Domsliced TetrominoesIn order to make symmetric figures with the 10 domsliced tetrominoes it is necessary to ensure that we have a design which may be possible. To check on possibility we need to look at a chessboard colouring of the pieces. If we look at the pieces below we see that nine have a colouring of 1¾ to 1¼ an excess of ½ whereas one piece has a 2¼ to ¾ colouring which gives an excess of 1½.

Providing that we have no

cross grainjoins as in the figure above, whenever two half dominoes join they will for a domino which has a 1-1 colouring. Thus any figure with a border purely along grid lines will have all half dominoes joined into a domino and so the resulting figure must have a colouring excess of 2 squares.If two of the edges of the border are formed by the sloping part of the half domino then we can have a colouring excess of 2 ± ½ ± ½ = 1, 2 or 3 and with four sloping edges we have 2 ± ½ ± ½± ½ ± ½ = 0, 1, 2, 3 or 4. Examples of each of these types are shown below.

From the above we can see immediately that no rectangle following grid lines is possible with the set. It might, however, be possible to for a sloping rectangle such as the one below which has an area of 30 units. This also, is seen to be impossible, however, if we consider the four edges. Firstly no piece can contribute to more than one edge and so if we remove the edge pieces we shall have four sloping edges and an area in the centre made up entirely of squares. We can now join the two opposite long edges to form another piece made up entirely of squares and finally the four pieces forming the short edges can be joined by their colouring to form two more balanced blocks. These four pieces will have a balanced colouring just as the figure itself did and so we can see that no solution is possible.

Despite all these restrictions, a number of symmetrical shapes is indeed possible as shown here.

It is also possible to make single and duplicate copies of hexominoes. The hexomino must be one with a 4-2 colouring and this is probably only possible for the two shown here.

If, however, we allow the duplicated hexomino to be different from the original then there are many more possibilities although the single sized one must still have a 4-2 colouring.

Similar constructions can be made with two sets of pieces. This is also solvable using the one sided set (bottom figure) but it seems likely that this may be the only solution.

Also with the one-sided set it is possible to form quadruplicated copies of a tetromino with a hole in the shape of the tetromino. The T-tetromino is impossible to use because of colouring considerations.