  The $^{12}$C nucleus with $N$=6 and $Z$=6 is a doubly closed-shell nucleus in
a toroidal potential. In the description of the ground state and the Hoyle
state of $^{12}$C in the resonating group method or the generator coordinate
method, a superposition of the orientations of Wheeler's triangular cluster on
the cluster plane would naturally generate an intrinsic toroidal density. A
toroidal state also has a probability amplitude to overlap with a 3-alpha
cluster, which is the dominant decay mode for the Hoyle state. For these
reasons, we study a toroidal description of the states of $^{12}$C in the
toroidal configuration both phenomenologically and microscopically. A toroidal
$^{12}$C nucleus distinguishes itself by toroidal particle-hole multiplet
excitations between one toroidal single-particle shell to another. From such a
signature and experimental data, we find phenomenologically that the Hoyle
state and many of its higher excited states may be tentatively attributed to
those of the $^{12}$C nucleus in a toroidal configuration. We then study the
$^{12}$C system from a microscopic mean-field approximation using variational
wave functions. We find that the equidensity surfaces of the $^{12}$C ground
state exhibit a dense toroidal core immersed in lower-density oblate spheroids
in the surface region. Furthermore, there are prominent toroidal features of
the equidensity surfaces for the state at the Hoyle excitation energy, at which
previous cluster model calculations indicate the presence of a 3-alpha cluster
state. A toroidal coexistence model therefore may emerge to suggest the
possibility that the physical Hoyle state may have probability amplitudes to be
in the toroidal configuration and the 3-alpha cluster configuration.
