This method retains the premise of stick crosses of studies 1-2, but with the notable difference in that this is not confined to a square or rectangular matrix, nor indeed reflected. Here, the stick crosses are placed as according to the orthogonals of the squared paper, with the premise of allowing the crosses to ‘drop’. As a consequence, this gives a most interesting Cairo-like tiling with the following properties:

1. One pentagon only

2. The number of 90° angles can vary, from two (mostly) to three (few), the latter being a ‘special case’ (involving, or what can be interpreted as a Greek cross)

As can be seen, this slightly ‘weakens’ the premise of the typical Cairo-type, in that the tiles, with the exception of the ‘special case’ is no longer symmetrical, but is still worthy in my opinion of the designator ‘Cairo-type’.

Tilings

As can be seen, this procedure gives rise to two different pentagons. Although these are not symmetrical, they nonetheless give cause of interest. Perhaps of most note is that the subsidiary hexagons are now no longer par for all three sides; only one is. As can be seen, this slightly ‘weakens’ the premise of the typical Cairo-type (i.e. of subsidiary par hexagons), in that the tiles are no longer as aesthetic with all sides par hexagon, but instead are obviously more irregular. However, even with this weakened ‘feature’, they are still worthy in my opinion of the designator ‘Cairo-type’, and in any case are of interest in their own right. One interesting feature of these is the distribution of the pentagons, in that the like pentagons are ‘contiguous’, rather than at opposite ends of the subsidiary hexagon as occurs with other, non-related example
Of note is that this is yet another example of an ‘infinite’ type of composing, of which those shown is just a sample of the first few. As can be seen, the pentagons arising (and subsidiary hexagons) are the ‘same’ in all instances, from which one can extrapolate that all others must be of the same type.

Another Variation

Premise
This method retains the premise of stick crosses of the type as evinced by the above, but with the notable difference of both equal arm and asymmetrical arm crosses (previously, the study above was of like equal arm crosses, of 2 and 4 units), placed on the vertices of a square with a gradient (i.e. ‘drop’). Here, the stick crosses are placed as according to the orthogonals of the squared paper, with the premise of allowing the crosses to ‘drop’.

Tilings
As these are of the infinite type, of necessity I restrict the drawings to the first few gradient types. Broadly, a few suffice to show the general principle, and so for reasons of conciseness further examples are not shown.
Of note is that all the examples are based on a single type of unit size of cross, namely the 4, 4 and 4, 6. Obviously, this is, to say the least, inadequate, of which one could be accused of poor scientific practice, and with good reason for such a restricted sample. However, due to the time consuming nature of the diagrams, all drawn by hand, I curtail this, at least for know.

Figure 17: 4, 4 and 4, 6 units, gradient 1:6, 4 asymmetrical pentagons, two subsidiary pentagons, one wholly par, the other just one side

Figure 18: 4, 4 and 4, 6 units, gradient 1:7, 4 asymmetrical pentagons, two subsidiary pentagons, one wholly par, the other just one side

Figure 19: 4, 4 and 4, 6 units, gradient 1:8, 4 asymmetrical pentagons, two subsidiary pentagons, one wholly par, the other just one side

Figure 20: 4, 4 and 4, 6 units, gradient 2:5, 4 asymmetrical pentagons, two subsidiary pentagons, one wholly par, the other just one side

Figure 21: 4, 4 and 4, 6 units, gradient 2:6, 4 asymmetrical pentagons, two subsidiary pentagons, one wholly par, the other just one side