Essay 01 · Foundations
What are
Myrions?
A plain-English introduction to a proposed number system with a direction for every new way to combine.
Myrions are a proposed way of extending ordinary and complex numbers into any finite number of dimensions.
The idea starts with something familiar. Ordinary real numbers use one number line. Complex numbers add a second direction, usually written using i, where i² = −1. Myrions keep going by adding further independent directions: 1, i, j, k, l, … - with a simple rule linking each new direction to the one before it.
The result is a family of number systems that behave like complex numbers in two dimensions, but become non-associative in three or more. That means the way a sequence of multiplications is grouped can affect the answer.
This unusual property may be useful in computing, particularly in neural networks that need to represent hierarchical or compositional relationships.
1. From real numbers to complex numbers - and beyond
A real number has one component: a. A complex number has two: a + bi. A three-dimensional myrion has three: a + bi + cj. A five-dimensional myrion has five: a + bi + cj + dk + el.
There is no requirement to stop at five. A myrion system can be defined with any finite number of dimensions greater than one. The first two dimensions are exactly the ordinary complex numbers. The new behaviour begins when j is introduced.
2. The basic rule
The defining pattern is i² = −1, j² = i, k² = j, l² = k, and so on. Each new basis element squares to the one immediately below it.
Products between different basis elements follow the same downward pattern: ij = −1, jk = i, and kl = j.
When two non-scalar basis elements are multiplied, the result drops to the level immediately below the lower of the two.
For example, j l = i because j is the lower of j and l; similarly, k l = j. Multiplying anything by 1 leaves it unchanged.
3. The basis elements form cycles
Repeatedly multiplying a basis element by itself produces a cycle. For i, the familiar complex-number cycle is i, −1, −i, 1, then back to i. For j, it is longer: j, i, −1, −j, −i, 1. For k: k, j, i, −1, −k, −j, −i, 1.
The higher the basis element, the longer the cycle. This cyclic behaviour was one of the original motivations for the system.
4. Each finite-dimensional myrion system is closed
An important property is that a chosen finite-dimensional myrion system stays within itself. Suppose we work only with 1, i, j, k, l. Multiplying any two numbers made from those five components can never create a new component such as a hypothetical m.
The multiplication rule always produces one of the original components, a lower component, or its negative. It never produces a higher one. So a five-dimensional myrion calculation remains five-dimensional. The same is true for every finite number of dimensions.
5. Myrions are commutative
Myrion multiplication is commutative: x ⋆ y = y ⋆ x. For example, j ⋆ k = k ⋆ j = i. Reversing two factors does not change their product. This is different from quaternions, where multiplication generally depends on order.
6. But they are not associative
With real and complex numbers, (ab)c = a(bc); the brackets do not matter. For myrions with three or more dimensions, they can.
Take i, j, and j. First calculate i ⋆ (j ⋆ j). Since j ⋆ j = i, this becomes i ⋆ i = −1. But if we group the same factors as (i ⋆ j) ⋆ j, then i ⋆ j = −1, so the result is −j.
Therefore i ⋆ (j ⋆ j) ≠ (i ⋆ j) ⋆ j. The same factors have produced different answers purely because multiplication was grouped differently. That is called non-associativity.
7. Why non-associativity matters
Non-associativity means a calculation has a structure as well as a list of ingredients. Four values a, b, c, d could be grouped as ((a ⋆ b) ⋆ c) ⋆ d, a ⋆ (b ⋆ (c ⋆ d)), or (a ⋆ b) ⋆ (c ⋆ d).
With ordinary numbers these groupings ultimately give the same result. With myrions they need not. A sequence of multiplications follows a particular path through intermediate values, giving the system a natural way to represent hierarchical structure.
8. The complex numbers appear naturally inside myrions
If we use only the first two dimensions, 1, i, the myrion rules reduce exactly to ordinary complex arithmetic. A two-dimensional myrion is simply a + bi, with i² = −1. Nothing new is introduced until a third component, j, is added.
The complex numbers are not merely similar to part of the myrion system: they are exactly the two-dimensional member of the family.
9. Division: zero becomes something larger
With ordinary real or complex numbers, division works for every value except zero. In higher-dimensional myrions, there can be whole sets of values for which unique division is impossible. The paper calls this the singular division locus: the set of myrions by which we cannot divide uniquely.
