The genome of the human immunodeficiency virus (HIV) is packaged within an unusual conical core particle located at the center of the infectious virion. The core is composed of a complex of the NC (nucleocapsid) protein and genomic RNA, surrounded by a shell of the CA (capsid) protein. A method was developed for assembling cones in vitro using pure recombinant HIV-1 CA-NC fusion proteins and RNA templates. These synthetic cores are capped at both ends and appear similar in size and morphology to authentic viral cores. It is proposed that both viral and synthetic cores are organized on conical hexagonal lattices, which by Euler's theorem requires quantization of their cone angles. Electron microscopic analyses revealed that the cone angles of synthetic cores were indeed quantized into the five allowed angles. The viral core and most synthetic cones exhibited cone angles of approximately 19 degrees (the narrowest of the allowed angles). These observations suggest that the core of HIV is organized on the principles of a fullerene cone, in analogy to structures recently observed for elemental carbon.