Example 3 On the plane, there are $n(n \geqslant 5)$ distinct points, each point is exactly at a distance of 1 from four other points. Find the minimum value of such $n$. [2]
Keep points $A, B, D, E, G$ in Figure 1, construct $\square E A B C, \square D A G F, \square B A G I$ to get points $C, F, I$, and construct $\square C I F H$ to get point $H$.
$$
\begin{array}{l}
\text { From } B I=A G=A E=B C, \\
\angle I B C=\angle A B C-\angle A B I \\
=\left(180^{\circ}-\angle B A E\right)-\left(180^{\circ}-\angle B A G\right) \\
=\angle E A G=60^{\circ},
\end{array}
$$

we know that $\triangle B C I$ is an equilateral triangle.
Similarly, $\triangle F G I$ is an equilateral triangle.
Thus, $C E=A B=G I=F I=C H$.
And $\angle E C H=\angle G I F=60^{\circ}$, so $\triangle C E H$ is an equilateral triangle.
Similarly, $\triangle D F H$ is an equilateral triangle.
Therefore, the points that are 1 unit away from point $A$ are $B, E, D, G$; the points that are 1 unit away from point $B$ are $A, D, I, C$; the points that are 1 unit away from point $C$ are $B, E, I, H$; the points that are 1 unit away from point $D$ are $B, A, F, H$; the points that are 1 unit away from point $E$ are $A, G, H, C$; the points that are 1 unit away from point $F$ are $G, D, I, H$; the points that are 1 unit away from point $G$ are $A, E, I, F$; the points that are 1 unit away from point $H$ are $C, E, D, F$; the points that are 1 unit away from point $I$ are $B, C, F, G$. This is an example of $n=9$, as shown in Figure 2.