# Trigintaduonion

In abstract algebra, the trigintaduonions, also known as the 32-ions, 32-nions, 2$^5$-nions, or sometimes pathions ($\mathbb {P}$), form a 32-dimensional noncommutative and nonassociative algebra over the real numbers, usually represented by the capital letter T, boldface T or blackboard bold $\mathbb {T}$.

## Names

The word trigintaduonion is derived from Latin triginta 'thirty' + duo 'two' + the suffix -nion, which is used for hypercomplex number systems.
Although trigintaduonion is typically the more widely used term, Robert P. C. de Marrais instead uses the term pathion in reference to the 32 paths of wisdom from the Kabbalistic (Jewish mystical) text Sefer Yetzirah, since pathion is shorter and easier to remember and pronounce. It is represented by blackboard bold $\mathbb {P}$. Other names include 32-ion, 32-nion, 2$^5$-ion, and 2$^5$-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 x$_i$.
The trigintaduonions can be obtained by applying the Cayley–Dickson construction to the sedenions, which can be mathematically expressed as $\mathbb {T} ={\mathcal {CD}}(\mathbb {S} ,1)$. Applying the Cayley–Dickson construction to the trigintaduonions yields a 64-dimensional algebra called the 64-ions, 64-nions, sexagintaquatronions, or sexagintaquattuornions, sometimes also known as the chingons.
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. 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). This can be also extended to PG(5,2) for the 64-nions, as explained in the abstract of Saniga, Holweck & Pracna (2015):
   Given a $2^{n}$-dimensional Cayley–Dickson algebra, where $3\leq n\leq 6$, we first observe that the multiplication table of its imaginary units $e_{a},1\leq a\leq 2^{n}-1$, is encoded in the properties of the projective space $PG(n-1,2)$ if these imaginary units are regarded as points and distinguished triads of them $\{e_{a},e_{b},e_{c}\},1\leq a<b<c\leq 2^{n}-1$ and $e_{a}\cdot e_{b}=\pm e_{c}$, as lines. This projective space is seen to feature two distinct kinds of lines according as $a+b=c$ or $a+b\neq c$.
Furthermore, Saniga, Holweck & Pracna (2015) state that:
   The corresponding point-line incidence structure is found to be a specific binomial configuration ${\mathcal {C}}_{n}$; in particular, ${\mathcal {C}}_{3}$ (octonions) is isomorphic to the Pasch (6$_2$,4$_3$)-configuration, ${\mathcal {C}}_{4}$ (sedenions) is the famous Desargues (10$_3$)-configuration, ${\mathcal {C}}_{5}$ (32-nions) coincides with the Cayley–Salmon (15$_4$,20$_3$)-configuration found in the well-known Pascal mystic hexagram and ${\mathcal {C}}_{6}$ (64-nions) is identical with a particular (21$_5$,35$_3$)-configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration.
The configuration of $2^{n}$-nions can thus be generalized as: ${\binom {n+1}{2}}_{n-1},{\binom {n+1}{3}}_{3}$

### 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.
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.
