Trigintaduonion

In abstract algebra, the trigintaduonions, also called the 32-ions, 32-nions, or 2⁵-nions, are a special type of mathematical structure. They exist in a space with 32 different directions, which mathematicians call dimensions.

These trigintaduonions are interesting because their math does not follow the usual rules for everyday numbers. For example, when you multiply these numbers, the order matters. In regular multiplication, 3 × 4 is the same as 4 × 3, but this is not true for trigintaduonions. This makes them a noncommutative algebra.

They also do not follow the associative rule. This means that (a × b) × c can give a different result than a × (b × c). These special numbers are studied using 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. 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, working with trigintaduonions is not as simple as with regular numbers. But it has a special rule called power associativity. This means that for any number in this system, raising it to different powers makes sense.

The trigintaduonions also have special numbers called zero divisors. These numbers do not act like regular numbers in every way. They can be grouped in many ways into sets of three, following different patterns.

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

Trigintaduonions are used to study quantum physics and other areas of modern physics. They have also helped researchers working with neural networks.