Here I describe the way I found my first tetrad with a hole in December 1970.
Step 1
At least one of the component regions of the tetrad has to be surrounded by the other regions.
Therefore a triangle shape for the whole tetrad should be suitable.
Inside this triangle an empty area is needed for the inner region.
In principle there are 3 ways to dissect the triangle frame into 3 congruent parts.
Step 2
The first frame is not suitable because the parts are too long and do not fit in the empty space.
For the other two frames tetrads can be constructed.
The shown tetrads have the maximal thickness for the component regions. (1/7 of the height of the whole tetrad)
Step 3
In 1971 I only considered the left solution with the bilateral symmetric tiles.
But I had the disection puzzle in mind and wanted a tetrad with only one simple hole.
Therefore the small triangular holes were added to the left and the right tile.
But then the upper (red) tile cuts off a part of the inner (yellow) tile.
This also has to be done for the other tiles.
Construction of one tile
Draw an equilateral triangle and dissect the base side into 4 equal parts.
With parallel and perpendicular lines the task can be completed.