| $e_{i}e_{j}$ | $e_{i}e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ |
| $e_{i}e_{j}$ | $e_{i}e_{j}$ | $e_{0}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$ | $e_{9}$ | $e_{10}$ | $e_{11}$ | $e_{12}$ | $e_{13}$ | $e_{14}$ | $e_{15}$ |
| $e_{i}$ | $e_{0}$ | $e_{0}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$ | $e_{9}$ | $e_{10}$ | $e_{11}$ | $e_{12}$ | $e_{13}$ | $e_{14}$ | $e_{15}$ |
| $e_{i}$ | $e_{1}$ | $e_{1}$ | $-e_{0}$ | $e_{3}$ | $-e_{2}$ | $e_{5}$ | $-e_{4}$ | $-e_{7}$ | $e_{6}$ | $e_{9}$ | $-e_{8}$ | $-e_{11}$ | $e_{10}$ | $-e_{13}$ | $e_{12}$ | $e_{15}$ | $-e_{14}$ |
| $e_{i}$ | $e_{2}$ | $e_{2}$ | $-e_{3}$ | $-e_{0}$ | $e_{1}$ | $e_{6}$ | $e_{7}$ | $-e_{4}$ | $-e_{5}$ | $e_{10}$ | $e_{11}$ | $-e_{8}$ | $-e_{9}$ | $-e_{14}$ | $-e_{15}$ | $e_{12}$ | $e_{13}$ |
| $e_{i}$ | $e_{3}$ | $e_{3}$ | $e_{2}$ | $-e_{1}$ | $-e_{0}$ | $e_{7}$ | $-e_{6}$ | $e_{5}$ | $-e_{4}$ | $e_{11}$ | $-e_{10}$ | $e_{9}$ | $-e_{8}$ | $-e_{15}$ | $e_{14}$ | $-e_{13}$ | $e_{12}$ |
| $e_{i}$ | $e_{4}$ | $e_{4}$ | $-e_{5}$ | $-e_{6}$ | $-e_{7}$ | $-e_{0}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{12}$ | $e_{13}$ | $e_{14}$ | $e_{15}$ | $-e_{8}$ | $-e_{9}$ | $-e_{10}$ | $-e_{11}$ |
| $e_{i}$ | $e_{5}$ | $e_{5}$ | $e_{4}$ | $-e_{7}$ | $e_{6}$ | $-e_{1}$ | $-e_{0}$ | $-e_{3}$ | $e_{2}$ | $e_{13}$ | $-e_{12}$ | $e_{15}$ | $-e_{14}$ | $e_{9}$ | $-e_{8}$ | $e_{11}$ | $-e_{10}$ |
| $e_{i}$ | $e_{6}$ | $e_{6}$ | $e_{7}$ | $e_{4}$ | $-e_{5}$ | $-e_{2}$ | $e_{3}$ | $-e_{0}$ | $-e_{1}$ | $e_{14}$ | $-e_{15}$ | $-e_{12}$ | $e_{13}$ | $e_{10}$ | $-e_{11}$ | $-e_{8}$ | $e_{9}$ |
| $e_{i}$ | $e_{7}$ | $e_{7}$ | $-e_{6}$ | $e_{5}$ | $e_{4}$ | $-e_{3}$ | $-e_{2}$ | $e_{1}$ | $-e_{0}$ | $e_{15}$ | $e_{14}$ | $-e_{13}$ | $-e_{12}$ | $e_{11}$ | $e_{10}$ | $-e_{9}$ | $-e_{8}$ |
| $e_{i}$ | $e_{8}$ | $e_{8}$ | $-e_{9}$ | $-e_{10}$ | $-e_{11}$ | $-e_{12}$ | $-e_{13}$ | $-e_{14}$ | $-e_{15}$ | $-e_{0}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ |
| $e_{i}$ | $e_{9}$ | $e_{9}$ | $e_{8}$ | $-e_{11}$ | $e_{10}$ | $-e_{13}$ | $e_{12}$ | $e_{15}$ | $-e_{14}$ | $-e_{1}$ | $-e_{0}$ | $-e_{3}$ | $e_{2}$ | $-e_{5}$ | $e_{4}$ | $e_{7}$ | $-e_{6}$ |
