Trigintaduonion

In abstract algebra, the trigintaduonions, also known as the 32-ions, 32-nions, or 2⁵-nions, are a special kind of mathematical structure. They exist in a space that has 32 different directions, which mathematicians call dimensions. These trigintaduonions are interesting because their math does not follow the usual rules that we use for everyday numbers. For example, when you multiply these numbers, the order in which you multiply them matters, unlike regular multiplication where 3 × 4 is the same as 4 × 3. This makes them a noncommutative algebra. Additionally, they also do not follow the associative rule, meaning that (a × b) × c can give a different result than a × (b × c). These special numbers are studied over the real numbers, which include all the numbers we use in everyday life, like whole numbers, decimals, and fractions.

Names

The word trigintaduonion comes from Latin words meaning "thirty" and "two," plus a special ending used for certain number systems. Other names for this concept include 32-ion, 32-nion, 2⁵-ion, and 2⁵-nion.

These names all refer to the same idea in a special area of math.

Definition

Every trigintaduonion is made by combining special units called e₀, e₁, e₂, and so on up to e₃₁. These units act like building blocks for trigintaduonions.

Trigintaduonions can also be created from another math system called sedenions. When this process is done again, it makes an even bigger system with 64 parts.

Multiplication

Like octonions and sedenions, multiplying trigintaduonions is not straightforward in the same way as regular numbers. However, it follows a special pattern called power associativity, meaning that for any number in this system, raising it to different powers is well-defined.

The trigintaduonions also include special numbers called zero divisors, which means they do not behave like regular numbers in all ways. There are many ways to group these special numbers into triples, with different patterns of grouping.

e i e j {\displaystyle e_{i}e_{j}}		e j {\displaystyle e_{j}}
		e 0 {\displaystyle e_{0}}	e 1 {\displaystyle e_{1}}	e 2 {\displaystyle e_{2}}	e 3 {\displaystyle e_{3}}	e 4 {\displaystyle e_{4}}	e 5 {\displaystyle e_{5}}	e 6 {\displaystyle e_{6}}	e 7 {\displaystyle e_{7}}	e 8 {\displaystyle e_{8}}	e 9 {\displaystyle e_{9}}	e 10 {\displaystyle e_{10}}	e 11 {\displaystyle e_{11}}	e 12 {\displaystyle e_{12}}	e 13 {\displaystyle e_{13}}	e 14 {\displaystyle e_{14}}	e 15 {\displaystyle e_{15}}
e i {\displaystyle e_{i}}	e 0 {\displaystyle e_{0}}	e 0 {\displaystyle e_{0}}	e 1 {\displaystyle e_{1}}	e 2 {\displaystyle e_{2}}	e 3 {\displaystyle e_{3}}	e 4 {\displaystyle e_{4}}	e 5 {\displaystyle e_{5}}	e 6 {\displaystyle e_{6}}	e 7 {\displaystyle e_{7}}	e 8 {\displaystyle e_{8}}	e 9 {\displaystyle e_{9}}	e 10 {\displaystyle e_{10}}	e 11 {\displaystyle e_{11}}	e 12 {\displaystyle e_{12}}	e 13 {\displaystyle e_{13}}	e 14 {\displaystyle e_{14}}	e 15 {\displaystyle e_{15}}
	e 1 {\displaystyle e_{1}}	e 1 {\displaystyle e_{1}}	− e 0 {\displaystyle -e_{0}}	e 3 {\displaystyle e_{3}}	− e 2 {\displaystyle -e_{2}}	e 5 {\displaystyle e_{5}}	− e 4 {\displaystyle -e_{4}}	− e 7 {\displaystyle -e_{7}}	e 6 {\displaystyle e_{6}}	e 9 {\displaystyle e_{9}}	− e 8 {\displaystyle -e_{8}}	− e 11 {\displaystyle -e_{11}}	e 10 {\displaystyle e_{10}}	− e 13 {\displaystyle -e_{13}}	e 12 {\displaystyle e_{12}}	e 15 {\displaystyle e_{15}}	− e 14 {\displaystyle -e_{14}}
	e 2 {\displaystyle e_{2}}	e 2 {\displaystyle e_{2}}	− e 3 {\displaystyle -e_{3}}	− e 0 {\displaystyle -e_{0}}	e 1 {\displaystyle e_{1}}	e 6 {\displaystyle e_{6}}	e 7 {\displaystyle e_{7}}	− e 4 {\displaystyle -e_{4}}	− e 5 {\displaystyle -e_{5}}	e 10 {\displaystyle e_{10}}	e 11 {\displaystyle e_{11}}	− e 8 {\displaystyle -e_{8}}	− e 9 {\displaystyle -e_{9}}	− e 14 {\displaystyle -e_{14}}	− e 15 {\displaystyle -e_{15}}	e 12 {\displaystyle e_{12}}	e 13 {\displaystyle e_{13}}
	e 3 {\displaystyle e_{3}}	e 3 {\displaystyle e_{3}}	e 2 {\displaystyle e_{2}}	− e 1 {\displaystyle -e_{1}}	− e 0 {\displaystyle -e_{0}}	e 7 {\displaystyle e_{7}}	− e 6 {\displaystyle -e_{6}}	e 5 {\displaystyle e_{5}}	− e 4 {\displaystyle -e_{4}}	e 11 {\displaystyle e_{11}}	− e 10 {\displaystyle -e_{10}}	e 9 {\displaystyle e_{9}}	− e 8 {\displaystyle -e_{8}}	− e 15 {\displaystyle -e_{15}}	e 14 {\displaystyle e_{14}}	− e 13 {\displaystyle -e_{13}}	e 12 {\displaystyle e_{12}}
