Hilbert's spacefilling curve described by automatic, regular, and synchronized sequences

TL;DR

This paper presents multiple representations of Hilbert's spacefilling curve, including automatic sequences, regular synchronized coordinate sequences, and bitmap images, providing diverse perspectives on its structure.

Contribution

It introduces novel descriptions of the Hilbert curve using automatic and synchronized sequences, expanding understanding of its computational and structural properties.

Findings

Hilbert's curve can be represented as an automatic sequence of directions.

The curve's coordinates form a regular synchronized sequence.

The curve can be depicted as an automatic bitmap image.

Abstract

We describe Hilbert's spacefilling curve in several different ways: as an automatic sequence of directions,as a regular and synchronized sequence of coordinates of lattice points encountered, and as an automatic bitmap image.