22. The diameter of a figure refers to the maximum distance between any two points in the figure. Given an equilateral triangle with a side length of 1, indicate how to cut it into two parts with a straight line so that when these two parts are reassembled into a figure, the figure has the maximum diameter.
(1) If the figure must be convex (i.e., the line segment connecting any two points in the figure is also in the figure).
(2) Other cases.
22. (1) The maximum value of the diameter is $\frac{\sqrt{13}}{2}$.

If the cutting line passes through one vertex and does not pass through the midpoint of the opposite side, there are the following three scenarios:

Since the diameter is generated by the distance between two endpoints, among the three figures, only $B C^{\prime}$ and $D D^{\prime}$ are new lengths, and
$$
\begin{aligned}
B C^{\prime} & \leqslant \frac{1}{2}\left(B D+D C^{\prime}+B A+A C^{\prime}\right) \\
& =\frac{3}{2}180^{\circ}, \\
\angle E C D+\angle A E D>180^{\circ} .
\end{aligned}
$$

Therefore, $\triangle E D C$ cannot be attached to the side $A E$ of the quadrilateral $A E D B$, otherwise the resulting shape is not convex. So $\triangle E D C$ can only be attached to $B D$. Regardless of how it is attached, the diameter is always $\leqslant \frac{3}{2}$.
$$
\begin{array}{l}
\text { If } B D=D E, \text { then } \\
\max \left\{A C^{\prime}, E C^{\prime}\right\} \\
\leqslant \frac{1}{2}\left(A B+B C^{\prime}+C^{\prime} D+D E+E A\right) \\
=\frac{1}{2}(A B+A C+B C)=\frac{3}{2}<\frac{\sqrt{13}}{2} .
\end{array}
$$

Therefore, if the resulting shape is convex, the maximum value of the diameter is $\frac{\sqrt{13}}{2}$.
(2) The maximum value of the diameter is 2.
Suppose a line divides $\triangle A B C$ into two parts $M, N$, and let $P$ be the common point of the two parts after they are folded. For any $X \in M, Y \in N$,
$$
X Y \leqslant X P+P Y \leqslant 1+1=2 .
$$

Thus, the diameter does not exceed 2.
Furthermore, according to the following assembly method, a shape with a diameter of 2 can be obtained, so the maximum value of the diameter is 2.