	e 4 {\displaystyle e_{4}}	e 4 {\displaystyle e_{4}}	− e 5 {\displaystyle -e_{5}}	− e 6 {\displaystyle -e_{6}}	− e 7 {\displaystyle -e_{7}}	− e 0 {\displaystyle -e_{0}}	e 1 {\displaystyle e_{1}}	e 2 {\displaystyle e_{2}}	e 3 {\displaystyle e_{3}}	e 12 {\displaystyle e_{12}}	e 13 {\displaystyle e_{13}}	e 14 {\displaystyle e_{14}}	e 15 {\displaystyle e_{15}}	− e 8 {\displaystyle -e_{8}}	− e 9 {\displaystyle -e_{9}}	− e 10 {\displaystyle -e_{10}}	− e 11 {\displaystyle -e_{11}}
	e 5 {\displaystyle e_{5}}	e 5 {\displaystyle e_{5}}	e 4 {\displaystyle e_{4}}	− e 7 {\displaystyle -e_{7}}	e 6 {\displaystyle e_{6}}	− e 1 {\displaystyle -e_{1}}	− e 0 {\displaystyle -e_{0}}	− e 3 {\displaystyle -e_{3}}	e 2 {\displaystyle e_{2}}	e 13 {\displaystyle e_{13}}	− e 12 {\displaystyle -e_{12}}	e 15 {\displaystyle e_{15}}	− e 14 {\displaystyle -e_{14}}	e 9 {\displaystyle e_{9}}	− e 8 {\displaystyle -e_{8}}	e 11 {\displaystyle e_{11}}	− e 10 {\displaystyle -e_{10}}
	e 6 {\displaystyle e_{6}}	e 6 {\displaystyle e_{6}}	e 7 {\displaystyle e_{7}}	e 4 {\displaystyle e_{4}}	− e 5 {\displaystyle -e_{5}}	− e 2 {\displaystyle -e_{2}}	e 3 {\displaystyle e_{3}}	− e 0 {\displaystyle -e_{0}}	− e 1 {\displaystyle -e_{1}}	e 14 {\displaystyle e_{14}}	− e 15 {\displaystyle -e_{15}}	− e 12 {\displaystyle -e_{12}}	e 13 {\displaystyle e_{13}}	e 10 {\displaystyle e_{10}}	− e 11 {\displaystyle -e_{11}}	− e 8 {\displaystyle -e_{8}}	e 9 {\displaystyle e_{9}}
	e 7 {\displaystyle e_{7}}	e 7 {\displaystyle e_{7}}	− e 6 {\displaystyle -e_{6}}	e 5 {\displaystyle e_{5}}	e 4 {\displaystyle e_{4}}	− e 3 {\displaystyle -e_{3}}	− e 2 {\displaystyle -e_{2}}	e 1 {\displaystyle e_{1}}	− e 0 {\displaystyle -e_{0}}	e 15 {\displaystyle e_{15}}	e 14 {\displaystyle e_{14}}	− e 13 {\displaystyle -e_{13}}	− e 12 {\displaystyle -e_{12}}	e 11 {\displaystyle e_{11}}	e 10 {\displaystyle e_{10}}	− e 9 {\displaystyle -e_{9}}	− e 8 {\displaystyle -e_{8}}
	e 8 {\displaystyle e_{8}}	e 8 {\displaystyle e_{8}}	− e 9 {\displaystyle -e_{9}}	− e 10 {\displaystyle -e_{10}}	− e 11 {\displaystyle -e_{11}}	− e 12 {\displaystyle -e_{12}}	− e 13 {\displaystyle -e_{13}}	− e 14 {\displaystyle -e_{14}}	− e 15 {\displaystyle -e_{15}}	− e 0 {\displaystyle -e_{0}}	e 1 {\displaystyle e_{1}}	e 2 {\displaystyle e_{2}}	e 3 {\displaystyle e_{3}}	e 4 {\displaystyle e_{4}}	e 5 {\displaystyle e_{5}}	e 6 {\displaystyle e_{6}}	e 7 {\displaystyle e_{7}}
	e 9 {\displaystyle e_{9}}	e 9 {\displaystyle e_{9}}	e 8 {\displaystyle e_{8}}	− e 11 {\displaystyle -e_{11}}	e 10 {\displaystyle e_{10}}	− e 13 {\displaystyle -e_{13}}	e 12 {\displaystyle e_{12}}	e 15 {\displaystyle e_{15}}	− e 14 {\displaystyle -e_{14}}	− e 1 {\displaystyle -e_{1}}	− e 0 {\displaystyle -e_{0}}	− e 3 {\displaystyle -e_{3}}	e 2 {\displaystyle e_{2}}	− e 5 {\displaystyle -e_{5}}	e 4 {\displaystyle e_{4}}	e 7 {\displaystyle e_{7}}	− e 6 {\displaystyle -e_{6}}
	e 10 {\displaystyle e_{10}}	e 10 {\displaystyle e_{10}}	e 11 {\displaystyle e_{11}}	e 8 {\displaystyle e_{8}}	− e 9 {\displaystyle -e_{9}}	− e 14 {\displaystyle -e_{14}}	− e 15 {\displaystyle -e_{15}}	e 12 {\displaystyle e_{12}}	e 13 {\displaystyle e_{13}}	− e 2 {\displaystyle -e_{2}}	e 3 {\displaystyle e_{3}}	− e 0 {\displaystyle -e_{0}}	− e 1 {\displaystyle -e_{1}}	− e 6 {\displaystyle -e_{6}}	− e 7 {\displaystyle -e_{7}}	e 4 {\displaystyle e_{4}}	e 5 {\displaystyle e_{5}}
	e 11 {\displaystyle e_{11}}	e 11 {\displaystyle e_{11}}	− e 10 {\displaystyle -e_{10}}	e 9 {\displaystyle e_{9}}	e 8 {\displaystyle e_{8}}	− e 15 {\displaystyle -e_{15}}	e 14 {\displaystyle e_{14}}	− e 13 {\displaystyle -e_{13}}	e 12 {\displaystyle e_{12}}	− e 3 {\displaystyle -e_{3}}	− e 2 {\displaystyle -e_{2}}	e 1 {\displaystyle e_{1}}	− e 0 {\displaystyle -e_{0}}	− e 7 {\displaystyle -e_{7}}	e 6 {\displaystyle e_{6}}	− e 5 {\displaystyle -e_{5}}	e 4 {\displaystyle e_{4}}
	e 12 {\displaystyle e_{12}}	e 12 {\displaystyle e_{12}}	e 13 {\displaystyle e_{13}}	e 14 {\displaystyle e_{14}}	e 15 {\displaystyle e_{15}}	e 8 {\displaystyle e_{8}}	− e 9 {\displaystyle -e_{9}}	− e 10 {\displaystyle -e_{10}}	− e 11 {\displaystyle -e_{11}}	− e 4 {\displaystyle -e_{4}}	e 5 {\displaystyle e_{5}}	e 6 {\displaystyle e_{6}}	e 7 {\displaystyle e_{7}}	− e 0 {\displaystyle -e_{0}}	− e 1 {\displaystyle -e_{1}}	− e 2 {\displaystyle -e_{2}}	− e 3 {\displaystyle -e_{3}}
