Another curious aspect concerning the Cairo tiling is its connection to the Type 13 convex pentagon (as discovered by Marjorie Rice), something that I haven’t seen mentioned before. Simply stated, by simply judiciously omitting and extending certain of the ‘Cairo lines’ from a specificangled Cairo tiling with collinearity properties, the Type 13 tiling is derived. The first example above, Figure 1a, is based on a Bailey pentagon with angles of 108° 43’ and 143° 13’ thus gives rise to the type 13 pentagon, with dashed lines showing how it is derived, and with for comparison purposes a line diagram on the right, Figure 1b. An alternative pentagon (by Macmillan) with angles of 108° 26’ and 143° 8’ is another possibility (not shown). Upon then drawing this, for general curiosity I then tried out variations with other aesthetic examples of the tiling, with the Cordovan pentagon, Figure 2a, angles of 112.5° and 135°; Equilateral, Figure 3a, with 114° 18’ and 131° 24’; and the Archimedean dual, Figure 4a, with 120°.
An open question therefore is to how Rice discovered her own example. Prima facie, it would seem likely that she would have used this simple process above, this being so simple, and so thus more likely. However, this is not necessarily so, as in an article in The Mathematical Gardner, ‘In Praise of Amateurs’, where Doris Schattsneider discusses her methods, her process, although not specifically addressed to the Type 13 pentagon, is largely one that can be described as of ‘vertex and angle concerns’ in an abstract sense, and certainly nothing like this procedure of my own. However, that is not to say that she didn't discover it this way (I asked her how she did this, but she didn't respond).
Such matters aside, what is significant about this is in relation to the convex pentagon problem is the sheer ease that a type 13 pentagon can indeed be discerned from this procedure, whether it was found by Rice with apparent angle consideration (although I have my doubts) or myself with judicious additions and extensions of existing lines. Digressing somewhat, with the above sheer ease of composing, and with in mind another simple composing with a different procedure of a type 9 by Richard James, in which he ‘rearranged’ a existing tiling, is a putative type 15 necessarily going to involve complex mathematics? Could it not be found by similar easy means as to the above? I see no reason why this shouldn't be so.

 
Figure 1a: Bailey pentagon   Figure 1b: Type 13 tiling as given by Marjorie Rice 
  
Figure 2a: Cordovan pentagon 
 Figure 2b: Variation of type 13 tiling 
  
Figure 3a: Equilateral pentagon   Figure 3b: Variation of type 13 tiling 
  
Figure 4a: Archimedean dual pentagon   Figure 4b: Variation of type 13 tiling 
Created: 24 November 2011