Understanding the fundamental excitations of many-fermion systems is of sig
nificant current interest. In atomic nuclei with even numbers of neutrons a
nd protons, the low-lying excitation spectrum is generally formed by nucleo
n pair breaking and nuclear vibrations or rotations. However, for certain n
umbers of protons and neutrons, a subtle rearrangement of only a few nucleo
ns among the orbitals at the Fermi surface can result in a different elemen
tary mode: a macroscopic shape change(1-3). The first experimental evidence
for this phenomenon came from the observation of shape coexistence in O-16
(ref. 4). Other unexpected examples came with the discovery of fission iso
mers(5) and superdeformed nuclei(6). Here we find experimentally that the l
owest three states in the energy spectrum of the neutron deficient nucleus
Pb-186 are spherical, oblate and prolate. The states are populated by the a
lpha-decay of a parent nucleus; to identify them, we combine knowledge of t
he particular features of this decay(7) with sensitive measurement techniqu
es (a highly efficient velocity filter(8) with strong background reduction,
and an extremely selective recoil-alpha-electron coincidence tagging metho
d(8-10)). The existence of this apparently unique shape triplet is permitte
d only by the specific conditions that are met around this particular nucle
us.