4. (40 points) 64 points on a plane form an $8 \times 8$ grid. The distance between any two adjacent points in the same row or column is 1. How many rectangles with an area of 12 can be formed using four of these 64 points as vertices?
4. Connecting points in the same row and column can form a grid.

First, consider rectangles with sides parallel to the grid lines. For a rectangle of shape $2 \times 6$, there are two orientation choices (horizontal and vertical). One pair of opposite sides has $8-6=2$ choices, and the other pair has $8-2=6$ choices. Therefore, there are $2 \times 2 \times 6=24$ such rectangles.

Similarly, for a rectangle of shape $3 \times 4$, there are $2 \times 4 \times 5=40$ such rectangles.

Next, consider rectangles with sides parallel to the diagonals of the grid, with shapes $\sqrt{2} \times 6 \sqrt{2}$ and $2 \sqrt{2} \times 3 \sqrt{2}$.

These rectangles also have two orientation choices. First, consider the direction where the longest side extends from the bottom left to the top right. For the first shape, there is only 1 such rectangle (as shown in Figure 12); for the second shape, there are 9 such rectangles, one of which is marked in Figure 12. The top-left vertices of these 9 rectangles are marked with black dots in Figure 12. Therefore, there are $2 \times (1+9)=20$ such rectangles.

Since $12=2^{2} \times 3$, and 3 cannot be expressed as the sum of two perfect squares, there are no other shapes of rectangles.
Therefore, the total number of rectangles that meet the criteria is $24+40+20=84$.