Brief Overview of a Complex Hexapolar Number System
by Ben Blohowiak
Last updated: 2025-09-12
A geometric line expresses properties of the real number system such that each point of a line may correspond to a unique real number. The real numbers as well as a geometric line may be regarded as a totally ordered continuum extending in two inverse directions.
Why two directions? Why not three (or more)?
The total order over the reals and the fact that addition associates imply each other and constrain additive inverses to unique pairs. (This has implications that make an object's chirality possible; such chirality is a function of the dipolarity of the axes of its embedding space, among other properties.) If you want additive inverses to come in triplets, quadruplets, etc. such that there are more than two inverse directions of extension per basis vector, axis, or "line," then you can generalize addition such that it does not necessarily associate (at the costs of a continuum's total order and the chirality of objects embedded in spaces defined over such systems): "consolidation" is a name for such a generalized addition.
Whereas the associative property of addition constrains additive inverses to unique pairs indicated by positive and negative signs, consolidative inverses may come in sets of any countable number greater than two such that, in principle, there is no inherent or necessary bound on the quantity of signs that a consolidative system may contain apart from the details of its construction. Number systems containing more than two signs may be termed "multipolar" with the quantity of "poles" mapped to the quantity of signs or consolidative inverses for a nonzero magnitude. (I also explore systems that imply an unbounded quantity of signs/orthants by means of Lie groups.)
Nontrivial multipolar number systems that embed the real line have a minimum of six signs (i.e., they are at minimum "hexapolar"). A given multipolar system that embeds the real line and has more than the minimum quantity of signs could instead have 10,14,18, etc. [Why? See section at bottom of page.]
Whereas a geometric interpretation of the complex numbers, the complex plane, contains a fourth root of unity (i) not contained in the real number system, the complex hexapolar plane embeds and extends the reals via a 12th root of unity (s) not contained in the reals or complex numbers such that s3=i and s9=-i. Accordingly, the complex hexapolar plane embeds and extends the conventional complex plane as well as the real line it contains.
Graphing escape-time fractals given the complex hexapolars as domain and codomain (e.g., z[i+1]=z[i]2+c) induces/reveals symmetry relationships among orthants such that, depending on the recursive function, they lend themselves to grouping into various four-quadrant subplanes whose escape values sometimes resemble so-called "absolute value variations" of fractals in the complex plane (e.g., ""burning ship"). Such variations, inherent in the logic of the complex multipolars, illustrate by contrast the role of addition's associativity in the shape of the tricorn or the Mandelbrot set, themselves also expressed in the complex hexapolar plane as per the latter's embedding of the complex plane.
Click image of a tricorn function in the (s10,s10) orthant to load larger size. Click above image of a tricorn function in the (s10,s10) orthant to load larger size.
Click here to access a bare-bones version of the cHex visualizer.
Computing power series functions without addition's associative property requires "chunking" strategies for consolidations such as "keeping a running total" in which (((1+x)+x2/2!)+x3/3!)... For the complex hexapolars, exp(s3θ)=eiθ and exp(s9θ)=e-iθ with conventional magnitude and periodicity. As θ grows in exp(sθ), however, it becomes increasingly difficult to compute as the numbers involved grow quite large; implementation challenges have prevented computation for larger θ such that whether the complex hexapolar exp() function is periodic has yet to be demonstrated or proven. As has thus far been computed, |exp(sθ)|=1 if θ=0 and the variations in
|exp(sθ)| (whether its natural modulus or Euclidean magnitude) seem to become periodic with phase offset between component magnitudes > π/2, though that curve is not differentiable everywhere; it contains cusps, points at which the curve changes direction as if instantaneously. Given a definition for natural exponents as per repeat multiplication such that s3=s*s*s, etc., although exp(s3θ) maps to a unique point on each ray, it remains unproven whether exp(sθ) maps to a point on each ray from the origin, more than one, or neither.
Via a Cayley-Dickson construction, one may define a hexapolar quaternion extension and similarly one may use 2x2 matrices containing complex hexapolars to generalize the Pauli matrices. Because the natural modulus of the complex hexapolars isn't necessarily positive-definite, such extensions of dimension four contain subspaces associated with ℝ4, ℝ(1,3), and ℝ(2,2). The indefinite modulus/inner product of the complex hexapolar system is related to the fact that multipolar systems may contain zero divisors that are invertible--that is, zero divisors that have multiplicative inverses (in contrast to, say, the split-complex number system).
[Why do multipolar systems that nontrivially embed the reals have 6, 10, 14, 18, etc. signs?]
For a multipolar system to embed the real line, one and only one of its nonpositive signs must be extensionally equivalent to the negative sign of the reals (and the rationals and integers they contain). Properties that distinguish negatively signed real numbers include squaring to positive numbers and lacking non-imaginary square roots.
The sign multiplication tables of multipolars correspond to Cayley tables of groups which are necessarily Abelian such that multipolar multiplication may be commutative. (If the sign multiplication table does not express the group property of unique invertibility, then multiplication does not necessarily distribute over consolidation or addition generalized such that it is not necessarily associative.) The groups associated with sign multiplication tables of multipolar systems that embed the reals need to be cyclic as well as Abelian such that only one nonpositive sign squares to positive; this permits a unique mapping to the negative numbers of the real line. There is only one possible table per sign quantity for a cyclic group (up to isomorphism). Of those, only cyclic groups with an even quantity of elements permit a nonpositive sign to produce a positive sign when squared. Multipolar sign multiplication tables containing an even quantity of elements might or might not contain signs that lack a non-imaginary square root even if they square positive; those that do contain a sign that squares positive and lacks a non-imaginary square root (i.e., those that contain a sign sharing properties with conventional negative numbers) may be characterized by total sign quantity (p) such that (p/2)mod2=1, which permits the reals (p=2) as a trivial instance of a multipolar system.
A less brief overview here.
An earlier draft with more intensive set-builder notation here.
A video playlist:
