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Another curious aspect concerning the Cairo tiling is its connection to the Type 13 convex pentagon (as discovered by the amateur mathematician 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 specific-angled 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 giving 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 how Rice discovered her example. Prima facie, it would seem likely that she would have used this simple process above, this being so simple, and thus more likely. However, this is not necessarily so, as in an article in The Mathematical Gardner, ‘In Praise of Amateurs’, where Doris Schattschneider 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, likely due to ill-health at the end of her life).
At the time of writing (2011), it was still an open problem whether there were more types of convex pentagons, where I made speculations about whether this simple procedure could lead to an undiscovered type being found. Subsequent to this analysis, in 2015, Casey Mann, Jennifer McLoud-Mann, and David Von Derau found such a 15th tiling and then in 2017, Michaël Rao found that the listing was indeed complete. Such matters aside, what is significant about this is in relation to the convex pentagon problem is the sheer ease with which 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.
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
References
Schattschneider, Doris. 'In Praise of Amateurs’. In The Mathematical Gardner, David Klarner, ed.
Page History
24 July 2025. Updated from the conversion. The conversion had left the captions detached from illustrations, which I now address. Also added references. An initial check of the text was made in Grammarly, with minor corrections, pending a more extended reappraisal. The page had dated somewhat, in that upon asking if a Type 15 pentagon was possible by such means, a 15th having been found subsequently by other means. I thus revise that paragraph.
Created: 24 November 2011
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