[Paper Review] Complex base numeral systems

Jarek Duda|ArXiv.org|Dec 10, 2007

Mathematical Dynamics and Fractals3 references3 citations

TL;DR

This paper introduces a comprehensive family of complex base numeral systems in 2D and 3D that are 'proper'—ensuring unique representations for almost all points in the complex plane, with only a zero-measure set having multiple representations. It establishes periodic tiling via self-similar fractal attractors (IFS) and provides analytical methods to compute convex hulls, boundary dimensions, and generalizations to higher dimensions using algebraic conditions on the base and digit set.

ABSTRACT

In this paper will be introduced large, probably complete family of complex base systems, which are 'proper' - for each point of the space there is a representation which is unique for all but some zero measure set. The condition defining this family is the periodicity - we get periodic covering of the plane by fractals in hexagonal-type structure, what can be used for example in image compression. There will be introduced full methodology of analyzing and using this approach - both for the integer part: periodic lattice and the fractional: attractor of some IFS, for which the convex hull or properties like dimension of the boundary can be found analytically. There will be also shown how to generalize this approach to higher dimensions and found some proper systems in dimension 3.

Motivation & Objective

• To develop a complete, systematic family of complex base numeral systems in 2D that are proper—surjective and pseudoinjective with zero-measure non-injective sets.
• To characterize periodic complex base systems where the base satisfies |z|² = n and z² ∈ ℤ + zℤ, enabling structured tiling of the plane.
• To provide analytical tools for computing the convex hull and boundary dimension of the fractional part attractor F_z.
• To generalize the framework to 3D, identifying conditions under which proper systems exist using algebraic relations and matrix-based representations.
• To explore applications in image compression through hexagonal-type tiling and improved correlation structures.

Proposed method

• Uses self-similarity equations: z·F_z = ∪_{i=0}^{n-1} (F_z + i) and ∪_{i=0}^{n-1} (z·I_z + i) = I_z to define the integer and fractional parts.
• Applies iterated function systems (IFS) to model F_z as a compact, central symmetric attractor with positive Lebesgue measure.
• Employs reduction techniques (analogous to right-shift) to analyze the integer part I_z and prove that I_z = ℤ + zℤ under periodicity conditions.
• Derives analytical expressions for the boundary ∂F_z by constructing approximations via infinite sums and solving functional equations in angular coordinates.
• Generalizes to 3D by modeling z as a 3D vector via Z = rO with orthogonal matrix O, and derives algebraic conditions such as (n, mA, A, 1)_z = 0 for proper systems.
• Uses width functions and boundary chain analysis to distinguish neighbor tiles in 3D, with each tile having 14 neighbors and a closed chain structure.

Experimental results

Research questions

• RQ1 What conditions on the complex base z and digit set {0, ..., n-1} ensure that the numeral system is proper, i.e., unique representation almost everywhere?
• RQ2 How can the boundary of the fractional part attractor F_z be analytically characterized, and what is its Hausdorff dimension in periodic cases?
• RQ3 Can the framework of complex base systems be generalized to 3D Euclidean space, and what algebraic constraints define proper systems in this setting?
• RQ4 What is the role of periodicity in ensuring that the integer part I_z forms a lattice ℤ + zℤ and that F_z tiles the plane with boundary-only intersections?
• RQ5 How do the geometric and topological properties of F_z (e.g., convex hull, connectivity, simple connectivity) relate to the base z and digit count n?

Key findings

• The condition |z|² = n is necessary and sufficient for proper systems, ensuring positive measure and non-overlapping tiling of the plane by copies of F_z.
• For periodic systems, F_z is a compact, simply connected, central symmetric set with boundary of dimension in [1,2), analytically computable via functional equations.
• In 2D, the boundary ∂F_z corresponds to points with multiple representations, forming a dense set S = ∪_{k∈ℤ} z^k (I_z + ∂F_z), with Hausdorff dimension strictly less than 2.
• In 3D, proper systems exist when the base satisfies a cubic algebraic relation (n, mA, A, 1)_z = 0, with r = -m and n = m³ for integer m ≥ 2.
• Each tile in the 3D tiling has 14 neighbors, and the boundary structure is analyzed via a closed chain of neighbors that splits the neighbor set into two distinguishable subsets.
• Numerical results show that Hausdorff dimension of F_z peaks at 2 for periodic cases and drops to 1 as the argument φ approaches π, indicating a transition to a line segment.