| $e_{i}$ | $e_{10}$ | $e_{10}$ | $e_{11}$ | $e_{8}$ | $-e_{9}$ | $-e_{14}$ | $-e_{15}$ | $e_{12}$ | $e_{13}$ | $-e_{2}$ | $e_{3}$ | $-e_{0}$ | $-e_{1}$ | $-e_{6}$ | $-e_{7}$ | $e_{4}$ | $e_{5}$ |
| $e_{i}$ | $e_{11}$ | $e_{11}$ | $-e_{10}$ | $e_{9}$ | $e_{8}$ | $-e_{15}$ | $e_{14}$ | $-e_{13}$ | $e_{12}$ | $-e_{3}$ | $-e_{2}$ | $e_{1}$ | $-e_{0}$ | $-e_{7}$ | $e_{6}$ | $-e_{5}$ | $e_{4}$ |
| $e_{i}$ | $e_{12}$ | $e_{12}$ | $e_{13}$ | $e_{14}$ | $e_{15}$ | $e_{8}$ | $-e_{9}$ | $-e_{10}$ | $-e_{11}$ | $-e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{0}$ | $-e_{1}$ | $-e_{2}$ | $-e_{3}$ |
| $e_{i}$ | $e_{13}$ | $e_{13}$ | $-e_{12}$ | $e_{15}$ | $-e_{14}$ | $e_{9}$ | $e_{8}$ | $e_{11}$ | $-e_{10}$ | $-e_{5}$ | $-e_{4}$ | $e_{7}$ | $-e_{6}$ | $e_{1}$ | $-e_{0}$ | $e_{3}$ | $-e_{2}$ |
| $e_{i}$ | $e_{14}$ | $e_{14}$ | $-e_{15}$ | $-e_{12}$ | $e_{13}$ | $e_{10}$ | $-e_{11}$ | $e_{8}$ | $e_{9}$ | $-e_{6}$ | $-e_{7}$ | $-e_{4}$ | $e_{5}$ | $e_{2}$ | $-e_{3}$ | $-e_{0}$ | $e_{1}$ |
| $e_{i}$ | $e_{15}$ | $e_{15}$ | $e_{14}$ | $-e_{13}$ | $-e_{12}$ | $e_{11}$ | $e_{10}$ | $-e_{9}$ | $e_{8}$ | $-e_{7}$ | $e_{6}$ | $-e_{5}$ | $-e_{4}$ | $e_{3}$ | $e_{2}$ | $-e_{1}$ | $-e_{0}$ |
| $e_{i}$ | $e_{16}$ | $e_{16}$ | $-e_{17}$ | $-e_{18}$ | $-e_{19}$ | $-e_{20}$ | $-e_{21}$ | $-e_{22}$ | $-e_{23}$ | $-e_{24}$ | $-e_{25}$ | $-e_{26}$ | $-e_{27}$ | $-e_{28}$ | $-e_{29}$ | $-e_{30}$ | $-e_{31}$ |
| $e_{i}$ | $e_{17}$ | $e_{17}$ | $e_{16}$ | $-e_{19}$ | $e_{18}$ | $-e_{21}$ | $e_{20}$ | $e_{23}$ | $-e_{22}$ | $-e_{25}$ | $e_{24}$ | $e_{27}$ | $-e_{26}$ | $e_{29}$ | $-e_{28}$ | $-e_{31}$ | $e_{30}$ |
| $e_{i}$ | $e_{18}$ | $e_{18}$ | $e_{19}$ | $e_{16}$ | $-e_{17}$ | $-e_{22}$ | $-e_{23}$ | $e_{20}$ | $e_{21}$ | $-e_{26}$ | $-e_{27}$ | $e_{24}$ | $e_{25}$ | $e_{30}$ | $e_{31}$ | $-e_{28}$ | $-e_{29}$ |
| $e_{i}$ | $e_{19}$ | $e_{19}$ | $-e_{18}$ | $e_{17}$ | $e_{16}$ | $-e_{23}$ | $e_{22}$ | $-e_{21}$ | $e_{20}$ | $-e_{27}$ | $e_{26}$ | $-e_{25}$ | $e_{24}$ | $e_{31}$ | $-e_{30}$ | $e_{29}$ | $-e_{28}$ |
| $e_{i}$ | $e_{20}$ | $e_{20}$ | $e_{21}$ | $e_{22}$ | $e_{23}$ | $e_{16}$ | $-e_{17}$ | $-e_{18}$ | $-e_{19}$ | $-e_{28}$ | $-e_{29}$ | $-e_{30}$ | $-e_{31}$ | $e_{24}$ | $e_{25}$ | $e_{26}$ | $e_{27}$ |
