On iterated circumcenter sequences

TL;DR

This paper characterizes the parameter space of iterated circumcenter sequences in any dimension, proves Goddyn's conjecture on periodic sequences, and establishes the existence of periodic ICSs in all dimensions.

Contribution

It fully determines the parameter space of ICSs, proves Goddyn's conjecture, and shows that periodic ICSs exist in every dimension, advancing understanding of geometric iterative processes.

Findings

Complete parameter space characterization of ICSs

Proof of Goddyn's conjecture on periodic ICSs

Existence of periodic ICSs in all dimensions

Abstract

An iterated circumcenter sequence (ICS) in dimension $d$ is a sequence of points in $\mathbb{R}^d$ where each point is the circumcenter of the preceding $d+1$ points. The purpose of this paper is to completely determine the parameter space of ICSs and its subspace consisting of periodic ICSs. In particular, we prove Goddyn's conjecture on periodic ICSs, which was independently proven recently by Ardanuy. We also prove the existence of a periodic ICS in any dimension.