I will describe how an affinity between projective spaces and bordism rings extends further than previously known. The well-known manifestations of this affinity are that real projective bundles generate the unoriented bordism ring; that complex projective bundles generate the oriented bordism ring; and that quaternionic projective bundles "almost" generate the spin bordism ring---they generate the kernel of the Atiyah invariant. The new manifestation of this affinity is that Cayley plane bundles---that is, octonionic projective plane bundles---almost generate string bordism after inverting 6---they generate the kernel of the Witten genus. The key behind this is that the arithmetic of Cayley plane bundle characteristic numbers arising in Borel-Hirzebruch Lie-group-theoretic calculations align with perfectly with arithmetic arising in the Hovey-Ravenel-Wilson BP-Hopf-ring-theoretic calculation of string bordism MO at primes >3. I will also point out that tmf is not a ring spectrum quotient of string bordism.