| $e_{i}$ | $e_{21}$ | $e_{21}$ | $-e_{20}$ | $e_{23}$ | $-e_{22}$ | $e_{17}$ | $e_{16}$ | $e_{19}$ | $-e_{18}$ | $-e_{29}$ | $e_{28}$ | $-e_{31}$ | $e_{30}$ | $-e_{25}$ | $e_{24}$ | $-e_{27}$ | $e_{26}$ |
| $e_{i}$ | $e_{22}$ | $e_{22}$ | $-e_{23}$ | $-e_{20}$ | $e_{21}$ | $e_{18}$ | $-e_{19}$ | $e_{16}$ | $e_{17}$ | $-e_{30}$ | $e_{31}$ | $e_{28}$ | $-e_{29}$ | $-e_{26}$ | $e_{27}$ | $e_{24}$ | $-e_{25}$ |
| $e_{i}$ | $e_{23}$ | $e_{23}$ | $e_{22}$ | $-e_{21}$ | $-e_{20}$ | $e_{19}$ | $e_{18}$ | $-e_{17}$ | $e_{16}$ | $-e_{31}$ | $-e_{30}$ | $e_{29}$ | $e_{28}$ | $-e_{27}$ | $-e_{26}$ | $e_{25}$ | $e_{24}$ |
| $e_{i}$ | $e_{24}$ | $e_{24}$ | $e_{25}$ | $e_{26}$ | $e_{27}$ | $e_{28}$ | $e_{29}$ | $e_{30}$ | $e_{31}$ | $e_{16}$ | $-e_{17}$ | $-e_{18}$ | $-e_{19}$ | $-e_{20}$ | $-e_{21}$ | $-e_{22}$ | $-e_{23}$ |
| $e_{i}$ | $e_{25}$ | $e_{25}$ | $-e_{24}$ | $e_{27}$ | $-e_{26}$ | $e_{29}$ | $-e_{28}$ | $-e_{31}$ | $e_{30}$ | $e_{17}$ | $e_{16}$ | $e_{19}$ | $-e_{18}$ | $e_{21}$ | $-e_{20}$ | $-e_{23}$ | $e_{22}$ |
| $e_{i}$ | $e_{26}$ | $e_{26}$ | $-e_{27}$ | $-e_{24}$ | $e_{25}$ | $e_{30}$ | $e_{31}$ | $-e_{28}$ | $-e_{29}$ | $e_{18}$ | $-e_{19}$ | $e_{16}$ | $e_{17}$ | $e_{22}$ | $e_{23}$ | $-e_{20}$ | $-e_{21}$ |
| $e_{i}$ | $e_{27}$ | $e_{27}$ | $e_{26}$ | $-e_{25}$ | $-e_{24}$ | $e_{31}$ | $-e_{30}$ | $e_{29}$ | $-e_{28}$ | $e_{19}$ | $e_{18}$ | $-e_{17}$ | $e_{16}$ | $e_{23}$ | $-e_{22}$ | $e_{21}$ | $-e_{20}$ |
| $e_{i}$ | $e_{28}$ | $e_{28}$ | $-e_{29}$ | $-e_{30}$ | $-e_{31}$ | $-e_{24}$ | $e_{25}$ | $e_{26}$ | $e_{27}$ | $e_{20}$ | $-e_{21}$ | $-e_{22}$ | $-e_{23}$ | $e_{16}$ | $e_{17}$ | $e_{18}$ | $e_{19}$ |
| $e_{i}$ | $e_{29}$ | $e_{29}$ | $e_{28}$ | $-e_{31}$ | $e_{30}$ | $-e_{25}$ | $-e_{24}$ | $-e_{27}$ | $e_{26}$ | $e_{21}$ | $e_{20}$ | $-e_{23}$ | $e_{22}$ | $-e_{17}$ | $e_{16}$ | $-e_{19}$ | $e_{18}$ |
| $e_{i}$ | $e_{30}$ | $e_{30}$ | $e_{31}$ | $e_{28}$ | $-e_{29}$ | $-e_{26}$ | $e_{27}$ | $-e_{24}$ | $-e_{25}$ | $e_{22}$ | $e_{23}$ | $e_{20}$ | $-e_{21}$ | $-e_{18}$ | $e_{19}$ | $e_{16}$ | $-e_{17}$ |
| $e_{i}$ | $e_{31}$ | $e_{31}$ | $-e_{30}$ | $e_{29}$ | $e_{28}$ | $-e_{27}$ | $-e_{26}$ | $e_{25}$ | $-e_{24}$ | $e_{23}$ | $-e_{22}$ | $e_{21}$ | $e_{20}$ | $-e_{19}$ | $-e_{18}$ | $e_{17}$ | $e_{16}$ |
Below is the trigintaduonion multiplication table for $e_{j},16\leq j\leq 31$.
