4. There are 57 small equilateral triangles with a side length of 1 that form a hexagon where all internal angles are no greater than 180 degrees. The small equilateral triangles fit together without gaps or overlaps. What is the minimum perimeter of this hexagon?
【Analysis】
To make the perimeter of the equilateral triangles stacked together the shortest while keeping the area unchanged, they should be arranged into a hexagon, with each internal angle of the hexagon being 120 degrees. Then construct a large triangle: divide each side of the large triangle into $n$ equal parts, and connect the $n$ equal division points on each side to form $n \times n$ small equilateral triangles.

【Solution】Solution: If we divide each side of an equilateral triangle into 2 equal parts, connecting the midpoints of each side will form $2 \times 2=4$ small equilateral triangles;

If we divide each side into 3 equal parts, connecting the trisection points of each side will form $3 \times 3=9$ small equilateral triangles; and so on, dividing each side into $n$ equal parts, connecting the $n$ equal division points on each side will form $n \times n$ small equilateral triangles.
$$
7 \times 7<57<8 \times 8<9 \times 9,8 \times 8=64,
$$
$64-57=7,7$ cannot be decomposed into the sum of three perfect squares,
$$
9 \times 9=81,81=4+4+16 \text {, }
$$

Therefore, we can place these 57 small triangles in an equilateral triangle as shown in the figure, with each side divided into 9 equal parts, $\triangle$ $A B C$ has a side length of 9, and a part of each angle is cut off,

At this point, $D E=5, E F=2, F G=3, G H=4, H I=3, D I=2$,
then $D E+E F+F G+G H+H I+D I=19$,
which means the perimeter of this hexagon is at least 19.
Answer: The perimeter of this hexagon is at least 19.
The answer is: 19.