A centriole is a cylindrical organelle with a characteristic ninefold organization of microtubules and associated proteins. In many animal cells, two centrioles lie within the surrounding material of a centrosome. Much of the microtubule nucleation occurs in that surrounding material: a centriole and a centrosome are therefore different structural levels. A centriole can also become the basal body that organizes a cilium. [ref]
Molecular Biology of the Cell, 4th edition by Bruce Alberts and others. Chapter 16, Microtubules and the Centrosome; Cilia and Flagella
The geometry has a molecular basis. Structural and reconstitution experiments showed that SAS-6 family proteins form elongated dimers whose head-to-head interactions can assemble a cartwheel-like hub with radiating spokes. Mutations interfering with oligomerization impaired centriole formation. This connects a repeated protein interaction to the symmetry of a larger cellular structure. [ref]
Structural basis of the 9-fold symmetry of centrioles by Daiju Kitagawa and others. 2011, summary: SAS-6 oligomerization, structural model and recombinant Bld12p assembly
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