| $e_{i}e_{j}$ | $e_{i}e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ | $e_{j}$ |
| $e_{i}e_{j}$ | $e_{i}e_{j}$ | $e_{16}$ | $e_{17}$ | $e_{18}$ | $e_{19}$ | $e_{20}$ | $e_{21}$ | $e_{22}$ | $e_{23}$ | $e_{24}$ | $e_{25}$ | $e_{26}$ | $e_{27}$ | $e_{28}$ | $e_{29}$ | $e_{30}$ | $e_{31}$ |
| $e_{i}$ | $e_{0}$ | $e_{16}$ | $e_{17}$ | $e_{18}$ | $e_{19}$ | $e_{20}$ | $e_{21}$ | $e_{22}$ | $e_{23}$ | $e_{24}$ | $e_{25}$ | $e_{26}$ | $e_{27}$ | $e_{28}$ | $e_{29}$ | $e_{30}$ | $e_{31}$ |
| $e_{i}$ | $e_{1}$ | $e_{17}$ | $-e_{16}$ | $-e_{19}$ | $e_{18}$ | $-e_{21}$ | $e_{20}$ | $e_{23}$ | $-e_{22}$ | $-e_{25}$ | $e_{24}$ | $e_{27}$ | $-e_{26}$ | $e_{29}$ | $-e_{28}$ | $-e_{31}$ | $e_{30}$ |
| $e_{i}$ | $e_{2}$ | $e_{18}$ | $e_{19}$ | $-e_{16}$ | $-e_{17}$ | $-e_{22}$ | $-e_{23}$ | $e_{20}$ | $e_{21}$ | $-e_{26}$ | $-e_{27}$ | $e_{24}$ | $e_{25}$ | $e_{30}$ | $e_{31}$ | $-e_{28}$ | $-e_{29}$ |
| $e_{i}$ | $e_{3}$ | $e_{19}$ | $-e_{18}$ | $e_{17}$ | $-e_{16}$ | $-e_{23}$ | $e_{22}$ | $-e_{21}$ | $e_{20}$ | $-e_{27}$ | $e_{26}$ | $-e_{25}$ | $e_{24}$ | $e_{31}$ | $-e_{30}$ | $e_{29}$ | $-e_{28}$ |
| $e_{i}$ | $e_{4}$ | $e_{20}$ | $e_{21}$ | $e_{22}$ | $e_{23}$ | $-e_{16}$ | $-e_{17}$ | $-e_{18}$ | $-e_{19}$ | $-e_{28}$ | $-e_{29}$ | $-e_{30}$ | $-e_{31}$ | $e_{24}$ | $e_{25}$ | $e_{26}$ | $e_{27}$ |
| $e_{i}$ | $e_{5}$ | $e_{21}$ | $-e_{20}$ | $e_{23}$ | $-e_{22}$ | $e_{17}$ | $-e_{16}$ | $e_{19}$ | $-e_{18}$ | $-e_{29}$ | $e_{28}$ | $-e_{31}$ | $e_{30}$ | $-e_{25}$ | $e_{24}$ | $-e_{27}$ | $e_{26}$ |
| $e_{i}$ | $e_{6}$ | $e_{22}$ | $-e_{23}$ | $-e_{20}$ | $e_{21}$ | $e_{18}$ | $-e_{19}$ | $-e_{16}$ | $e_{17}$ | $-e_{30}$ | $e_{31}$ | $e_{28}$ | $-e_{29}$ | $-e_{26}$ | $e_{27}$ | $e_{24}$ | $-e_{25}$ |
| $e_{i}$ | $e_{7}$ | $e_{23}$ | $e_{22}$ | $-e_{21}$ | $-e_{20}$ | $e_{19}$ | $e_{18}$ | $-e_{17}$ | $-e_{16}$ | $-e_{31}$ | $-e_{30}$ | $e_{29}$ | $e_{28}$ | $-e_{27}$ | $-e_{26}$ | $e_{25}$ | $e_{24}$ |
| $e_{i}$ | $e_{8}$ | $e_{24}$ | $e_{25}$ | $e_{26}$ | $e_{27}$ | $e_{28}$ | $e_{29}$ | $e_{30}$ | $e_{31}$ | $-e_{16}$ | $-e_{17}$ | $-e_{18}$ | $-e_{19}$ | $-e_{20}$ | $-e_{21}$ | $-e_{22}$ | $-e_{23}$ |
