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Content in this edit is translated from the existing Chinese Wikipedia article at [[:zh:三十二元數]]; see its history for attribution.
{{Translated|zh|三十二元數}}
In abstract algebra, the trigintaduonions, also known as the 32-ions, 32-nions, 25-ions, or sometimes pathions ( P {\displaystyle \mathbb {P} } ),[1][2] form a 32-dimensional noncommutative and nonassociative algebra over the real numbers,[3][4] usually represented by the capital letter T, boldface T or blackboard bold T {\displaystyle \mathbb {T} } .[5]
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 P {\displaystyle \mathbb {P} } .[1] Other alternative names include 32-ion, 32-nion, and 25-ion.
Every trigintaduonion is a linear combination of the unit trigintaduonions e 0 {\displaystyle e_{0}} , e 1 {\displaystyle e_{1}} , e 2 {\displaystyle e_{2}} , e 3 {\displaystyle e_{3}} , ..., e 31 {\displaystyle e_{31}} , which form a basis of the vector space of trigintaduonions. Every trigintaduonion can be represented in the form
with real coefficients xi.
The trigintaduonions can be obtained by applying the Cayley–Dickson construction to the sedenions, which can be mathematically expressed as T = C D ( S , 1 ) {\displaystyle \mathbb {T} ={\mathcal {CD}}(\mathbb {S} ,1)} .[6] Applying the Cayley–Dickson construction to the sedenions yields a 64-dimensional algebra called the 64-ions, 64-nions, sexagintaquatronions, or sexagintaquattuornions, sometimes also known as the chingons.[7][8][9]
As a result, the trigintaduonions can also be defined as the following.[6]
An algebra of dimension 4 over the octonions O {\displaystyle \mathbb {O} } :
An algebra of dimension 8 over quaternions H {\displaystyle \mathbb {H} } :
An algebra of dimension 16 over the complex numbers C {\displaystyle \mathbb {C} } :
An algebra of dimension 32 over the real numbers R {\displaystyle \mathbb {R} } :
R , C , H , R , O , S {\displaystyle \mathbb {R} ,\mathbb {C} ,\mathbb {H} ,\mathbb {R} ,\mathbb {O} ,\mathbb {S} } are all subsets of T {\displaystyle \mathbb {T} } . This relation can be expressed as:
Like octonions and sedenions, multiplication of trigintaduonions is neither commutative nor associative. As with the sedenions, the trigintaduonions contain zero divisors and are thus not a division algebra.
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):[10]
Given a 2 n {\displaystyle 2^{n}} -dimensional Cayley–Dickson algebra, where 3 ≤ n ≤ 6 {\displaystyle 3\leq n\leq 6} , we first observe that the multiplication table of its imaginary units e a , 1 ≤ a ≤ 2 n − 1 {\displaystyle e_{a},1\leq a\leq 2^{n}-1} , is encoded in the properties of the projective space P G ( n − 1 , 2 ) {\displaystyle PG(n-1,2)} if these imaginary units are regarded as points and distinguished triads of them { e a , e b , e c } , 1 ≤ a < b < c ≤ 2 n − 1 {\displaystyle \{e_{a},e_{b},e_{c}\},1\leq a<b<c\leq 2^{n}-1} and e a ⋅ e b = ± e c {\displaystyle 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 {\displaystyle a+b=c} or a + b ≠ c {\displaystyle a+b\neq c} .
The multiplication of the unit trigintaduonions is illustrated in the two tables below. Combined, they form a single 32×32 table with 1024 cells.[11][6]
Below is the trigintaduonion multiplication table for e j , 0 ≤ j ≤ 15 {\displaystyle e_{j},0\leq j\leq 15} . The top half of this table corresponds to the multiplication table for the sedenions.
Below is the trigintaduonion multiplication table for e j , 16 ≤ j ≤ 31 {\displaystyle e_{j},16\leq j\leq 31} .
The first computational algorithm for the multiplication of trigintaduonions was developed by Cariow & Cariowa in 2014.[12]
The trigintaduonions have applications in particle physics,[13] quantum mechanics, and other branches of modern physics.[11] More recently, the trigintaduonions and other hypercomplex numbers have also been used in neural network research[14] and cryptography.
Robert de Marrais' terms for the algebras immediately following the sedenions are the pathions (i.e. trigintaduonions), chingons, routons, and voudons.[9][15] They are summarized as follows.[1]