Clifford’s and Grace’s chains and their connection with n-dimensional polytopes

During the first half of the 19th century, a number of mathematicians turned their attention to two theorems regarding elementary geometry, theorems that were in some way interconnected: the Wallace-Steiner question and Miquel’s triangle theorem. These two theorems constituted the starting point for a few chains of configurations of points, lines and circles devised by Clifford and Grace. Furthermore, in more recent times several mathematicians have studied possible generalisations in n-dimensional spaces of both the Wallace-Steiner question and Miquel’s theorem. The present paper aims to examine the historical origins of the Clifford and Grace chains and their extension into spaces of more than two dimensions, and to analyse the correspondence that Longuet-Higgins identified at the end of the 20th century between these configurations and n-dimensional polytopes. This will highlight the unexpected connections between different theories: elementary geometry, the study of higher-order curves, and the connection with polytopes and block designs.