There is a tiling comprised of pure T4's, in which they lie along diagonals as shown in the left hand mosaic of Figure 9. They can share a half-plane with T1's, producing a fuse, or fill the plane by themselves alone. Another combination, shown on the right in Figure 9, mixes T1's and T4's, stacking the combination in vertical columns.
There is some delicacy involved in aligning vertical columns of tiles, because three ones in a row evolve to zero. Avoiding the combination while retainng the vertical margin means that an additional one sits either on the left of the spine or the right, but not both. Thus any top margin of a triangle at the right can only attach where there is a zero on the left.
In the case of the T1-T4 combination, there are two umbilical points, leading to the two different alignments which alternate in Figure 9. But there is no reason for the full tilling of the plane to be so regular, ammounting to an infinitude of phases.
No further mosaics depend on one single T tile, because of the difficulty in evenly filling the lower half of the implicit tile square. Smaller pieces can be incorporated and more elaborate staggering can be arranged, for which there is no lack at all of regular or semiregular tilings for the plane which incorporate at least one of the larger tiles.
Figure 9: To tile the plane, T4 triangles can be stacked diagonally,
an arrangement which is compatible with the T1 lattice. Otherwise they have
to be paired with some other, smaller, triangle such as a T1 diagonally
situated. Once triangles of disparate sizes occupy a unit cell, many
combinations still having similar areas are possible.