| $e_{i}$ | $e_{9}$ | $e_{25}$ | $-e_{24}$ | $e_{27}$ | $-e_{26}$ | $e_{29}$ | $-e_{28}$ | $-e_{31}$ | $e_{30}$ | $e_{17}$ | $-e_{16}$ | $e_{19}$ | $-e_{18}$ | $e_{21}$ | $-e_{20}$ | $-e_{23}$ | $e_{22}$ |
| $e_{i}$ | $e_{10}$ | $e_{26}$ | $-e_{27}$ | $-e_{24}$ | $e_{25}$ | $e_{30}$ | $e_{31}$ | $-e_{28}$ | $-e_{29}$ | $e_{18}$ | $-e_{19}$ | $-e_{16}$ | $e_{17}$ | $e_{22}$ | $e_{23}$ | $-e_{20}$ | $-e_{21}$ |
| $e_{i}$ | $e_{11}$ | $e_{27}$ | $e_{26}$ | $-e_{25}$ | $-e_{24}$ | $e_{31}$ | $-e_{30}$ | $e_{29}$ | $-e_{28}$ | $e_{19}$ | $e_{18}$ | $-e_{17}$ | $-e_{16}$ | $e_{23}$ | $-e_{22}$ | $e_{21}$ | $-e_{20}$ |
| $e_{i}$ | $e_{12}$ | $e_{28}$ | $-e_{29}$ | $-e_{30}$ | $-e_{31}$ | $-e_{24}$ | $e_{25}$ | $e_{26}$ | $e_{27}$ | $e_{20}$ | $-e_{21}$ | $-e_{22}$ | $-e_{23}$ | $-e_{16}$ | $e_{17}$ | $e_{18}$ | $e_{19}$ |
| $e_{i}$ | $e_{13}$ | $e_{29}$ | $e_{28}$ | $-e_{31}$ | $e_{30}$ | $-e_{25}$ | $-e_{24}$ | $-e_{27}$ | $e_{26}$ | $e_{21}$ | $e_{20}$ | $-e_{23}$ | $e_{22}$ | $-e_{17}$ | $-e_{16}$ | $-e_{19}$ | $e_{18}$ |
| $e_{i}$ | $e_{14}$ | $e_{30}$ | $e_{31}$ | $e_{28}$ | $-e_{29}$ | $-e_{26}$ | $e_{27}$ | $-e_{24}$ | $-e_{25}$ | $e_{22}$ | $e_{23}$ | $e_{20}$ | $-e_{21}$ | $-e_{18}$ | $e_{19}$ | $-e_{16}$ | $-e_{17}$ |
| $e_{i}$ | $e_{15}$ | $e_{31}$ | $-e_{30}$ | $e_{29}$ | $e_{28}$ | $-e_{27}$ | $-e_{26}$ | $e_{25}$ | $-e_{24}$ | $e_{23}$ | $-e_{22}$ | $e_{21}$ | $e_{20}$ | $-e_{19}$ | $-e_{18}$ | $e_{17}$ | $-e_{16}$ |
| $e_{i}$ | $e_{16}$ | $-e_{0}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$ | $e_{9}$ | $e_{10}$ | $e_{11}$ | $e_{12}$ | $e_{13}$ | $e_{14}$ | $e_{15}$ |
| $e_{i}$ | $e_{17}$ | $-e_{1}$ | $-e_{0}$ | $-e_{3}$ | $e_{2}$ | $-e_{5}$ | $e_{4}$ | $e_{7}$ | $-e_{6}$ | $-e_{9}$ | $e_{8}$ | $e_{11}$ | $-e_{10}$ | $e_{13}$ | $-e_{12}$ | $-e_{15}$ | $e_{14}$ |
| $e_{i}$ | $e_{18}$ | $-e_{2}$ | $e_{3}$ | $-e_{0}$ | $-e_{1}$ | $-e_{6}$ | $-e_{7}$ | $e_{4}$ | $e_{5}$ | $-e_{10}$ | $-e_{11}$ | $e_{8}$ | $e_{9}$ | $e_{14}$ | $e_{15}$ | $-e_{12}$ | $-e_{13}$ |