	e 13 {\displaystyle e_{13}}	e 13 {\displaystyle e_{13}}	− e 12 {\displaystyle -e_{12}}	e 15 {\displaystyle e_{15}}	− e 14 {\displaystyle -e_{14}}	e 9 {\displaystyle e_{9}}	e 8 {\displaystyle e_{8}}	e 11 {\displaystyle e_{11}}	− e 10 {\displaystyle -e_{10}}	− e 5 {\displaystyle -e_{5}}	− e 4 {\displaystyle -e_{4}}	e 7 {\displaystyle e_{7}}	− e 6 {\displaystyle -e_{6}}	e 1 {\displaystyle e_{1}}	− e 0 {\displaystyle -e_{0}}	e 3 {\displaystyle e_{3}}	− e 2 {\displaystyle -e_{2}}
	e 14 {\displaystyle e_{14}}	e 14 {\displaystyle e_{14}}	− e 15 {\displaystyle -e_{15}}	− e 12 {\displaystyle -e_{12}}	e 13 {\displaystyle e_{13}}	e 10 {\displaystyle e_{10}}	− e 11 {\displaystyle -e_{11}}	e 8 {\displaystyle e_{8}}	e 9 {\displaystyle e_{9}}	− e 6 {\displaystyle -e_{6}}	− e 7 {\displaystyle -e_{7}}	− e 4 {\displaystyle -e_{4}}	e 5 {\displaystyle e_{5}}	e 2 {\displaystyle e_{2}}	− e 3 {\displaystyle -e_{3}}	− e 0 {\displaystyle -e_{0}}	e 1 {\displaystyle e_{1}}
	e 15 {\displaystyle e_{15}}	e 15 {\displaystyle e_{15}}	e 14 {\displaystyle e_{14}}	− e 13 {\displaystyle -e_{13}}	− e 12 {\displaystyle -e_{12}}	e 11 {\displaystyle e_{11}}	e 10 {\displaystyle e_{10}}	− e 9 {\displaystyle -e_{9}}	e 8 {\displaystyle e_{8}}	− e 7 {\displaystyle -e_{7}}	e 6 {\displaystyle e_{6}}	− e 5 {\displaystyle -e_{5}}	− e 4 {\displaystyle -e_{4}}	e 3 {\displaystyle e_{3}}	e 2 {\displaystyle e_{2}}	− e 1 {\displaystyle -e_{1}}	− e 0 {\displaystyle -e_{0}}
	e 16 {\displaystyle e_{16}}	e 16 {\displaystyle e_{16}}	− e 17 {\displaystyle -e_{17}}	− e 18 {\displaystyle -e_{18}}	− e 19 {\displaystyle -e_{19}}	− e 20 {\displaystyle -e_{20}}	− e 21 {\displaystyle -e_{21}}	− e 22 {\displaystyle -e_{22}}	− e 23 {\displaystyle -e_{23}}	− e 24 {\displaystyle -e_{24}}	− e 25 {\displaystyle -e_{25}}	− e 26 {\displaystyle -e_{26}}	− e 27 {\displaystyle -e_{27}}	− e 28 {\displaystyle -e_{28}}	− e 29 {\displaystyle -e_{29}}	− e 30 {\displaystyle -e_{30}}	− e 31 {\displaystyle -e_{31}}
	e 17 {\displaystyle e_{17}}	e 17 {\displaystyle e_{17}}	e 16 {\displaystyle e_{16}}	− e 19 {\displaystyle -e_{19}}	e 18 {\displaystyle e_{18}}	− e 21 {\displaystyle -e_{21}}	e 20 {\displaystyle e_{20}}	e 23 {\displaystyle e_{23}}	− e 22 {\displaystyle -e_{22}}	− e 25 {\displaystyle -e_{25}}	e 24 {\displaystyle e_{24}}	e 27 {\displaystyle e_{27}}	− e 26 {\displaystyle -e_{26}}	e 29 {\displaystyle e_{29}}	− e 28 {\displaystyle -e_{28}}	− e 31 {\displaystyle -e_{31}}	e 30 {\displaystyle e_{30}}
	e 18 {\displaystyle e_{18}}	e 18 {\displaystyle e_{18}}	e 19 {\displaystyle e_{19}}	e 16 {\displaystyle e_{16}}	− e 17 {\displaystyle -e_{17}}	− e 22 {\displaystyle -e_{22}}	− e 23 {\displaystyle -e_{23}}	e 20 {\displaystyle e_{20}}	e 21 {\displaystyle e_{21}}	− e 26 {\displaystyle -e_{26}}	− e 27 {\displaystyle -e_{27}}	e 24 {\displaystyle e_{24}}	e 25 {\displaystyle e_{25}}	e 30 {\displaystyle e_{30}}	e 31 {\displaystyle e_{31}}	− e 28 {\displaystyle -e_{28}}	− e 29 {\displaystyle -e_{29}}
	e 19 {\displaystyle e_{19}}	e 19 {\displaystyle e_{19}}	− e 18 {\displaystyle -e_{18}}	e 17 {\displaystyle e_{17}}	e 16 {\displaystyle e_{16}}	− e 23 {\displaystyle -e_{23}}	e 22 {\displaystyle e_{22}}	− e 21 {\displaystyle -e_{21}}	e 20 {\displaystyle e_{20}}	− e 27 {\displaystyle -e_{27}}	e 26 {\displaystyle e_{26}}	− e 25 {\displaystyle -e_{25}}	e 24 {\displaystyle e_{24}}	e 31 {\displaystyle e_{31}}	− e 30 {\displaystyle -e_{30}}	e 29 {\displaystyle e_{29}}	− e 28 {\displaystyle -e_{28}}
	e 20 {\displaystyle e_{20}}	e 20 {\displaystyle e_{20}}	e 21 {\displaystyle e_{21}}	e 22 {\displaystyle e_{22}}	e 23 {\displaystyle e_{23}}	e 16 {\displaystyle e_{16}}	− e 17 {\displaystyle -e_{17}}	− e 18 {\displaystyle -e_{18}}	− e 19 {\displaystyle -e_{19}}	− e 28 {\displaystyle -e_{28}}	− e 29 {\displaystyle -e_{29}}	− e 30 {\displaystyle -e_{30}}	− e 31 {\displaystyle -e_{31}}	e 24 {\displaystyle e_{24}}	e 25 {\displaystyle e_{25}}	e 26 {\displaystyle e_{26}}	e 27 {\displaystyle e_{27}}
	e 21 {\displaystyle e_{21}}	e 21 {\displaystyle e_{21}}	− e 20 {\displaystyle -e_{20}}	e 23 {\displaystyle e_{23}}	− e 22 {\displaystyle -e_{22}}	e 17 {\displaystyle e_{17}}	e 16 {\displaystyle e_{16}}	e 19 {\displaystyle e_{19}}	− e 18 {\displaystyle -e_{18}}	− e 29 {\displaystyle -e_{29}}	e 28 {\displaystyle e_{28}}	− e 31 {\displaystyle -e_{31}}	e 30 {\displaystyle e_{30}}	− e 25 {\displaystyle -e_{25}}	e 24 {\displaystyle e_{24}}	− e 27 {\displaystyle -e_{27}}	e 26 {\displaystyle e_{26}}
