On Bricardʹs doubly collapsible octahedron and its planar, spherical and skew counterparts

The most intriguing of Bricardʹs “paradoxical” linkages are those manifested by his three solutions of the “deformable” octahedron. Bennettʹs perception of planar and spherical counterparts to the spatial networks elicits further marvelling at these remarkable chains. Recent studies have provided the necessary algebraic bases for establishment of a metrical correspondence among the analogues, and the two symmetrical forms of the octahedron have been fully analysed in that context. This paper completes the task by examining the “doubly collapsible” case, considerably more complex than the others but far richer in its associations. In doing so, there is revealed a new six-bar kinematic loop which incorporates two categories of linkages previously regarded as distinct.