We discuss the similarities between BPS monopoles and Skyrmions, and point to an underlying connection in terms of rational maps between Riemann spheres. This involves the introduction of a new ansatz for Skyrme fields. We use this to construct good approximations to several known Skyrmions, including all the minimal energy configurations up to baryon number nine, and some new solutions such as a baryon number seventeen Skyrme field with the truncated icosahedron structure of a buckyball. The new approach is also used to understand the low-lying vibrational modes of Skyrmions, which are required for quantization. Along the way we discover an interesting Morse function on the space of rational maps which may be of use in understanding the Sen forms on the monopole moduli spaces.
Bibliographical note35pp including four figures, typos corrected, appearing in Nuclear Physics B