	e 22 {\displaystyle e_{22}}	e 22 {\displaystyle e_{22}}	− e 23 {\displaystyle -e_{23}}	− e 20 {\displaystyle -e_{20}}	e 21 {\displaystyle e_{21}}	e 18 {\displaystyle e_{18}}	− e 19 {\displaystyle -e_{19}}	e 16 {\displaystyle e_{16}}	e 17 {\displaystyle e_{17}}	− e 30 {\displaystyle -e_{30}}	e 31 {\displaystyle e_{31}}	e 28 {\displaystyle e_{28}}	− e 29 {\displaystyle -e_{29}}	− e 26 {\displaystyle -e_{26}}	e 27 {\displaystyle e_{27}}	e 24 {\displaystyle e_{24}}	− e 25 {\displaystyle -e_{25}}
	e 23 {\displaystyle e_{23}}	e 23 {\displaystyle e_{23}}	e 22 {\displaystyle e_{22}}	− e 21 {\displaystyle -e_{21}}	− e 20 {\displaystyle -e_{20}}	e 19 {\displaystyle e_{19}}	e 18 {\displaystyle e_{18}}	− e 17 {\displaystyle -e_{17}}	e 16 {\displaystyle e_{16}}	− e 31 {\displaystyle -e_{31}}	− e 30 {\displaystyle -e_{30}}	e 29 {\displaystyle e_{29}}	e 28 {\displaystyle e_{28}}	− e 27 {\displaystyle -e_{27}}	− e 26 {\displaystyle -e_{26}}	e 25 {\displaystyle e_{25}}	e 24 {\displaystyle e_{24}}
	e 24 {\displaystyle e_{24}}	e 24 {\displaystyle e_{24}}	e 25 {\displaystyle e_{25}}	e 26 {\displaystyle e_{26}}	e 27 {\displaystyle e_{27}}	e 28 {\displaystyle e_{28}}	e 29 {\displaystyle e_{29}}	e 30 {\displaystyle e_{30}}	e 31 {\displaystyle e_{31}}	e 16 {\displaystyle e_{16}}	− e 17 {\displaystyle -e_{17}}	− e 18 {\displaystyle -e_{18}}	− e 19 {\displaystyle -e_{19}}	− e 20 {\displaystyle -e_{20}}	− e 21 {\displaystyle -e_{21}}	− e 22 {\displaystyle -e_{22}}	− e 23 {\displaystyle -e_{23}}
	e 25 {\displaystyle e_{25}}	e 25 {\displaystyle e_{25}}	− e 24 {\displaystyle -e_{24}}	e 27 {\displaystyle e_{27}}	− e 26 {\displaystyle -e_{26}}	e 29 {\displaystyle e_{29}}	− e 28 {\displaystyle -e_{28}}	− e 31 {\displaystyle -e_{31}}	e 30 {\displaystyle e_{30}}	e 17 {\displaystyle e_{17}}	e 16 {\displaystyle e_{16}}	e 19 {\displaystyle e_{19}}	− e 18 {\displaystyle -e_{18}}	e 21 {\displaystyle e_{21}}	− e 20 {\displaystyle -e_{20}}	− e 23 {\displaystyle -e_{23}}	e 22 {\displaystyle e_{22}}
	e 26 {\displaystyle e_{26}}	e 26 {\displaystyle e_{26}}	− e 27 {\displaystyle -e_{27}}	− e 24 {\displaystyle -e_{24}}	e 25 {\displaystyle e_{25}}	e 30 {\displaystyle e_{30}}	e 31 {\displaystyle e_{31}}	− e 28 {\displaystyle -e_{28}}	− e 29 {\displaystyle -e_{29}}	e 18 {\displaystyle e_{18}}	− e 19 {\displaystyle -e_{19}}	e 16 {\displaystyle e_{16}}	e 17 {\displaystyle e_{17}}	e 22 {\displaystyle e_{22}}	e 23 {\displaystyle e_{23}}	− e 20 {\displaystyle -e_{20}}	− e 21 {\displaystyle -e_{21}}
	e 27 {\displaystyle e_{27}}	e 27 {\displaystyle e_{27}}	e 26 {\displaystyle e_{26}}	− e 25 {\displaystyle -e_{25}}	− e 24 {\displaystyle -e_{24}}	e 31 {\displaystyle e_{31}}	− e 30 {\displaystyle -e_{30}}	e 29 {\displaystyle e_{29}}	− e 28 {\displaystyle -e_{28}}	e 19 {\displaystyle e_{19}}	e 18 {\displaystyle e_{18}}	− e 17 {\displaystyle -e_{17}}	e 16 {\displaystyle e_{16}}	e 23 {\displaystyle e_{23}}	− e 22 {\displaystyle -e_{22}}	e 21 {\displaystyle e_{21}}	− e 20 {\displaystyle -e_{20}}
	e 28 {\displaystyle e_{28}}	e 28 {\displaystyle e_{28}}	− e 29 {\displaystyle -e_{29}}	− e 30 {\displaystyle -e_{30}}	− e 31 {\displaystyle -e_{31}}	− e 24 {\displaystyle -e_{24}}	e 25 {\displaystyle e_{25}}	e 26 {\displaystyle e_{26}}	e 27 {\displaystyle e_{27}}	e 20 {\displaystyle e_{20}}	− e 21 {\displaystyle -e_{21}}	− e 22 {\displaystyle -e_{22}}	− e 23 {\displaystyle -e_{23}}	e 16 {\displaystyle e_{16}}	e 17 {\displaystyle e_{17}}	e 18 {\displaystyle e_{18}}	e 19 {\displaystyle e_{19}}
	e 29 {\displaystyle e_{29}}	e 29 {\displaystyle e_{29}}	e 28 {\displaystyle e_{28}}	− e 31 {\displaystyle -e_{31}}	e 30 {\displaystyle e_{30}}	− e 25 {\displaystyle -e_{25}}	− e 24 {\displaystyle -e_{24}}	− e 27 {\displaystyle -e_{27}}	e 26 {\displaystyle e_{26}}	e 21 {\displaystyle e_{21}}	e 20 {\displaystyle e_{20}}	− e 23 {\displaystyle -e_{23}}	e 22 {\displaystyle e_{22}}	− e 17 {\displaystyle -e_{17}}	e 16 {\displaystyle e_{16}}	− e 19 {\displaystyle -e_{19}}	e 18 {\displaystyle e_{18}}
