What is known about Pathions, Chingons, Routons and Voudons?

Question

It is well known that the quaternions, octonions, and sedenions are well studied, but I don't find any articles or books in which other hypercomplex numbers are studied. Does anyone know a book or an article in which other hypercomplex numbers are studied? And if so, what is the usage of numbers like pathions, voudons, etc...?

Answer 1

After a quick search on Math Sci Net, I have found:

• 1.229 articles containing the word "octonion"
• 56 articles containing the word "sedenion"
• 0 articles containing the word "pathion"
• 0 articles containing the word "chingon"
• 0 articles containing the word "routon"
• 1 article containing the word "voudon", whose author is named Voudon, Benoît

(this search includes title, abstract, and names of the authors).

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Edit. If you type "Cayley-Dickson", these are the references with the highest number of citations according to Math Sci Net:

Answer 2

Hint

You can derive the sets of numbers using the Cayley–Dickson Construction. Algebras produced by this process are known as Cayley–Dickson Algebras and these are better studied aka you can find more about them under Cayley–Dickson Algebras.

In addition, the names after the complex numbers are linked to numbers in latin, which in the graphic does not apply to those wich come after Sedenions. Normally the continuation would be Sedenion ($16$-ions) $\to$ Trigintaduonions / Tricenibinions ($32$-ions) $\to$ Sexageniquaternions / Sexagintaquaternions ($64$-ions) $\to \dots$... (there are multible similar names for this and some of them are not even numbers - see here for more)

You will find much more with these more often used names.

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Depending on the definition, there are many more such hypercomplex numbers. In the following, by hypercomplex numbers I only mean the numbers that can be created using the Cayley–Dickson construction (for the real numbers) and I will write their set as $\mathbb{\tilde{C}}$.

Applications

Actual applications of hypercomplex numbers with dimension above $16$ (on your graph adter $\mathbb{S}$) in the natural sciences are rare, but there are some in mathematics, e.g. in Hypercomplex Analysis or proves.

In Mathematics

Hypercomplex Analysis: There is the field in mathematics called Hypercomplex Analysis, which deals with the study of functions with arguments in $\tilde{\mathbb{C}}$.

• (reference:) A book on that matter is Hypercomplex Analysis: New Perspectives and Applications by Swanhild Bernstein, Uwe Kähler, Irene Sabadini and Frank Sommen, which does not deal directly with these sets of numbers, but works a lot with Clifford algebras, which in turn can be used to form these sets. So far I've only seen uses in mathematics in the book (for the sets of numbers), but I haven't read through it yet.

Geometry

In geometry you could use them to represent spaces / objects (points, vectors, ...) in $D > 16$. With a lot of work, it might be possible to represent the movement of a point around an axis in this space by a certain angle, which might be helpful in some theoretical model of physics, after all, hypercomplex numbers usually have representations as vectors and matrices.

From this, geometric properties could be derived (e.g. Euler's formula for these sets of numbers) with which proofs for geometric relationships and analysis analogous to the sedenions could be led, which could be used in such a theoretical model in physics.

Calculus

In calculus, certain differential equations could be solved or certain operations could be carried out. An example would be differential operators of hypercomplex order or solving describing movements of objects in spaces in $D > 16$. But the whole thing would really only have a use in mathematics.

Hypercomplex Algebra

Equations can be constructed which do not have solutions in $\mathbb{S}$, but in the larger sets that follow.

• $x^{2} = -1$ has no real solution but two complex ones: $x \in \left\{ -i,\, i \right\} \in \mathbb{C}$
• $x \cdot i = -i \cdot x \ne 0$ has not complex solution but in $\mathbb{H}$: $x \in \left\{ a \cdot j,\, a \cdot k \mid a \in \mathbb{R} \right\} \in \mathbb{H}$
• $x \cdot \left( j \cdot \left( k \right) \right) \ne \left( \left( x \right) \cdot j \right) \cdot k$ has no solution in $\mathbb{H}$ but multible in $\mathbb{O}$ ones

...

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In Physics

There are a few applications e.g. in quantum physics.

For example, the Trigintaduonion Emanation Theory allows a mathematical description of certain physical constants with $32$-ions (and you can find some of them in the standard model of particle physics).

The chiral trigintaduonion emanation described here gives a precise derivation for the mysterious physics constant $\alpha$ (the fine-structure constant) from the mathematical physics formalism providing maximal information propagation, with $\alpha$ being the maximal perturbation amount (a fractal limit), and $\pi$ being the maximum amount of overall imaginary component contributing to that maximal perturbation. - Stephen Winters-Hilt in Fiat Numero: Trigintaduonion Emanation Theory and its Relation to the Fine-Structure Constant $\alpha$, the Feigenbaum Constant $c^{\infty}$, and $\pi$

$32$-ion spaces have some applications too:

A new insight on the problem of the sub-quark movement and their interactions can be given by the concept of trigintaduonion space. - Zihua Weng in Compounding Fields and Their Quantum Equations in the Trigintaduonion Space

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Studies

There are some studies in certain areas of mathematics. For example, there is a proof that all numbers in $\tilde{C}$ continue to have the property of power associativity.

Since calculating in $\tilde{\mathbb{C}}$ can be quite tiring if you constantly have such long formulas, there are also some algorithms for calculating in $\tilde{\mathbb{C}}$, e.g. An algorithm for multiplication of trigintaduonions by Aleksandr Cariow and Galina Cariowa.

For more precise or specific answers or references, a second question would probably be good.

Answer 3

These terms are ad-hoc names for higher Cayley-Dickson algebras after the sedenions, however they are introduced in a formal mathematical paper published by ArXiv in 2002. These terms most notably appear in a Maple implementation of hypercomplex numbers up to the 256-D case (the 9th Cayley-Dickson algebra). They are actually proposed by Tony Smith (for the voudons, the highest of the nine) and Robert P.C. de Marrais (from 32-ions to the 128-ions).

Sequence	Dimension	Name	Symbol	Etymology
5	32	pathion	ℙ	for the "32 Paths" of Kabbalah
6	64	chingon		for the 64 Hexagrams of the I Ching
7	128	routon		after that legendary source of high-tech innovativeness, Route 128 of the "Massachusetts Miracle" that paralleled Silicon Valley's on the "Left Coast" of this country.
8	256	voudon		after the 256 deities of the Ifa pantheon of Voodoo or Voudon

The reason why the author limited the common "un-Latinate" names to such higher hypercomplex number systems is to study Clifford algebras, which has uses in representing rotations in high-dimensional space, the periodicity theorem, and monstrous moonshine. The authors claimed that all higher hypercomplex number systems can be broken down into copies of "voudon" space.

De Marrais continued to use the word "pathion" and "chingon" in his further papers. The word "routon" and "voudon", however only appeared twice and thrice and are usually still called $2^7$-ions and $2^8$-ions respectively. His paper that introduces these terms has been cited by at least two other journals not authored by him, including In Defense of Octonions by Jonathan J. Dickau.

Source notes

• The de Marrais source is introduced in Page 7, the Tony Smith link is now dead