Essay 03 · Comparisons
Beyond the
complex plane
Four different ways to extend arithmetic beyond the real number line — and the distinct structures they preserve.
Complex numbers, quaternions, octonions and myrions all extend real arithmetic, but they do not form one simple ladder.
Complex numbers, quaternions and octonions are often presented as the sequence ℝ → ℂ → ℍ → 𝕆, doubling its number of real components at each step. Myrions begin with the same complex plane, then take a separate route: there is a finite myrion algebra in every dimension.
Quaternions and octonions make the order of factors matter. Higher-dimensional myrions retain commutativity, but make the grouping of factors matter.
Starting with the complex numbers
A real number has one component, a. A complex number has two: z = a + bi, where i² = −1. Complex multiplication is both commutative and associative: zw = wz and (zw)u = z(wu).
Every non-zero complex number has a unique inverse, z⁻¹ = (a − bi)/(a² + b²). Since the denominator is zero only at the origin, zero is the sole exceptional value for division. Complex numbers also have a multiplicative magnitude, |zw| = |z||w|.
The two-dimensional myrion algebra is not merely analogous to this system: M₂ ≅ ℂ. The families differ only when more directions are introduced.
Quaternions: four dimensions, but order matters
A quaternion has four real components: q = a + bi + cj + dk. Its basis elements satisfy i² = j² = k² = ijk = −1. Quaternion multiplication remains associative, so brackets can be moved, but it is not commutative: ij = k while ji = −k.
This order sensitivity is useful for three-dimensional rotations, where successive rotations generally depend on the order in which they are performed. Quaternions still have a multiplicative norm and every non-zero quaternion has an inverse.
Octonions: brackets matter too
An octonion has eight real components. Like quaternions, octonions are generally non-commutative; unlike quaternions, they are also non-associative, so (xy)z need not equal x(yz).
They retain alternativity: expressions such as x(xy) = (xx)y remain valid, and the subalgebra generated by two octonions is associative. Repeated powers of one octonion are therefore unambiguous. Octonions also retain a multiplicative norm and inverses for all non-zero elements.
Myrions: a different branch
An n-dimensional myrion has the form x = a₀ + a₁e₁ + ⋯ + aₙ₋₁eₙ₋₁, with e₁ = i, e₂ = j, e₃ = k, and so on. There is a separate algebra Mₙ for every finite n ≥ 2.
Its basis multiplication descends: i² = −1, j² = i, k² = j. For two non-scalar basis elements, the product lands one level below the lower-ranked factor: j ⋆ k = i, j ⋆ l = i, and k ⋆ l = j. The rule is symmetric, so myrion multiplication is commutative.
The main comparison
| Property | Complex | Quaternions | Octonions | Myrions |
|---|---|---|---|---|
| Real dimensions | 2 | 4 | 8 | Any finite n ≥ 2 |
| Commutative | Yes | No | No | Yes |
| Associative | Yes | Yes | No | Only for n = 2 |
| Alternative / power-associative | Yes | Yes | Yes | No for n > 2 |
| Every non-zero element invertible | Yes | Yes | Yes | No for n > 2 |
| Construction | Complex field | Cayley–Dickson | Cayley–Dickson | Descending-axis rule |
Equal dimension does not imply equal algebra: M₄ ≠ ℍ and M₈ ≠ 𝕆. Their multiplication laws are different. No counterpart of the multiplicative norm of the classical three systems has currently been established for myrions.
Order and grouping are different information
Commutativity asks whether x ⋆ y equals y ⋆ x. Associativity asks whether (x ⋆ y) ⋆ z equals x ⋆ (y ⋆ z). Complex numbers have both properties; quaternions lose commutativity; octonions lose both; higher-dimensional myrions lose associativity while keeping commutativity.
In M₃, i ⋆ (j ⋆ j) = i ⋆ i = −1, whereas (i ⋆ j) ⋆ j = (−1) ⋆ j = −j. The same factors have been used; only the binary tree changed.
A stronger loss for repeated myrion powers
Both octonions and higher myrions are non-associative, but octonionic alternativity keeps repeated powers unambiguous. Higher myrions are not power-associative. For the basis element j, (j ⋆ j) ⋆ (j ⋆ j) = −1, while ((j ⋆ j) ⋆ j) ⋆ j = −j.
A higher myrion power such as x⁴ therefore needs explicit brackets or a stated association convention.
Division and dimension
Complex numbers, quaternions and octonions are division algebras: every non-zero element has an inverse. For a myrion x, unique division is determined by the linear multiplication operator Lₓ(y) = x ⋆ y. The singular division locus is Σₙ = {x : det Lₓ = 0}.
For complex numbers this is only the origin. In higher myrion algebras it includes non-zero zero divisors; for example (j − i) ⋆ i = 0. Away from the locus, a myrion has a unique inverse. On it, an inverse equation may have no solution or many solutions.
Different approaches to dimension
The Cayley–Dickson route doubles: 1, 2, 4, 8, 16, …. Myrions add a single basis direction at a time: 2, 3, 4, 5, 6, …. Each Mₙ is closed: multiplication never requires a component above the highest basis level already present. The family is nested, M₂ ⊂ M₃ ⊂ M₄ ⊂ ⋯.
The distinction
A separate family, not a missing rung
Myrions are best understood as an alternative continuation from the complex numbers. They preserve commutative binary interaction, exist in every finite dimension, and make the association tree significant above two dimensions.
These are different algebraic design choices, not points on a scale from weaker to stronger. Whether myrions offer a useful computational bias remains an experimental question.