Journal Article - Published
Acta Mathematica Hungarica · Vol. 167, No. 1 (2022) · Real Analysis
* Equal contribution
St. Olaf College
Theorems 4 & 10
Lemma 2 & Theorem 3
Theorems 13 & 14
Overview
Peano's 1890 space-filling curve \(f_P : [0,1] \to [0,1]^2\) was a landmark in mathematics, demonstrating a continuous surjection from an interval onto a square. Despite its historical significance, the original arithmetic proof was opaque, offering little geometric insight into how individual points in the domain map to points in the range.
This paper gives a fairly complete arithmetization of the Peano curve using the geometric framework Hilbert introduced. We show that the inductive pattern assignment governing \(f_P\) is controlled by the action of the Klein Four-Group \(K_4\) via matrix multiplication, enabling closed-form computation of patterns at any level without resorting to full induction.
The main contribution is a complete point-by-point classification of the preimage structure of \(f_P\). We prove that \(f_P\) is at most 4-to-1 everywhere on \([0,1]^2\), that 3-to-1 points do not exist, and that the preimages of triadic line segments are unions of Cantor sets each of Hausdorff dimension \(1/2\).
Method
Formalize the Peano curve via adjacency and nesting conditions on correspondences between subintervals of [0,1] and triadic subsquares of [0,1]². Rose's theorem guarantees a unique continuous surjection from any such correspondence.
Define fP inductively using four Peano patterns and base-9 interval labeling. Each level assigns one of four vector patterns to every triadic subsquare, ordering the next level's subsquares and fully determining the curve's geometry.
Show that the pattern of any triadic subsquare at grid position (k, m) is determined directly by P = A^{k mod 2} B^{m mod 2} ⟨1,1⟩, where K⊂4 = {I, A, B, AB} acts via matrix multiplication - no induction needed.
Classify preimage multiplicity at every point in [0,1]². Use the Klein 4-group structure and Peano sequence analysis to characterize 1-to-1, 2-to-1, and 4-to-1 points, and identify Cantor set preimages of all triadic line segments.
Results
The Peano function is at most 4-to-1 everywhere. It is 1-to-1 on a residual subset of [0,1]², 2-to-1 on triadic line segment points, and 4-to-1 precisely at odd interior triadic nodes. Notably, no 3-to-1 points exist - a property not shared by all space-filling curves.
The Klein Four-Group governs pattern assignment at every level. This gives a closed-form formula for the Peano pattern of any triadic subsquare from its grid coordinates alone, enabling direct computation without stepping through the inductive construction.
The preimage of every triadic line segment is a union of Cantor-type sets, each with Hausdorff dimension 1/2. This matches the analogous result for the Hilbert curve, suggesting a structural property shared across Hilbert-type space-filling curves.
Resources
Citation
@article{humke2022peano,
title = {Finding Keys to the {Peano} Curve},
author = {Humke, Paul D. and Huynh, Khang Vo},
journal = {Acta Mathematica Hungarica},
volume = {167},
number = {1},
pages = {255--277},
year = {2022}
}