Supplemental Table 1Database of figured and reposited Helicoprion specimens. Column abbreviation and notes: #, database index number; Inst Acro, Institutional acronym for specimen; Spec., Specimen number; Notes & Alt name, notation on synonymous specimen numbers and counterparts (cp); Locality#, registered locality number; Lat, latitude in decimal degrees; Long, longitude in decimal degrees; Place, town or region; Country, country; Museum, institutional abbreviations listed below; Stratigraphy, formation and member; Stage, based on current stratigraphy, Art=Artinskian, Kung=Kungurian, Leon= Leonardian, Olen=Olenekian, Road=Roadian, Wolf=Wolfcampanian; Original ID, specimen diagnosis according to reference or repository, [H]=holotype, [L]=lectotype; Whorls, specimen quality including number of volutions (w), or number of teeth (t), whether partial (p) or a complete whorl (c), frags=fragments; Diagnosis, our assessment according to the emended diagnoses presented in this paper. Institutional Abbreviations.—AMNH, American Museum of Natural History, New York, USA; BANFF, Banff Park, Alberta, Canada; ER, Ekaterinburg Regional Natural History Museum, Russia; FCMP, Museo de Paleontologia de la Facultad de Ciencias, Universidad Nacional Autonoma de Mexico; FH, Fort Hall Reservation, Shoshone Bannock Tribal Museum, Idaho, USA; FIELD, Field Museum, Chicago, USA; GSA, Geological Society of America Office, Denver, USA; IMNH, Idaho Museum of Natural History, Pocatello, USA; KCL, Kesen-numa City Library, Japan; KSU, Geological Museum of Kazan State University, Russia ; LACM , Los Angeles County Museum, USA; MNH, Museum of Natural History, University of Kansas, Lawrence, USA; NHMU, Natural History Museum of Utah, Salt Lake City, USA; NMC, Canadian Museum of Nature, Ottawa, Canada; PIN, Palaeontological Institute of the Russian Academy of Sciences, Moscow, Russia; PERM, Perm Regional Natural History Museum, Russia; PMO, Paleontologisk Museum, Oslo, Norway; PRIV, Private Collection; TsNIGR, Central Research Geological Museum, St.-Petersburg, Russia; UCMP, University of California Museum of Paleontology, Berkeley, USA; UFH, Utah Field House of Natural History State Park Museum, Vernal, USA; UMUT, University Museum, University of Tokyo, Japan; UNR, W.M. Keck Earth Science and Mineral Engineering Museum, University of Nevada, Reno, USA; USNM, National Museum of Natural History, Washington DC, USA; YIGM, Yichang Institute of Geology and Mineral Resources, China; WAMAG, Western Australian Museum, Perth, Australia.SupplentalTable1.xlsxSupplemental table 2First two principal components for PC analyses in Figures 9 and 10. Current and previous species diagnosis reported where appropriate. *Abbrev: (H) holotype, (L) lectotype, (T) type; **ital.=estimated tooth number from Table 7.

The genus Helicoprion (Chondricthyes: Euchondrocephali) is preserved primarily by a continuous spiral root that forms the base for more than 130 teeth. Helicoprion is found globally in Lower Permian marine rocks and at least 100 specimens exist in public collections worldwide. Ten species of the genus are reviewed in the context of new morphometric analyses conducted on dozens of specimens. Helicoprion whorls share a common inner spiral geometry that results in exponential growth of tooth and root dimensions. Novel growth equations permit calculation of whorl diameter, volution number and tooth count from incomplete specimens. The results of meristic and geometric analyses identify taxobases that distinguish three emended species concepts. Differentiation of form is evident among specimens only after the second volution or roughly the 85th tooth. Helicoprion davisii has widely-spaced, stout teeth with tall cutting surfaces and is distinguished from H. bessonowi, whi ch has narrow, closely-spaced teeth with short cutting surfaces. H. ergassaminon is an intermediate form, having narrow, closely-spaced teeth with tall cutting surfaces. Several large specimens in the study are too dissimilar to place in the new emended species concepts.