For complex numbers this set contains only 0. In higher-dimensional myrion systems, it can extend through the space rather than collapsing to a single point.
10. A three-dimensional example
A three-dimensional myrion is x = a + bi + cj. For this system, the set of non-invertible values is described by a³ + ab² + 2abc − bc² − c³ = 0.
You do not need to follow the algebra behind that equation to understand its significance. For complex numbers the equivalent condition, a² + b² = 0, is possible only when both components are zero. The three-dimensional equation is different: non-zero values can satisfy it, so division fails on a larger set.
11. Near the singular division locus
There is a difference between being safely invertible and being almost non-invertible. A myrion close to its singular division locus may technically have an inverse, but division can become highly sensitive: a small change in the original number can cause a large change in the result.
This is similar to a nearly singular matrix in numerical computing. Practical implementations may need to measure not only whether a myrion is invertible, but how close it is to losing invertibility.
12. How myrions differ from other higher-dimensional number systems
Myrions are not the first attempt to extend arithmetic beyond complex numbers. Quaternions have four components; they are associative but not commutative. Octonions have eight components; they are neither commutative nor fully associative.
Myrions take a different route:
- available in any finite number of dimensions;
- commutative;
- closed within each chosen finite dimension;
- identical to complex numbers in two dimensions;
- non-associative from three dimensions upwards.
The key difference is the particular combination of commutative multiplication and association-dependent multi-step calculations.
13. Why this may be interesting for neural networks
Neural networks usually work with arrays of real numbers. Researchers have also experimented with complex numbers, quaternions, and other hypercomplex systems because their multiplication rules can impose useful relationships between groups of values.
A five-dimensional myrion a + bi + cj + dk + el can simply be stored as [a, b, c, d, e]. The unusual behaviour comes from the multiplication rule used when two such groups are multiplied.
13.1 Hierarchical structure
With ordinary multiplication, changing operation grouping may make no mathematical difference. With myrions it can. A neural architecture could make the shape of its computation tree part of what is learned - potentially useful for language structure, nested reasoning, compositional relationships, and tree-structured data. This is a hypothesis to test, not a demonstrated advantage.
13.2 Fixed or learned grouping
A network could use a fixed grouping pattern, such as ((x₁ ⋆ x₂) ⋆ x₃) ⋆ x₄. Or it could compare several groupings and learn how much weight to give each one, making grouping part of the model.
13.3 The singular division locus as a computational feature
A network could monitor how close a myrion weight or activation is to becoming non-invertible. Potential experiments include using this information for gating, routing, regularisation, stability control, or identifying computational regimes.
14. Could myrions be used in a transformer or language model?
In principle, yes. Some real-valued transformations used to form queries, keys, values, and intermediate representations could be replaced by myrion-valued ones. Five real channels could be treated as one five-dimensional myrion, with its multiplication rule coupling them together.
The important question is not whether this can be programmed - it can - but whether it gives a measurable advantage over ordinary real, five-channel, complex, quaternion, or other structured hypercomplex models. That requires experiments.
15. What has been established, and what has not
For every finite-dimensional myrion system, multiplication is closed and commutative; addition and multiplication are distributive; 1 acts as the identity; the two-dimensional system is exactly the complex numbers; systems with three or more dimensions are non-associative; and unique division exists away from the singular division locus.
Open questions include the exact shape of that locus in higher dimensions, the full classification of the algebras, useful identities and symmetries, numerical implementations, and whether neural networks can exploit these properties.
16. Why study the system?
Myrions are interesting for two separate reasons. First, they are a relatively simple extension of complex numbers with an unusual combination of properties. The system remains commutative and closed in every finite dimension, but becomes non-associative as soon as the third dimension is added.
Second, those properties have a possible computational interpretation. With ordinary arithmetic, a long multiplication can often be rearranged without changing its meaning. With myrions, the arrangement itself can carry information.
That suggests a different kind of numerical building block for systems in which how information is combined matters as much as which information is combined. Whether this proves practically useful remains an experimental question, but the system is mathematically defined well enough for that question to be tested.
Further reading
The formal paper
A mathematical treatment with definitions, proofs, coordinate multiplication formulae and references is available in The Myrion Number System: A Commutative Non-Associative Algebra with a Singular Division Locus.
Read the research paper ↗