ}: {1, 2, 3}, {1, 4, 5}, {1, 8, 9}, {1, 16, 17}, {1, 30, 31}, {2, 4, 6}, {2, 8, 10}, {2, 16, 18}, {2, 29, 31}, {4, 8, 12}, {4, 16, 20}, {4, 27, 31}, {8, 16, 24}, {8, 23, 31}, {5, 16, 31}

Applications

{{Primary sources section|date=April 2025}}

The trigintaduonions have applications in quantum physics, and other branches of modern physics. More recently, the trigintaduonions and other hypercomplex numbers have also been used in neural network research.{{cite journal | last1=Baluni | first1=Sapna | last2=Yadav | first2=Vijay K. | last3=Das | first3=Subir | title=Lagrange stability criteria for hypercomplex neural networks with time varying delays | journal=Communications in Nonlinear Science and Numerical Simulation | publisher=Elsevier BV | volume=131 | year=2024 | issn=1007-5704 | doi=10.1016/j.cnsns.2023.107765 | page=107765| bibcode=2024CNSNS.13107765B }}

References

{{reflist}}

External links

{{wikt|trigintaduonion}}

  • [https://pypi.org/project/hypercomplex/ Hypercomplex Python 3 package]

{{Number systems}}

{{Dimension topics}}

{{Authority control}}

Category:Hypercomplex numbers

Category:Non-associative algebras