A new simple family of Cantor sets in $\mathbb{R}^3$ all of whose projections are one-dimensional

TL;DR

This paper introduces a new, simple family of self-similar Cantor sets in three-dimensional space whose projections onto any plane are connected and one-dimensional, expanding the understanding of projection properties of fractal sets.

Contribution

It presents a novel family of Cantor sets in ^3 with all projections being connected and one-dimensional, building on classical self-similar constructions.

Findings

All projections are connected and one-dimensional.

The sets are self-similar and related to Antoine's work.

The construction celebrates a centenary milestone.

Abstract

In 1994, J.Cobb described a Cantor set in $\mathbb{R}^3$ each of whose projections into 2-planes is one-dimensional. A series of works by other authors developing this field followed. We present another very simple series of Cantor sets in $\mathbb{R}^3$ all of whose projections are connected and one-dimensional. These are self-similar Cantor sets which go back to the work of Louis Antoine, and we celebrate their centenary birthday in 2020-2021.