Apollonian sets in taxicab geometry

TL;DR

This paper explores the properties of Apollonian sets within taxicab geometry, revealing their complex structure and relationships to simpler sets, extending classical Euclidean results to a non-Euclidean context.

Contribution

The paper characterizes Apollonian sets in taxicab geometry, showing they are not circles but can be described through simpler related sets, extending classical geometric concepts.

Findings

Apollonian sets in taxicab geometry are not circles.

Complex Apollonian sets can be characterized using simpler sets.

The study extends classical Euclidean results to taxicab geometry.

Abstract

Fix two points $p$ and $q$ in the plane and a positive number $k \neq 1$. A result credited to Apollonius of Perga states that the set of points $x$ that satisfy $d(x, p)/d(x, q) = k$ forms a circle. In this paper we study the analogous set in taxicab geometry. We find that while Apollonian sets are not taxicab circles, more complicated Apollonian sets can be characterized in terms of simpler ones.