| $e_{i}$ | $e_{19}$ | $-e_{3}$ | $-e_{2}$ | $e_{1}$ | $-e_{0}$ | $-e_{7}$ | $e_{6}$ | $-e_{5}$ | $e_{4}$ | $-e_{11}$ | $e_{10}$ | $-e_{9}$ | $e_{8}$ | $e_{15}$ | $-e_{14}$ | $e_{13}$ | $-e_{12}$ |
| $e_{i}$ | $e_{20}$ | $-e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{0}$ | $-e_{1}$ | $-e_{2}$ | $-e_{3}$ | $-e_{12}$ | $-e_{13}$ | $-e_{14}$ | $-e_{15}$ | $e_{8}$ | $e_{9}$ | $e_{10}$ | $e_{11}$ |
| $e_{i}$ | $e_{21}$ | $-e_{5}$ | $-e_{4}$ | $e_{7}$ | $-e_{6}$ | $e_{1}$ | $-e_{0}$ | $e_{3}$ | $-e_{2}$ | $-e_{13}$ | $e_{12}$ | $-e_{15}$ | $e_{14}$ | $-e_{9}$ | $e_{8}$ | $-e_{11}$ | $e_{10}$ |
| $e_{i}$ | $e_{22}$ | $-e_{6}$ | $-e_{7}$ | $-e_{4}$ | $e_{5}$ | $e_{2}$ | $-e_{3}$ | $-e_{0}$ | $e_{1}$ | $-e_{14}$ | $e_{15}$ | $e_{12}$ | $-e_{13}$ | $-e_{10}$ | $e_{11}$ | $e_{8}$ | $-e_{9}$ |
| $e_{i}$ | $e_{23}$ | $-e_{7}$ | $e_{6}$ | $-e_{5}$ | $-e_{4}$ | $e_{3}$ | $e_{2}$ | $-e_{1}$ | $-e_{0}$ | $-e_{15}$ | $-e_{14}$ | $e_{13}$ | $e_{12}$ | $-e_{11}$ | $-e_{10}$ | $e_{9}$ | $e_{8}$ |
| $e_{i}$ | $e_{24}$ | $-e_{8}$ | $e_{9}$ | $e_{10}$ | $e_{11}$ | $e_{12}$ | $e_{13}$ | $e_{14}$ | $e_{15}$ | $-e_{0}$ | $-e_{1}$ | $-e_{2}$ | $-e_{3}$ | $-e_{4}$ | $-e_{5}$ | $-e_{6}$ | $-e_{7}$ |
| $e_{i}$ | $e_{25}$ | $-e_{9}$ | $-e_{8}$ | $e_{11}$ | $-e_{10}$ | $e_{13}$ | $-e_{12}$ | $-e_{15}$ | $e_{14}$ | $e_{1}$ | $-e_{0}$ | $e_{3}$ | $-e_{2}$ | $e_{5}$ | $-e_{4}$ | $-e_{7}$ | $e_{6}$ |
| $e_{i}$ | $e_{26}$ | $-e_{10}$ | $-e_{11}$ | $-e_{8}$ | $e_{9}$ | $e_{14}$ | $e_{15}$ | $-e_{12}$ | $-e_{13}$ | $e_{2}$ | $-e_{3}$ | $-e_{0}$ | $e_{1}$ | $e_{6}$ | $e_{7}$ | $-e_{4}$ | $-e_{5}$ |
| $e_{i}$ | $e_{27}$ | $-e_{11}$ | $e_{10}$ | $-e_{9}$ | $-e_{8}$ | $e_{15}$ | $-e_{14}$ | $e_{13}$ | $-e_{12}$ | $e_{3}$ | $e_{2}$ | $-e_{1}$ | $-e_{0}$ | $e_{7}$ | $-e_{6}$ | $e_{5}$ | $-e_{4}$ |
| $e_{i}$ | $e_{28}$ | $-e_{12}$ | $-e_{13}$ | $-e_{14}$ | $-e_{15}$ | $-e_{8}$ | $e_{9}$ | $e_{10}$ | $e_{11}$ | $e_{4}$ | $-e_{5}$ | $-e_{6}$ | $-e_{7}$ | $-e_{0}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ |
