Poncelet's Theorem in the four nonisomorphic finite projective planes of order 9

TL;DR

This paper investigates Poncelet's Theorem within the four non-isomorphic finite projective planes of order 9, revealing that only the Desarguesian plane satisfies the theorem, unlike the others constructed over a miniquaternion near-field.

Contribution

It provides a complete analysis of Poncelet's Theorem in finite projective planes of order 9, distinguishing the Desarguesian plane as the unique Poncelet plane among them.

Findings

Only the Desarguesian plane satisfies Poncelet's Theorem.

The other three planes constructed over the miniquaternion near-field do not satisfy the theorem.

The study completes the classification of Poncelet's Theorem in these finite planes.

Abstract

We study Poncelet's Theorem in the four non-isomorphic finite projective planes of order 9. Among these planes, only the Desarguesian plane turns out to be a Poncelet plane, while the other three planes which are constructed over the miniquaternion near-field of order 9, are not. This gives a complete discussion of Poncelet's Theorem in finite projective planes of order 9.