  The carbon atom provides the backbone for the complex organic chemistry
composing the building blocks of life. The physics of the carbon nucleus in its
predominant isotope, $^{12}$C, is similarly full of multifaceted complexity.
Some nuclear states of $^{12}$C can be preferentially treated as a collection
of independent particles held by the mean field of the nucleus, while other
states behave more as a collection of three alpha-particle clusters. But these
two pictures are not mutually exclusive, and some states can be described in
either fashion$^{1,2}$. In this work, we provide the first model-independent
tomographic scan of the three-dimensional geometry of the nuclear states of
$^{12}$C using the {\it ab initio} framework of nuclear lattice effective field
theory. We find that the well-known but enigmatic Hoyle state is composed of a
"bent-arm" or obtuse triangular arrangement of alpha clusters. We identify all
of the low-lying nuclear states of $^{12}$C as having an intrinsic shape
composed of three alpha clusters forming either an equilateral triangle or an
obtuse triangle. From these basic structural formations, the various nuclear
states correspond to different rotational and vibrational excitations as well
as either distortions or large-amplitude displacements of the alpha clusters.
The states with the equilateral triangle formation also have a dual description
in terms of particle-hole excitations in the mean-field picture. We compare our
theoretical calculations with experimental data for binding energies,
quadrupole moments, electromagnetic transitions, charge densities, and form
factors. The overall agreement is good, and further studies using
higher-fidelity interactions are planned.
