Trade-offs between tasks leads to phenotypes arranged in a polyhedron, whose vertices are archetypes that specialize in one task, in situations ranging from bacterial gene expression to animal morphology to cancer. a 90% of the variation in E. coli gene expression falls on a line due to a trade-off between tasks of growth and survival. Axes are percent of total promoter activity. b Morphology of Darwin’s ground-finch species falls on a triangle. Specialists in different diets are found near the three archetypes, and generalist are near the center of the triangle. A and B adapted with permission from AAAS from Shoval et al.23. c Single-cell gene expression of mouse intestinal progenitor cells fall on a tetrahedron, shown in principal components (PC) space. The four archetypal gene expression profiles correspond to fundamental progenitor cell tasks. Adapted from Korem et al.29. d Tumor gene expression profiles of eight cancer types fall on polyhedra. Individual tumors (dots) plotted in the space spanned by the first three gene expression PCs (TCGA, breast cancer from Metabric). Archetype (colored dots) number and position were inferred using ParTI. Inset: shuffled data has a convex hull (CH, pink) that fills less of the minimal enclosing (ME) triangle than the real data. The ratio of the CH area (or volume) and ME triangle area (or tetrahedron volume) was used to compute statistical significance.