| $e_{i}$ | $e_{29}$ | $-e_{13}$ | $e_{12}$ | $-e_{15}$ | $e_{14}$ | $-e_{9}$ | $-e_{8}$ | $-e_{11}$ | $e_{10}$ | $e_{5}$ | $e_{4}$ | $-e_{7}$ | $e_{6}$ | $-e_{1}$ | $-e_{0}$ | $-e_{3}$ | $e_{2}$ |
| $e_{i}$ | $e_{30}$ | $-e_{14}$ | $e_{15}$ | $e_{12}$ | $-e_{13}$ | $-e_{10}$ | $e_{11}$ | $-e_{8}$ | $-e_{9}$ | $e_{6}$ | $e_{7}$ | $e_{4}$ | $-e_{5}$ | $-e_{2}$ | $e_{3}$ | $-e_{0}$ | $-e_{1}$ |
| $e_{i}$ | $e_{31}$ | $-e_{15}$ | $-e_{14}$ | $e_{13}$ | $e_{12}$ | $-e_{11}$ | $-e_{10}$ | $e_{9}$ | $-e_{8}$ | $e_{7}$ | $-e_{6}$ | $e_{5}$ | $e_{4}$ | $-e_{3}$ | $-e_{2}$ | $e_{1}$ | $-e_{0}$ |

### 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 (See 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}

### Computational algorithms

The first computational algorithm for the multiplication of trigintaduonions was developed by Cariow & Cariowa (2014).

## Applications

The trigintaduonions have applications in particle physics, quantum physics, and other branches of modern physics. More recently, the trigintaduonions and other hypercomplex numbers have also been used in neural network research and cryptography.

## Further algebras

Robert de Marrais's terms for the algebras immediately following the sedenions are the pathions (i.e. trigintaduonions), chingons, routons, and voudons. They are summarized as follows.
| Name | Dimension | Symbol | Etymology | Other names |
| Pathions | 32 = 2$^5$ | $\mathbb {P}$, ${\mathcal {C}}_{5}$ | 32 paths of wisdom of Kabbalah, from the Sefer Yetzirah | Trigintaduonions ($\mathbb {T}$), 32-nions |
| Chingons | 64 = 2$^6$ | $\mathbb {X}$, ${\mathcal {C}}_{6}$ | 64 hexagrams of the I Ching | Sexagintaquatronions, 64-nions |
| Routons | 128 = 2$^7$ | $\mathbb {U}$, ${\mathcal {C}}_{7}$ | Massachusetts Route 128, of the "Massachusetts Miracle" | Centumduodetrigintanions, 128-nions |
| Voudons | 256 = 2$^8$ | $\mathbb {V}$, ${\mathcal {C}}_{8}$ | 256 deities of the Ifá pantheon of Voodoo or Voudon | Ducentiquinquagintasexions, 256-nions |