	e 30 {\displaystyle e_{30}}	e 30 {\displaystyle e_{30}}	e 31 {\displaystyle e_{31}}	e 28 {\displaystyle e_{28}}	− e 29 {\displaystyle -e_{29}}	− e 26 {\displaystyle -e_{26}}	e 27 {\displaystyle e_{27}}	− e 24 {\displaystyle -e_{24}}	− e 25 {\displaystyle -e_{25}}	e 22 {\displaystyle e_{22}}	e 23 {\displaystyle e_{23}}	e 20 {\displaystyle e_{20}}	− e 21 {\displaystyle -e_{21}}	− e 18 {\displaystyle -e_{18}}	e 19 {\displaystyle e_{19}}	e 16 {\displaystyle e_{16}}	− e 17 {\displaystyle -e_{17}}
	e 31 {\displaystyle e_{31}}	e 31 {\displaystyle e_{31}}	− e 30 {\displaystyle -e_{30}}	e 29 {\displaystyle e_{29}}	e 28 {\displaystyle e_{28}}	− e 27 {\displaystyle -e_{27}}	− e 26 {\displaystyle -e_{26}}	e 25 {\displaystyle e_{25}}	− e 24 {\displaystyle -e_{24}}	e 23 {\displaystyle e_{23}}	− e 22 {\displaystyle -e_{22}}	e 21 {\displaystyle e_{21}}	e 20 {\displaystyle e_{20}}	− e 19 {\displaystyle -e_{19}}	− e 18 {\displaystyle -e_{18}}	e 17 {\displaystyle e_{17}}	e 16 {\displaystyle e_{16}}

e i e j {\displaystyle e_{i}e_{j}}		e j {\displaystyle e_{j}}
		e 16 {\displaystyle e_{16}}	e 17 {\displaystyle e_{17}}	e 18 {\displaystyle e_{18}}	e 19 {\displaystyle e_{19}}	e 20 {\displaystyle e_{20}}	e 21 {\displaystyle e_{21}}	e 22 {\displaystyle e_{22}}	e 23 {\displaystyle e_{23}}	e 24 {\displaystyle e_{24}}	e 25 {\displaystyle e_{25}}	e 26 {\displaystyle e_{26}}	e 27 {\displaystyle e_{27}}	e 28 {\displaystyle e_{28}}	e 29 {\displaystyle e_{29}}	e 30 {\displaystyle e_{30}}	e 31 {\displaystyle e_{31}}
e i {\displaystyle e_{i}}	e 0 {\displaystyle e_{0}}	e 16 {\displaystyle e_{16}}	e 17 {\displaystyle e_{17}}	e 18 {\displaystyle e_{18}}	e 19 {\displaystyle e_{19}}	e 20 {\displaystyle e_{20}}	e 21 {\displaystyle e_{21}}	e 22 {\displaystyle e_{22}}	e 23 {\displaystyle e_{23}}	e 24 {\displaystyle e_{24}}	e 25 {\displaystyle e_{25}}	e 26 {\displaystyle e_{26}}	e 27 {\displaystyle e_{27}}	e 28 {\displaystyle e_{28}}	e 29 {\displaystyle e_{29}}	e 30 {\displaystyle e_{30}}	e 31 {\displaystyle e_{31}}
	e 1 {\displaystyle e_{1}}	e 17 {\displaystyle e_{17}}	− e 16 {\displaystyle -e_{16}}	− e 19 {\displaystyle -e_{19}}	e 18 {\displaystyle e_{18}}	− e 21 {\displaystyle -e_{21}}	e 20 {\displaystyle e_{20}}	e 23 {\displaystyle e_{23}}	− e 22 {\displaystyle -e_{22}}	− e 25 {\displaystyle -e_{25}}	e 24 {\displaystyle e_{24}}	e 27 {\displaystyle e_{27}}	− e 26 {\displaystyle -e_{26}}	e 29 {\displaystyle e_{29}}	− e 28 {\displaystyle -e_{28}}	− e 31 {\displaystyle -e_{31}}	e 30 {\displaystyle e_{30}}
	e 2 {\displaystyle e_{2}}	e 18 {\displaystyle e_{18}}	e 19 {\displaystyle e_{19}}	− e 16 {\displaystyle -e_{16}}	− e 17 {\displaystyle -e_{17}}	− e 22 {\displaystyle -e_{22}}	− e 23 {\displaystyle -e_{23}}	e 20 {\displaystyle e_{20}}	e 21 {\displaystyle e_{21}}	− e 26 {\displaystyle -e_{26}}	− e 27 {\displaystyle -e_{27}}	e 24 {\displaystyle e_{24}}	e 25 {\displaystyle e_{25}}	e 30 {\displaystyle e_{30}}	e 31 {\displaystyle e_{31}}	− e 28 {\displaystyle -e_{28}}	− e 29 {\displaystyle -e_{29}}
	e 3 {\displaystyle e_{3}}	e 19 {\displaystyle e_{19}}	− e 18 {\displaystyle -e_{18}}	e 17 {\displaystyle e_{17}}	− e 16 {\displaystyle -e_{16}}	− e 23 {\displaystyle -e_{23}}	e 22 {\displaystyle e_{22}}	− e 21 {\displaystyle -e_{21}}	e 20 {\displaystyle e_{20}}	− e 27 {\displaystyle -e_{27}}	e 26 {\displaystyle e_{26}}	− e 25 {\displaystyle -e_{25}}	e 24 {\displaystyle e_{24}}	e 31 {\displaystyle e_{31}}	− e 30 {\displaystyle -e_{30}}	e 29 {\displaystyle e_{29}}	− e 28 {\displaystyle -e_{28}}
	e 4 {\displaystyle e_{4}}	e 20 {\displaystyle e_{20}}	e 21 {\displaystyle e_{21}}	e 22 {\displaystyle e_{22}}	e 23 {\displaystyle e_{23}}	− e 16 {\displaystyle -e_{16}}	− e 17 {\displaystyle -e_{17}}	− e 18 {\displaystyle -e_{18}}	− e 19 {\displaystyle -e_{19}}	− e 28 {\displaystyle -e_{28}}	− e 29 {\displaystyle -e_{29}}	− e 30 {\displaystyle -e_{30}}	− e 31 {\displaystyle -e_{31}}	e 24 {\displaystyle e_{24}}	e 25 {\displaystyle e_{25}}	e 26 {\displaystyle e_{26}}	e 27 {\displaystyle e_{27}}
	e 5 {\displaystyle e_{5}}	e 21 {\displaystyle e_{21}}	− e 20 {\displaystyle -e_{20}}	e 23 {\displaystyle e_{23}}	− e 22 {\displaystyle -e_{22}}	e 17 {\displaystyle e_{17}}	− e 16 {\displaystyle -e_{16}}	e 19 {\displaystyle e_{19}}	− e 18 {\displaystyle -e_{18}}	− e 29 {\displaystyle -e_{29}}	e 28 {\displaystyle e_{28}}	− e 31 {\displaystyle -e_{31}}	e 30 {\displaystyle e_{30}}	− e 25 {\displaystyle -e_{25}}	e 24 {\displaystyle e_{24}}	− e 27 {\displaystyle -e_{27}}	e 26 {\displaystyle e_{26}}
