A History of Phylum Apicalia: Tapering Solids from Stone Tools to the Miluk ptsi·nł

The circle-to-line solid, a body tapering from a circular base to a linear apex, has been independently discovered, manufactured, and analyzed across multiple disciplines and millennia without any need for taxonomic classification. Yet there is value in shared language. This paper traces the dual history of this geometric family: first as a shape in material culture, present from the earliest known Acheulean industries (≈1.95 Ma) through modern industrial production; then as a subject of mathematical inquiry, from Archimedes' classification of conoids (c. 250 BCE) through Wallis's conocuneus (1684), Gardner's cork plug puzzle (1958), Laurentini's Visual Hull formalization (1994), and the recent Phylum Apicalia taxonomy [Anderson, 2026c].

We show that the same mathematical object (the intersection of orthogonally extruded silhouettes) was independently constructed in at least four disciplines (recreational mathematics, computer vision, engineering drawing reconstruction, and computational geometry) without cross-citation in the sources we surveyed, and that shared language across these traditions is now available as a cross-silo taxonomy. The attempt to translate the Miluk term ptsi·nł into its precise mathematical context catalyzed the taxonomic research presented in the companion papers, and revealed an opportunity to use systematic classification for disambiguation in CAD and other endeavors.