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Analysis of the In Situ Tiles as Regards Collinearity

Introduction
An aspect of interest concerning the in situ pavings is the matter of their analysis, in regards as to what angles the pentagons actually are, as well their possessing a interesting condition of collinearity that the subsidiary hexagons possess, the latter as noted by Robert H. Macmillan in his paper ‘Pyramids and pavements: some thoughts from Cairo’, in the Mathematical Gazette, December 1979, pages 251-255, a first-hand account. This study thus examines both of these aspects, of which in a sense they are to a certain extent intertwined.

Collinearity Premise
Of note is that in Macmillan’s paper, he refers directly to in situ collinearity of the tiling:
‘[if certain angles] … are collinear, the tessellation is especially pleasing to the eye, and this in fact is the proportion often* adopted in Cairo’
He then gives a mathematical explanation, in which he gives a pentagon with angles of 108° 26’ and 143° 8’ (I disregard the fixed angle of 90° here, as well as the figures repeating twice), albeit his presentation is not ideal, in that the diagram shown, Fig. 4 does not show this aspect, as well as there being no in situ pictures with overlays showing collinearity. (Simply stated, I now believe Macmillan is referring to a possibility of such an occurrence, and not holding Fig. 4 as an exemplar. This is the crucial aspect. Therefore, the diagram is not in error per se.) Essentially this thus sets the scene for collinearity, in which such a feature is established, if not entirely illustrated satisfactorily by Macmillan for reasons as above.

Collinearity Analysis
Of interest, without actual evidence, is thus to substantiate this claim, of which I examine the in situ tilings to see if this collinearity is indeed a feature, by the simple process of overlaying such lines and seeing if these are indeed collinear. However, although at first glance the task is complicated somewhat, given that the tiles in the in situ pictures differ considerably as regards format and colouring, much of this can be dismissed as irrelevant. Broadly, I differentiate two different types, of single pentagons and of pentagons in a square matrix. Both examples have colour variations. The single pentagons are in black, white, yellow and red, whilst the square format example are dull yellow, dull red, and a plain grey, the latter likely left being the same type as the ‘colours’, but left uncoloured. Furthermore, much of the colour variations are trivial, for example the square matrix tiles are arranged in stripes, of different unit thickness. Also, different colouring arrangements can be observed for the single pentagons. However, such colouring is an inconsequential matter here, in that it does not play a part in the analysis. What is important is determining this is the inherent distinctness of the tiles. Obviously, with single pentagons and square matrices these are obviously different. However, one could just possibly forward an argument that the plain grey is of a different type, despite the like format, and so for analysis purposes I include this as well, giving three types.
Furthermore, due to the tiles being actual physical objects (rather than the geometer’s theoretical lines), and with these being put in place not necessarily with the strictest degree of carefulness as regards the ‘perfect alignment’, especially of the square format type, the task of determining collinearity is thus made more difficult. Indeed, it can be seen that in some instances the proposed ‘correct’ collinearity is not seen, for this very reason. Therefore, one could thus quibble at being categorical here as to the existence of collinearity. However, as the individual coloured pentagons are placed (of necessity) considerably more accurately, from which one can then surmise the collinearity condition.

Figure 1: Collinearity

Figure 4: No collinearity likely due to improper placement of the tiles

As can be seen, each of the three types does indeed possess collinearity, as outlined by Macmillan. Therefore, at face value, it would indeed appear that the in situ tiles are as given by Macmillan, with angles of 108° 26’ and 143° 8’. Indeed, indisputably, he sets out a ‘fair case’ for the pentagons as to exactly the type and angles they consist of, almost beyond all reasonable doubt. However, this finding is not necessarily so! An alternative argument can also be for in situ tilings with tiles based on squared intersections. The example below of a square intersection type, of a 3 x 3 format, but can be scaled up, with 6 x 6, 9 x 9…), with a like collinearity, shows a tiling of a remarkably like appearance, with angles of 108° 43’ and 143° 13’ (as against Macmillan’s 108° 26’ and 143° 8’). For the sake of brevity and ease of reference, the two pentagon types here are referred to below as by Macmillan and Bailey. I might just add that I’ve examined larger formats, up to 12 x 12, and this is the only other example of collinearity I have found. Furthermore, in favour of this conjecture is that such square-based intersections are by far the easier construction; indeed, childlike in ease of drawing. Could one not suppose that the designer opted for the most obvious and easier option?

Summary

As such, with the collinear feature now firmly established, with all the in situ pictures possessing this feature, there can be no doubt whatsoever that the in situ pentagons were designed with collinear aspects in mind. Furthermore, as a corollary of this, as the commonly quoted examples of the equilateral or dual as being the Cairo pentagon do not possess collinearity, they cannot thus be the in situ pentagons!
Therefore, having established the collinearity principle, the one aspect remaining unresolved is to the exact angles of the pentagons. Although it might be thought that Macmillan has established this definitively, I now have doubts, in that I have shown in my studies that there is at least a second possibility. Indeed, one could put forward a case for this as being more likely, given its sheer simplicity of construction, as against Macmillan’s more mathematical example. An open question is to how he arrived at his angle determination; was it by seeing the blueprints, measuring or by mathematical argument alone? It would seem doubtful to say the least that he would have access to the blueprints or could indeed measure sufficiently accurately enough with a protractor as to the angles given. Therefore, it would seem likely he arrived at this purely mathematically. Therefore, if the latter, as would seem likely, he seems to have missed this alternative possibility.

*Another interesting aspect to Macmillan’s work on this is an indirect reference to possibly Cairo tiling in his and E. H. Lockwood’s book Geometric symmetry, of 1978, page 88, of which the following discusses, somewhat as an side or adjunct to collinearity. However, although specific as to types of tiling found, his account is frustratingly vague in terms of location, with

… the reciprocals of the tessellations 3². 4. 3. 4 and 3⁴. 6 are patterns of congruent pentagons such as often used for street paving in Moslem countries.

However, upon searching for such examples, I can find none. Given their relative simplicity, one would have thought that these would be common. The reference to ‘Moslem countries’ is decidedly vague; is he referring to Cairo, which would seem a fair supposition, given that he has clearly visited? Oddly, he makes no direct references here to the in situ pentagons as specifically described in his later pyramids and pavements 1979 paper, although he does refer to the 3². 4. 3. 4 tiling as a ‘special case’ there (amongst others), but apparently just as an abstract sense. However, possible support for this conjecture of such reciprocals existing is that amongst the in situ observations in the 1979 paper he gives, in regards to the Macmillan pentagon:

… and this is in fact the proportion often adopted in Cairo….

The operative word here is often, when he could have stated always. This suggests the possibility of variety. But perhaps I am reading too much into this choice of word; was this just a throwaway comment? It would now appear likely, given that there are no sightings of such types, and of the prevailing collinear types. Of note is that the in situ pictures are likely of same pentagon throughout, of either the Macmillan or Bailey types, albeit in two different formats. What could explain this dichotomy? Likely, the above sentence, of a relatively lightweight nature, was just a relatively throwaway comment, this being in nature of decidedly less exhaustive account than with the 1979 paper. Therefore, I find it highly unlikely that these two reciprocals (duals) are to be found. But that said, why would Macmillan refer to these if he had no evidence? It’s all rather mysterious.

Created 7 October 2011