As the most primitive organisms, occupying the gray area between the living and nonliving, viruses are the least complex biological system. One can begin to think about them in a quantitative way, while still being at some level faithful to biochemical processes. We make some observations about their structure, formalizing in mathematical terms some rules-of-construction discovered by Watson and Crick and Caspar and Klug. We call the resulting structures objective structures. It is then seen that objective structures include many of the most important structures studied in science today: carbon nanotubes, the capsids, necks, tails and other parts of many viruses, the cilia of some bacteria, DNA octahedra, buckyballs, actin and collagen and many other common proteins, and numerous atomic-scale rods, springs and wires now being synthesized. Objective structures also have an intriguing relation to the crystalline and noncrystalline structures adopted by elements in the Periodic Table. The rules defining them relate to the basic invariance group of quantum mechanics. We develop a methodology for computing such structures. Some of the nonperiodic structures revealed by the formulas exhibit beautifully subtle relations of symmetry. This common mathematical structure paves the way toward many interesting calculations for such structures: the likelihood of unusual electromagnetic and other collective properties, simplified schemes for exact molecular dynamics of such structures, phase transformations between them, defects and failure, new x-ray methods of determination of structure not relying on crystallization, and their growth by self-assembly.