4. Let $\mathrm{i}=\sqrt{-1}$, define a complex number sequence as follows:
$$
z_{1}=0, z_{n+1}=z_{n}^{2}+\mathrm{i}, n \geqslant 1 \text {. }
$$

Then, in the complex plane, the distance from $z_{111}$ to the origin is
$\qquad$
$4 \cdot \sqrt{2}$.
After calculation, we get
$$
z_{1}=0, z_{2}=\mathrm{i}, z_{3}=\mathrm{i}-1, z_{4}=-\mathrm{i}, z_{5}=\mathrm{i}-1 \text {. }
$$

Since $z_{3}=z_{5}$, then
$$
z_{111}=z_{109}=z_{107}=\cdots=z_{5}=z_{3}=\mathrm{i}-1 .
$$

Therefore, the distance from $\approx_{111}$ to the origin is $\sqrt{2}$.