	e 6 {\displaystyle e_{6}}	e 22 {\displaystyle e_{22}}	− e 23 {\displaystyle -e_{23}}	− e 20 {\displaystyle -e_{20}}	e 21 {\displaystyle e_{21}}	e 18 {\displaystyle e_{18}}	− e 19 {\displaystyle -e_{19}}	− e 16 {\displaystyle -e_{16}}	e 17 {\displaystyle e_{17}}	− e 30 {\displaystyle -e_{30}}	e 31 {\displaystyle e_{31}}	e 28 {\displaystyle e_{28}}	− e 29 {\displaystyle -e_{29}}	− e 26 {\displaystyle -e_{26}}	e 27 {\displaystyle e_{27}}	e 24 {\displaystyle e_{24}}	− e 25 {\displaystyle -e_{25}}
	e 7 {\displaystyle e_{7}}	e 23 {\displaystyle e_{23}}	e 22 {\displaystyle e_{22}}	− e 21 {\displaystyle -e_{21}}	− e 20 {\displaystyle -e_{20}}	e 19 {\displaystyle e_{19}}	e 18 {\displaystyle e_{18}}	− e 17 {\displaystyle -e_{17}}	− e 16 {\displaystyle -e_{16}}	− e 31 {\displaystyle -e_{31}}	− e 30 {\displaystyle -e_{30}}	e 29 {\displaystyle e_{29}}	e 28 {\displaystyle e_{28}}	− e 27 {\displaystyle -e_{27}}	− e 26 {\displaystyle -e_{26}}	e 25 {\displaystyle e_{25}}	e 24 {\displaystyle e_{24}}
	e 8 {\displaystyle e_{8}}	e 24 {\displaystyle e_{24}}	e 25 {\displaystyle e_{25}}	e 26 {\displaystyle e_{26}}	e 27 {\displaystyle e_{27}}	e 28 {\displaystyle e_{28}}	e 29 {\displaystyle e_{29}}	e 30 {\displaystyle e_{30}}	e 31 {\displaystyle e_{31}}	− e 16 {\displaystyle -e_{16}}	− e 17 {\displaystyle -e_{17}}	− e 18 {\displaystyle -e_{18}}	− e 19 {\displaystyle -e_{19}}	− e 20 {\displaystyle -e_{20}}	− e 21 {\displaystyle -e_{21}}	− e 22 {\displaystyle -e_{22}}	− e 23 {\displaystyle -e_{23}}
	e 9 {\displaystyle e_{9}}	e 25 {\displaystyle e_{25}}	− e 24 {\displaystyle -e_{24}}	e 27 {\displaystyle e_{27}}	− e 26 {\displaystyle -e_{26}}	e 29 {\displaystyle e_{29}}	− e 28 {\displaystyle -e_{28}}	− e 31 {\displaystyle -e_{31}}	e 30 {\displaystyle e_{30}}	e 17 {\displaystyle e_{17}}	− e 16 {\displaystyle -e_{16}}	e 19 {\displaystyle e_{19}}	− e 18 {\displaystyle -e_{18}}	e 21 {\displaystyle e_{21}}	− e 20 {\displaystyle -e_{20}}	− e 23 {\displaystyle -e_{23}}	e 22 {\displaystyle e_{22}}
	e 10 {\displaystyle e_{10}}	e 26 {\displaystyle e_{26}}	− e 27 {\displaystyle -e_{27}}	− e 24 {\displaystyle -e_{24}}	e 25 {\displaystyle e_{25}}	e 30 {\displaystyle e_{30}}	e 31 {\displaystyle e_{31}}	− e 28 {\displaystyle -e_{28}}	− e 29 {\displaystyle -e_{29}}	e 18 {\displaystyle e_{18}}	− e 19 {\displaystyle -e_{19}}	− e 16 {\displaystyle -e_{16}}	e 17 {\displaystyle e_{17}}	e 22 {\displaystyle e_{22}}	e 23 {\displaystyle e_{23}}	− e 20 {\displaystyle -e_{20}}	− e 21 {\displaystyle -e_{21}}
	e 11 {\displaystyle e_{11}}	e 27 {\displaystyle e_{27}}	e 26 {\displaystyle e_{26}}	− e 25 {\displaystyle -e_{25}}	− e 24 {\displaystyle -e_{24}}	e 31 {\displaystyle e_{31}}	− e 30 {\displaystyle -e_{30}}	e 29 {\displaystyle e_{29}}	− e 28 {\displaystyle -e_{28}}	e 19 {\displaystyle e_{19}}	e 18 {\displaystyle e_{18}}	− e 17 {\displaystyle -e_{17}}	− e 16 {\displaystyle -e_{16}}	e 23 {\displaystyle e_{23}}	− e 22 {\displaystyle -e_{22}}	e 21 {\displaystyle e_{21}}	− e 20 {\displaystyle -e_{20}}
	e 12 {\displaystyle e_{12}}	e 28 {\displaystyle e_{28}}	− e 29 {\displaystyle -e_{29}}	− e 30 {\displaystyle -e_{30}}	− e 31 {\displaystyle -e_{31}}	− e 24 {\displaystyle -e_{24}}	e 25 {\displaystyle e_{25}}	e 26 {\displaystyle e_{26}}	e 27 {\displaystyle e_{27}}	e 20 {\displaystyle e_{20}}	− e 21 {\displaystyle -e_{21}}	− e 22 {\displaystyle -e_{22}}	− e 23 {\displaystyle -e_{23}}	− e 16 {\displaystyle -e_{16}}	e 17 {\displaystyle e_{17}}	e 18 {\displaystyle e_{18}}	e 19 {\displaystyle e_{19}}
	e 13 {\displaystyle e_{13}}	e 29 {\displaystyle e_{29}}	e 28 {\displaystyle e_{28}}	− e 31 {\displaystyle -e_{31}}	e 30 {\displaystyle e_{30}}	− e 25 {\displaystyle -e_{25}}	− e 24 {\displaystyle -e_{24}}	− e 27 {\displaystyle -e_{27}}	e 26 {\displaystyle e_{26}}	e 21 {\displaystyle e_{21}}	e 20 {\displaystyle e_{20}}	− e 23 {\displaystyle -e_{23}}	e 22 {\displaystyle e_{22}}	− e 17 {\displaystyle -e_{17}}	− e 16 {\displaystyle -e_{16}}	− e 19 {\displaystyle -e_{19}}	e 18 {\displaystyle e_{18}}
	e 14 {\displaystyle e_{14}}	e 30 {\displaystyle e_{30}}	e 31 {\displaystyle e_{31}}	e 28 {\displaystyle e_{28}}	− e 29 {\displaystyle -e_{29}}	− e 26 {\displaystyle -e_{26}}	e 27 {\displaystyle e_{27}}	− e 24 {\displaystyle -e_{24}}	− e 25 {\displaystyle -e_{25}}	e 22 {\displaystyle e_{22}}	e 23 {\displaystyle e_{23}}	e 20 {\displaystyle e_{20}}	− e 21 {\displaystyle -e_{21}}	− e 18 {\displaystyle -e_{18}}	e 19 {\displaystyle e_{19}}	− e 16 {\displaystyle -e_{16}}	− e 17 {\displaystyle -e_{17}}
