Defining a Cairo-type Tiling

To begin, I have to admit, I am not entirely satisfied with this page; the sheer complexity of ‘defining a Cairo-type’ tiling became rather involved, with so many intricacies involved, and so the mathematician here may have considerable qualms and reservations about this page, to put it mildly. So be it. With this in mind, with considerable shortcomings and reservations, I offer the following as a first impression; it is at least a starting point for debate as to defining a Cairo-type’ tiling (note that I differentiate between Cairo-type and Cairo-like: ‘Cairo-like’ (defined below) is that of the standard instance commonly given, ‘Cairo-type’ is an extension of the premise.

How to exactly define a 'Cairo-type' tiling is not as straightforward a task as might otherwise be imagined. Although the ‘basic framework’ is established by convention, of a single, symmetric pentagon, with opposite 90° angles that in a tiling can be interpreted as of four pentagons that are suitably combined form two intersecting par hexagons at right angles, one can then vary this condition somewhat. By relaxing the above conditions, one can obtain a whole host of ‘interesting’ tilings that can loosely be described as of a ‘Cairo-type’, albeit admittedly weakened the further and further one strays from the above. For example, even with just the ‘one pentagon model’, one can have variations, with sides of different lengths, and change the proportion of the base from ‘long’ to ‘short’. One can also change the premise more dramatically:

Relax the core number of pentagons, with two, three, or four pentagons

Relax the sides conditions

Relax the 90° angles conditions

Relax the symmetry condition, with asymmetric pentagons.

Relax the convexity of the par hexagon condition, with concave par hexagons.

Relax the par hexagon condition altogether, with par octagons and par decagons

An open question is how best to present the material. One could have different formats, as arising from the varied conditions. However, what seems the most obvious, and is indeed adopted, is that of ordering as to the most basic condition, namely that of the number of pentagons involved. The following studies, grouped under 1, 2, 3, or 4 pentagons, are broadly some examples that arguably qualify as ‘Cairo-type’ tiling.

As alluded to in the introduction, this is not presented as perhaps as ideal as I would like, in that it is not a considered, systematic study per se, but rather with examples included as a result of various individual studies, in which I select individual tilings for their various properties as above. As ever, I am open to suggestions.

1 Pentagon

Figure 1

Typical ‘Cairo tiling’:

One pentagon, symmetrical, 4, 1 sides (four short, one long), two 90° opposite angles. As a tiling, two subsidiary convex par hexagons at right angles. As an aside, this example has lines of collinearity. A whole host of ‘special cases’ of the 1-pentagon can be formed, as discussed on the ‘aesthetics’ page.

Figure 2

Symmetrical, 3, 2 sides (two short, three long), three 90° angles. Two subsidiary convex par hexagons at right angles

2 Pentagons

Figure 3

Two subsidiary convex par hexagons at right angles

Figure 4

2 subsidiary concave par hexagons

Figure 5

1 subsidiary hexagon, only one side par

Figure 6

2 subsidiary convex hexagons, all sides par

Figure 7

2 subsidiary par hexagons, 1 convex, 1 concave

Figure 8

2 subsidiary par hexagons; 1 convex, 1 concave

Figure 9

Subsidiary par hexagons and par decagons; 1 subsidiary convex par hexagon, 1 subsidiary concave par decagon

Figure 10

Subsidiary par hexagons and par decagons; 1 subsidiary convex par hexagon, 1 subsidiary concave par decagon

3 Pentagons

Figure 11

2 subsidiary convex par hexagons, 1 completely par, 1 partial, with 1 side

4 Pentagons

Figure 12

2 subsidiary convex par hexagons, 1 completely par, 1partial, with 1 side

Figure 13

1 subsidiary convex par hexagon, 1 par octagon

Figure 14

2 subsidiary convex par hexagons, 1 completely par, the other one partial, 1 side

Created: Revised 25 January 2013