The quaternionic cobordism ring $/varOmega_*^{Sp}$ is sandwiched between the stable homotopy groups of spheres and the complex cobordism ring $/varOmega_*^{U}$. When the prime 2 is inverted it becomes a polynomial algebra with one generator in each dimension $/equiv 0$ mod $4$, but its 2-primary structure is still unknown, despite more than 50 years of effort. Its torsion subring is particularly interesting, and contains a fascinating sequence of elements $/phi_i$, of order 2, in dimensions $/equiv 5$ mod $8$. I shall survey the definition, elementary properties, and geometric realisation of these elements, and explain how they have recently reappeared (in joint work with Yumi Boote) as structure constants in a Hopf algebra associated to the quaternionic bordism of one of the eight homogeneous spaces involved in Bott periodicity over $/mathbb{R}$. Rather than emphasising technical details, I shall try to make the material accessible to a wider topological audience. 

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