	e 15 {\displaystyle e_{15}}	e 31 {\displaystyle e_{31}}	− e 30 {\displaystyle -e_{30}}	e 29 {\displaystyle e_{29}}	e 28 {\displaystyle e_{28}}	− e 27 {\displaystyle -e_{27}}	− e 26 {\displaystyle -e_{26}}	e 25 {\displaystyle e_{25}}	− e 24 {\displaystyle -e_{24}}	e 23 {\displaystyle e_{23}}	− e 22 {\displaystyle -e_{22}}	e 21 {\displaystyle e_{21}}	e 20 {\displaystyle e_{20}}	− e 19 {\displaystyle -e_{19}}	− e 18 {\displaystyle -e_{18}}	e 17 {\displaystyle e_{17}}	− e 16 {\displaystyle -e_{16}}
	e 16 {\displaystyle e_{16}}	− e 0 {\displaystyle -e_{0}}	e 1 {\displaystyle e_{1}}	e 2 {\displaystyle e_{2}}	e 3 {\displaystyle e_{3}}	e 4 {\displaystyle e_{4}}	e 5 {\displaystyle e_{5}}	e 6 {\displaystyle e_{6}}	e 7 {\displaystyle e_{7}}	e 8 {\displaystyle e_{8}}	e 9 {\displaystyle e_{9}}	e 10 {\displaystyle e_{10}}	e 11 {\displaystyle e_{11}}	e 12 {\displaystyle e_{12}}	e 13 {\displaystyle e_{13}}	e 14 {\displaystyle e_{14}}	e 15 {\displaystyle e_{15}}
	e 17 {\displaystyle e_{17}}	− e 1 {\displaystyle -e_{1}}	− e 0 {\displaystyle -e_{0}}	− e 3 {\displaystyle -e_{3}}	e 2 {\displaystyle e_{2}}	− e 5 {\displaystyle -e_{5}}	e 4 {\displaystyle e_{4}}	e 7 {\displaystyle e_{7}}	− e 6 {\displaystyle -e_{6}}	− e 9 {\displaystyle -e_{9}}	e 8 {\displaystyle e_{8}}	e 11 {\displaystyle e_{11}}	− e 10 {\displaystyle -e_{10}}	e 13 {\displaystyle e_{13}}	− e 12 {\displaystyle -e_{12}}	− e 15 {\displaystyle -e_{15}}	e 14 {\displaystyle e_{14}}
	e 18 {\displaystyle e_{18}}	− e 2 {\displaystyle -e_{2}}	e 3 {\displaystyle e_{3}}	− e 0 {\displaystyle -e_{0}}	− e 1 {\displaystyle -e_{1}}	− e 6 {\displaystyle -e_{6}}	− e 7 {\displaystyle -e_{7}}	e 4 {\displaystyle e_{4}}	e 5 {\displaystyle e_{5}}	− e 10 {\displaystyle -e_{10}}	− e 11 {\displaystyle -e_{11}}	e 8 {\displaystyle e_{8}}	e 9 {\displaystyle e_{9}}	e 14 {\displaystyle e_{14}}	e 15 {\displaystyle e_{15}}	− e 12 {\displaystyle -e_{12}}	− e 13 {\displaystyle -e_{13}}
	e 19 {\displaystyle e_{19}}	− e 3 {\displaystyle -e_{3}}	− e 2 {\displaystyle -e_{2}}	e 1 {\displaystyle e_{1}}	− e 0 {\displaystyle -e_{0}}	− e 7 {\displaystyle -e_{7}}	e 6 {\displaystyle e_{6}}	− e 5 {\displaystyle -e_{5}}	e 4 {\displaystyle e_{4}}	− e 11 {\displaystyle -e_{11}}	e 10 {\displaystyle e_{10}}	− e 9 {\displaystyle -e_{9}}	e 8 {\displaystyle e_{8}}	e 15 {\displaystyle e_{15}}	− e 14 {\displaystyle -e_{14}}	e 13 {\displaystyle e_{13}}	− e 12 {\displaystyle -e_{12}}
	e 20 {\displaystyle e_{20}}	− e 4 {\displaystyle -e_{4}}	e 5 {\displaystyle e_{5}}	e 6 {\displaystyle e_{6}}	e 7 {\displaystyle e_{7}}	− e 0 {\displaystyle -e_{0}}	− e 1 {\displaystyle -e_{1}}	− e 2 {\displaystyle -e_{2}}	− e 3 {\displaystyle -e_{3}}	− e 12 {\displaystyle -e_{12}}	− e 13 {\displaystyle -e_{13}}	− e 14 {\displaystyle -e_{14}}	− e 15 {\displaystyle -e_{15}}	e 8 {\displaystyle e_{8}}	e 9 {\displaystyle e_{9}}	e 10 {\displaystyle e_{10}}	e 11 {\displaystyle e_{11}}
	e 21 {\displaystyle e_{21}}	− e 5 {\displaystyle -e_{5}}	− e 4 {\displaystyle -e_{4}}	e 7 {\displaystyle e_{7}}	− e 6 {\displaystyle -e_{6}}	e 1 {\displaystyle e_{1}}	− e 0 {\displaystyle -e_{0}}	e 3 {\displaystyle e_{3}}	− e 2 {\displaystyle -e_{2}}	− e 13 {\displaystyle -e_{13}}	e 12 {\displaystyle e_{12}}	− e 15 {\displaystyle -e_{15}}	e 14 {\displaystyle e_{14}}	− e 9 {\displaystyle -e_{9}}	e 8 {\displaystyle e_{8}}	− e 11 {\displaystyle -e_{11}}	e 10 {\displaystyle e_{10}}
	e 22 {\displaystyle e_{22}}	− e 6 {\displaystyle -e_{6}}	− e 7 {\displaystyle -e_{7}}	− e 4 {\displaystyle -e_{4}}	e 5 {\displaystyle e_{5}}	e 2 {\displaystyle e_{2}}	− e 3 {\displaystyle -e_{3}}	− e 0 {\displaystyle -e_{0}}	e 1 {\displaystyle e_{1}}	− e 14 {\displaystyle -e_{14}}	e 15 {\displaystyle e_{15}}	e 12 {\displaystyle e_{12}}	− e 13 {\displaystyle -e_{13}}	− e 10 {\displaystyle -e_{10}}	e 11 {\displaystyle e_{11}}	e 8 {\displaystyle e_{8}}	− e 9 {\displaystyle -e_{9}}
