Journal Article - Published
Journal of Mathematical Analysis and Applications · Vol. 562 (2026) · Real Analysis
* Equal contribution
St. Olaf College · University of Virginia
Theorem 2.12
Theorems 3.2 & 3.21
Theorem 5.5
Overview
Peano's original 1890 space-filling curve was defined purely arithmetically via a base-3 shuffling formula. A year later, Hilbert revealed the underlying geometry. For the classical \(f_P : [0,1] \to [0,1]^2\), both approaches are known to coincide - but the story in higher dimensions was incomplete.
De Freitas, de Lima, and dos Santos (2019) generalized Peano's arithmetic formula to \(n\) dimensions, producing a family \(f_n : [0,1] \to [0,1]^n\). Independently, this paper constructs a family \(g_n\) using Hilbert's geometric framework - proper sequences of linked partitions of \([0,1]^n\) with cube markings governed by generalized Klein 4-groups \(H_n\). The central result is that \(f_n = g_n\) for every \(n \in \mathbb{N}\).
The geometric approach explains a previously mysterious aspect of the arithmetic formula: the \(\pm 1\) sign coefficients in the functional equation of de Freitas et al. (Theorem 4.1) emerge naturally as cube markings under the group \(H_n\), and the self-affine relation (Theorem 2.12) immediately implies their functional equation as a corollary.
Method
Define gn : [0,1] → [0,1]^n via Hilbert's framework: a proper sequence of linked partitions of [0,1]^n using 3^n congruent subcubes at each level, each marked with a diagonal matrix Φ(C) recording its orientation relative to [0,1]^n.
Prove (Theorem 2.12) that gn is self-affine with base 3^m for any m: the action of gn on any ternary interval, however small, isometrically determines the function globally via the affine maps L_{σ(m)(t)}. Corollary: gn(I_{σ}) = C_{σ}^n exactly.
Show (Theorems 3.2 and 3.21) that the cube marking Φ(S) depends only on the Cartesian position of S via the group Hn generated by {A_k^n}. When n is even Hn = Gn (full octahedral group); when n is odd, Hn is a proper Klein 4-group subgroup of Gn.
Identify fn and gn on a dense set using Lemma 5.1 (fn(.) = v0() for each ternary point) and Proposition 2.10. Both are continuous, so equality on the dense set of ternary rationals forces fn = gn everywhere on [0,1].
Results
The geometric construction gn is self-affine at every scale: the action of gn on any ternary interval is an isometric copy of its global behavior (Theorem 2.12). This provides an efficient framework for computing gn on any ternary subcube using only the cube markings.
The cube marking group Hn - a generalization of the Klein 4-group from the planar case - governs orientation of every subcube in [0,1]^n. Its structure depends on the parity of n: Hn equals the full octahedral group Gn when n is even, and is a proper subgroup when n is odd.
Peano's arithmetic shuffling (de Freitas, de Lima, dos Santos) and Hilbert's geometric nested-partition method produce identical space-filling curves in every dimension. The geometric view explains the sign pattern in the arithmetic functional equation as a direct consequence of cube markings.
Resources
Citation
@article{humke2026coincidence,
title = {A {Peano} Coincidence},
author = {Humke, Paul D. and Huynh, Khang Vo},
journal = {Journal of Mathematical Analysis and Applications},
volume = {562},
pages = {130657},
year = {2026}
}