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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, or sometimes pathions ( P {\displaystyle \mathbb {P} } ),[1][2][3] form a 32-dimensional algebra over the real numbers,[4] usually represented by the capital letter T, boldface T or blackboard bold T {\displaystyle \mathbb {T} } .[5]
The trigintaduonions are obtained by applying the Cayley–Dickson construction to the sedenions. Applying the Cayley–Dickson construction to the sedenions yields a 64-dimensional algebra called the 64-ions or sexagintaquattuornions, sometimes also known as the chingons.[6][7][8]
Whereas octonion unit multiplication can be visually represented by PG(2,2) (also known as the Fano plane) and sedenion unit multiplication by PG(3,2), trigintaduonion unit multiplication can be visually represented by PG(4,2).[9]
The multiplication of the unit trigintaduonions is illustrated in the two tables below.[10]
Below is the trigintaduonion multiplication table for e j , j = {\displaystyle e_{j},j=} 0 to 15. The top half of this table corresponds to the multiplication table for the sedenions.
Below is the trigintaduonion multiplication table for e j , j = {\displaystyle e_{j},j=} 16 to 31.
The first computational algorithm for the multiplication of trigintaduonions was developed by Cariow & Cariowa in 2014.[11]
The trigintaduonions have applications in particle physics,[12] quantum mechanics, and other branches of modern physics.[10] More recently, the trigintaduonions and other hypercomplex numbers have also been used in neural network research.[13]
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