	e 23 {\displaystyle e_{23}}	− e 7 {\displaystyle -e_{7}}	e 6 {\displaystyle e_{6}}	− e 5 {\displaystyle -e_{5}}	− e 4 {\displaystyle -e_{4}}	e 3 {\displaystyle e_{3}}	e 2 {\displaystyle e_{2}}	− e 1 {\displaystyle -e_{1}}	− e 0 {\displaystyle -e_{0}}	− e 15 {\displaystyle -e_{15}}	− e 14 {\displaystyle -e_{14}}	e 13 {\displaystyle e_{13}}	e 12 {\displaystyle e_{12}}	− e 11 {\displaystyle -e_{11}}	− e 10 {\displaystyle -e_{10}}	e 9 {\displaystyle e_{9}}	e 8 {\displaystyle e_{8}}
	e 24 {\displaystyle e_{24}}	− e 8 {\displaystyle -e_{8}}	e 9 {\displaystyle e_{9}}	e 10 {\displaystyle e_{10}}	e 11 {\displaystyle e_{11}}	e 12 {\displaystyle e_{12}}	e 13 {\displaystyle e_{13}}	e 14 {\displaystyle e_{14}}	e 15 {\displaystyle e_{15}}	− e 0 {\displaystyle -e_{0}}	− e 1 {\displaystyle -e_{1}}	− e 2 {\displaystyle -e_{2}}	− e 3 {\displaystyle -e_{3}}	− e 4 {\displaystyle -e_{4}}	− e 5 {\displaystyle -e_{5}}	− e 6 {\displaystyle -e_{6}}	− e 7 {\displaystyle -e_{7}}
	e 25 {\displaystyle e_{25}}	− e 9 {\displaystyle -e_{9}}	− e 8 {\displaystyle -e_{8}}	e 11 {\displaystyle e_{11}}	− e 10 {\displaystyle -e_{10}}	e 13 {\displaystyle e_{13}}	− e 12 {\displaystyle -e_{12}}	− e 15 {\displaystyle -e_{15}}	e 14 {\displaystyle e_{14}}	e 1 {\displaystyle e_{1}}	− e 0 {\displaystyle -e_{0}}	e 3 {\displaystyle e_{3}}	− e 2 {\displaystyle -e_{2}}	e 5 {\displaystyle e_{5}}	− e 4 {\displaystyle -e_{4}}	− e 7 {\displaystyle -e_{7}}	e 6 {\displaystyle e_{6}}
	e 26 {\displaystyle e_{26}}	− e 10 {\displaystyle -e_{10}}	− e 11 {\displaystyle -e_{11}}	− e 8 {\displaystyle -e_{8}}	e 9 {\displaystyle e_{9}}	e 14 {\displaystyle e_{14}}	e 15 {\displaystyle e_{15}}	− e 12 {\displaystyle -e_{12}}	− e 13 {\displaystyle -e_{13}}	e 2 {\displaystyle e_{2}}	− e 3 {\displaystyle -e_{3}}	− e 0 {\displaystyle -e_{0}}	e 1 {\displaystyle e_{1}}	e 6 {\displaystyle e_{6}}	e 7 {\displaystyle e_{7}}	− e 4 {\displaystyle -e_{4}}	− e 5 {\displaystyle -e_{5}}
	e 27 {\displaystyle e_{27}}	− e 11 {\displaystyle -e_{11}}	e 10 {\displaystyle e_{10}}	− e 9 {\displaystyle -e_{9}}	− e 8 {\displaystyle -e_{8}}	e 15 {\displaystyle e_{15}}	− e 14 {\displaystyle -e_{14}}	e 13 {\displaystyle e_{13}}	− e 12 {\displaystyle -e_{12}}	e 3 {\displaystyle e_{3}}	e 2 {\displaystyle e_{2}}	− e 1 {\displaystyle -e_{1}}	− e 0 {\displaystyle -e_{0}}	e 7 {\displaystyle e_{7}}	− e 6 {\displaystyle -e_{6}}	e 5 {\displaystyle e_{5}}	− e 4 {\displaystyle -e_{4}}
	e 28 {\displaystyle e_{28}}	− e 12 {\displaystyle -e_{12}}	− e 13 {\displaystyle -e_{13}}	− e 14 {\displaystyle -e_{14}}	− e 15 {\displaystyle -e_{15}}	− e 8 {\displaystyle -e_{8}}	e 9 {\displaystyle e_{9}}	e 10 {\displaystyle e_{10}}	e 11 {\displaystyle e_{11}}	e 4 {\displaystyle e_{4}}	− e 5 {\displaystyle -e_{5}}	− e 6 {\displaystyle -e_{6}}	− e 7 {\displaystyle -e_{7}}	− e 0 {\displaystyle -e_{0}}	e 1 {\displaystyle e_{1}}	e 2 {\displaystyle e_{2}}	e 3 {\displaystyle e_{3}}
	e 29 {\displaystyle e_{29}}	− e 13 {\displaystyle -e_{13}}	e 12 {\displaystyle e_{12}}	− e 15 {\displaystyle -e_{15}}	e 14 {\displaystyle e_{14}}	− e 9 {\displaystyle -e_{9}}	− e 8 {\displaystyle -e_{8}}	− e 11 {\displaystyle -e_{11}}	e 10 {\displaystyle e_{10}}	e 5 {\displaystyle e_{5}}	e 4 {\displaystyle e_{4}}	− e 7 {\displaystyle -e_{7}}	e 6 {\displaystyle e_{6}}	− e 1 {\displaystyle -e_{1}}	− e 0 {\displaystyle -e_{0}}	− e 3 {\displaystyle -e_{3}}	e 2 {\displaystyle e_{2}}
	e 30 {\displaystyle e_{30}}	− e 14 {\displaystyle -e_{14}}	e 15 {\displaystyle e_{15}}	e 12 {\displaystyle e_{12}}	− e 13 {\displaystyle -e_{13}}	− e 10 {\displaystyle -e_{10}}	e 11 {\displaystyle e_{11}}	− e 8 {\displaystyle -e_{8}}	− e 9 {\displaystyle -e_{9}}	e 6 {\displaystyle e_{6}}	e 7 {\displaystyle e_{7}}	e 4 {\displaystyle e_{4}}	− e 5 {\displaystyle -e_{5}}	− e 2 {\displaystyle -e_{2}}	e 3 {\displaystyle e_{3}}	− e 0 {\displaystyle -e_{0}}	− e 1 {\displaystyle -e_{1}}
	e 31 {\displaystyle e_{31}}	− e 15 {\displaystyle -e_{15}}	− e 14 {\displaystyle -e_{14}}	e 13 {\displaystyle e_{13}}	e 12 {\displaystyle e_{12}}	− e 11 {\displaystyle -e_{11}}	− e 10 {\displaystyle -e_{10}}	e 9 {\displaystyle e_{9}}	− e 8 {\displaystyle -e_{8}}	e 7 {\displaystyle e_{7}}	− e 6 {\displaystyle -e_{6}}	e 5 {\displaystyle e_{5}}	e 4 {\displaystyle e_{4}}	− e 3 {\displaystyle -e_{3}}	− e 2 {\displaystyle -e_{2}}	e 1 {\displaystyle e_{1}}	− e 0 {\displaystyle -e_{0}}

Applications

The trigintaduonions are used in the study of quantum physics and other areas of modern physics. Recently, they have also helped researchers working with neural networks.