  The spatial structure of the lowest 0$_1^+$, 0$_2^+$, 2$_1^+$ and 2$_2^+$
states of the $^{12}$C nucleus is studied within the 3$\alpha$ model with the
Buck, Friedrich, and Wheatley $\alpha \alpha$ potential with Pauli forbidden
states in the $S$ and $D$ waves. The Pauli forbidden states in the three-body
system are treated by the exact orthogonalization method. The largest
contributions to the ground and excited 2$_1^+$ bound states energies come from
the partial waves $(\lambda, \ell)=(2,2)$ and $(\lambda, \ell)=(4,4)$. As was
found earlier, these bound states are created by the critical eigenstates of
the three-body Pauli projector in the 0$^+$ and 2$^+$ functional spaces,
respectively. These special eigenstates of the Pauli projector are responsible
for the quantum phase transitions from a weakly bound "gas-like" phase to a
deep "quantum liquid" phase. In contrast to the bound states, for the Hoyle
resonance 0$_2^+$ and its analog state 2$_2^+$, dominant contributions come
from the $(\lambda, \ell)=(0,0)$ and $(\lambda, \ell)=(2,2)$ configurations,
respectively. The estimated probability density functions for the
$^{12}$C(0$_1^+$) ground and 2$_1^+$ excited bound states show mostly a
triangular structure, where the $\alpha$ particles move at a distance of about
2.5 fm from each other. However, the spatial structure of the Hoyle resonance
and its analog state have a strongly different structure, like $^8$Be +
$\alpha$. In the Hoyle state, the last $\alpha$ particle moves far from the
doublet at the distance between $R=3.0$ fm and $R=5.0$ fm. In the Hoyle analog
2$_2^+$ state the two alpha particles move at a distance of about 15 fm, but
the last $\alpha$ particle can move far from the doublet at the distance up to
$R=30.0$ fm.
