When Gr\"unbaum meets Poncelet -- Infinite Classes of Movable $n_4$ Configurations

TL;DR

This paper explores the connection between Poncelet's Porism and infinite classes of movable $(n_4)$ configurations, demonstrating that all trivial celestial configurations are movable through Poncelet's properties and related geometric methods.

Contribution

It establishes that all trivial celestial $(n_4)$ configurations are movable, linking classical Poncelet theory with the geometry of these configurations and expanding understanding of their flexibility.

Findings

All trivial celestial configurations are movable via Poncelet's Porism.

The Gr"unbaum-Rigby configuration admits nontrivial motions.

Connections to billiards, in-circle nets, and pentagram maps are discussed.

Abstract

We study relations between $(n_4)$ incidence configurations and the classical Poncelet Porism. Poncelet's result studies two conics and a sequence of points and lines that inscribes one conic and circumscribes the other. Poncelet's Porism states that whether this sequence closes up after $m$ steps only depends on the conics and not on the initial point of the sequence. In other words: Poncelet polygons are movable. We transfer this motion into a flexibility statement about a large class of $(n_4)$ configurations, which are configurations where 4 (straight) lines pass through each point and four points lie on each line. A first instance of such configurations in real geometry had been given by Gr\"unbaum and Rigby in their classical 1990 paper where they constructed the first known real geometric realisation of a well known combinatorial $(21_4)$ configuration (which had been studied by Felix Klein), now called the Gr\"unbaum-Rigby configuration. Since then, there has been an intensive search for movable $(n_4)$ configurations, but it is very surprising that the Gr\"unbaum-Rigby $(21_4)$ configuration admits nontrivial motions. It is well-known that the Gr\"unbaum-Rigby configuration is the smallest example of an infinite class of $(n_4)$ configurations, the trivial celestial configurations. A major result of this paper is that we show that all trivial celestial configurations are movable via Poncelet's Porism and results about properties of Poncelet grids. Alternative approaches via geometry of billiards, in-circle nets, and pentagram maps that relate the subject to discrete integrable systems are given as well.