In abstract algebra, the trigintaduonions, also known as the 32-ions, 32-nions, 25-nions, or sometimes pathions (

),[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

[2] 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.

, 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

[5] 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.

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

They are also flexible, and multiplication is distributive over addition.

[9] 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.

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

, we first observe that the multiplication table of its imaginary units

, is encoded in the properties of the projective space

if these imaginary units are regarded as points and distinguished triads of them

This projective space is seen to feature two distinct kinds of lines according as

[10] Furthermore, Saniga, Holweck & Pracna (2015) state that: The corresponding point-line incidence structure is found to be a specific binomial configuration

(32-nions) coincides with the Cayley–Salmon (154,203)-configuration found in the well-known Pascal mystic hexagram and

(64-nions) is identical with a particular (215,353)-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 multiplication of the unit trigintaduonions is illustrated in the two tables below.

Combined, they form a single 32×32 table with 1024 cells.

The top left quadrant of the table, for

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).

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

[11] More recently, the trigintaduonions and other hypercomplex numbers have also been used in neural network research[13] and cryptography.

Robert de Marrais's terms for the algebras immediately following the sedenions are the pathions (i.e. trigintaduonions), chingons, routons, and voudons.