Here I describe the way I found my first tetrad with no hole in January 1971.
My first attempt did not make use of a square grid or anything similar. Probably I was unfamiliar with triangular or hexagonal grids at that time.
Therefore my first tetrad is rather complicated, but the construction strategy is straightforward.
Idea 1
Start with 4 triangles because each pair already has at least one point in common.
Idea 2
Add a rectangle. Then the left and the right triangles have a common border.
Idea 3
Give the triangles a greater altitude, and change the orientation of the upper triangle.
Idea 4
Move the upper triangle downwards in order to touch all other triangles.
Problem
A part of the area of the upper (red) triangle was cut off.
The same has to be done with the other triangles.
Problem
There are two holes in the tetrad now.
The unique shape of the hole has to be pasted on each triangle.
Done! This was my first tetrad without holes.
It looks rather strange compared to the polyamond solution of Scott Kim and Frank Rubin.
Explanation of a detail
In general you obtain the tetrad shown below. Here the edge e is shorter than the edge d.
If e = d the tetrad looks a little nicer. 6 vertices vanish. This is the solution shown above.