abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, the trigintaduonions, also known as the , , form a

noncommutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a p ...

and nonassociative

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

over the

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

Names

The word ''trigintaduonion'' is derived from Latin ' 'thirty' + ' 'two' + the suffix -''nion'', which is used for

hypercomplex number In mathematics, hypercomplex number is a traditional term for an element (mathematics), element of a finite-dimensional Algebra over a field#Unital algebra, unital algebra over a field, algebra over the field (mathematics), field of real numbers. ...

systems. Other names include , , , and .

Definition

Every trigintaduonion is a

linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...

of the unit trigintaduonions

e_0

e_1

e_2

e_3

, ...,

e_

, which form a

basis Basis is a term used in mathematics, finance, science, and other contexts to refer to foundational concepts, valuation measures, or organizational names; here, it may refer to: Finance and accounting * Adjusted basis, the net cost of an asse ...

of the

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

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_ e_ + x_ e_

with real coefficients . The trigintaduonions can be obtained by applying the

Cayley–Dickson construction In mathematics, the Cayley–Dickson construction, sometimes also known as the Cayley–Dickson process or the Cayley–Dickson procedure produces a sequence of algebra over a field, algebras over the field (mathematics), field of real numbers, eac ...

to the

sedenion In abstract algebra, the sedenions form a 16-dimension of a vector space, dimensional commutative property, noncommutative and associative property, nonassociative algebra over a field, algebra over the real numbers, usually represented by the cap ...

s. 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

octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...

\mathbb

: :

\sum_^ a_i \cdot e_i

where

a_i \in \mathbb

and

e_i \notin \mathbb

An algebra of dimension 8 over

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...

\mathbb

: :

\sum_^ a_i \cdot e_i

where

a_i \in \mathbb

and

e_i \notin \mathbb

An algebra of dimension 16 over the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

\mathbb

: :

\sum_^ a_i \cdot e_i

where

a_i \in \mathbb

and

e_i \notin \mathbb

An algebra of dimension 32 over the

\mathbb

: :

\sum_^ a_i \cdot e_i

where

a_i \in \mathbb

and

e_i \notin \mathbb

\mathbb, \mathbb, \mathbb, \mathbb, \mathbb

are all

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

s of

\mathbb

. This relation can be expressed as:

\mathbb \subset \mathbb \subset \mathbb \subset \mathbb \subset \mathbb \subset \mathbb \subset \cdots

Multiplication

Properties

s and

multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...

of trigintaduonions is neither

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

nor

associative In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...

. However, being products of a Cayley–Dickson construction, trigintaduonions have the property of

power associativity In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. Definition An algebra (or more generally a magma) is said to be power-associative if the subalgebra g ...

, which can be stated as that, for any element

x

\mathbb

, the power

x^n

is well defined. They are also flexible, and multiplication is distributive over addition. As with the sedenions, the trigintaduonions contain

zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...

s and are thus not a

division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...

. Furthermore, in contrast to the octonions, both algebras do not even have the property of being

alternative Alternative or alternate may refer to: Arts, entertainment and media * Alternative (Kamen Rider), Alternative (''Kamen Rider''), a character in the Japanese TV series ''Kamen Rider Ryuki'' * Alternative comics, or independent comics are an altern ...

Geometric representations

Whereas

unit multiplication patterns can be geometrically represented by PG(2,2) (also known as the

Fano plane In finite geometry, the Fano plane (named after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and ...

) and

unit multiplication by

PG(3,2) In finite geometry, PG(3, 2) is the smallest three-dimensional projective space. It can be thought of as an extension of the Fano plane, ''PG(2, 2)''. Elements It has 15 points, 35 lines, and 15 planes. Each point is contained in 7 lines and 7 pl ...

, trigintaduonion unit multiplication can be geometrically represented by PG(4,2). Cayley-Salmon configuration g012

Multiplication tables

The

of the unit trigintaduonions is illustrated in the two tables below. Combined, they form a single 32×32 table with 1024 cells. 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 In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication binary operation, operation for an algebraic system. The decimal multiplication table was traditionally tau ...

for the

s. 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

s. {, 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. *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

The trigintaduonions have applications in

quantum physics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

, and other branches of modern physics. More recently, the trigintaduonions and other hypercomplex numbers have also been used in

neural network A neural network is a group of interconnected units called neurons that send signals to one another. Neurons can be either biological cells or signal pathways. While individual neurons are simple, many of them together in a network can perfor ...

research.

References

External links

Hypercomplex Python 3 package
{{Authority control Hypercomplex numbers Non-associative algebras