Collinearity

Or

An Examination of the In‑Situ Cairo Tiles: Alignments and Angles 

Introduction
An aspect of interest concerning the in‑situ pavings is their analysis with respect to collinearity. This was first reported in 1979 by Robert H. Macmillan in his paper Pyramids and pavements: some thoughts from Cairo. His account is notable as a first‑hand observation, one of the few, and the second after James Dunn (in the same journal) of 1971 (who did not discuss collinearity). A further matter of interest is the question of what angles the pentagons possess. This study examines both aspects, with the emphasis on collinearity, although the two are, to a certain extent, intertwined.

Earliest Known Collinearity Attribution
Macmillan’s reference to collinearity is the earliest known. Dunn’s omission is likely explained by his misapprehension that the pentagon was equilateral — which it is not — and an equilateral model does not exhibit collinearity, as shown below. Martin Gardner, basing his discussion on Dunn, likewise does not mention collinearity. In the same vein of misattribution, Doris Schattschneider gives the model as 3.3.4.3.4, which again does not possess collinearity. It was ever thus. Incidentally, Macmillan appears to have been unaware of Dunn’s account; he does not cite it.

Macmillan’s Paper Drawbacks
A drawback of Macmillan’s paper is its presentation, which is unclear and vague in places, despite some rigorous mathematics. It would have been interesting to hear his account directly. To this end, having found his phone details, Brian Wichmann — an intermediary with an interest in the Cairo tiling and who had met him previously — contacted him in January 2011. He found Macmillan hazy and unable to recall details, and nothing was established. At this stage he was ninety. I then phoned myself, but likewise found him hazy, and although he invited me to visit his home in Milton Keynes, I decided against it, believing nothing could be achieved. He died in 2015.

Macmillan’s Cairo Visit
Of interest, really only for curiosity, would be to know the circumstances and date of Macmillan’s visit. The article does not state this (unsurprisingly), merely noting: “On a recent visit to Cairo I was struck by two matters…” — referring to the pyramids and the tiling. Nor does his earlier Geometric symmetry, 1978, which he co‑authored with E. H. Lockwood (allthough Lockwood was the senior partner; see afterword), give any detail; almost certainly this part of the book came from Macmillan. His Wikipedia page likewise gives no information, nor do his other writings that I have examined. From the wording “recent”, one may infer that the visit occurred within a year or two of publication, so around 1976–1977 would be likely.
Incidentally, I am most impressed with Macmillan the man. His list of achievements, as given on his Wikipedia page, is formidable. I am duly humbled.

The Collinearity Condition
Of note is that in Macmillan's paper he refers directly to the in‑situ collinearity of the tiling, page 255, (iv), and refers back to p. 253, Figs. 1a and 1b:
(iv) If 𝝓 is such that, in Fig. 4, AB and CD are collinear, the tessellation is especially pleasing to the eye, and this in fact is the proportion often* adopted in Cairo.

Figure 1. Pages from The Mathematical Gazette discussing collinearity

He then gives a mathematical explanation, specifying a pentagon with angles of 108°26′ and 143°8′ (disregarding here the fixed angle of 90°, together with the repeated occurrences of those angles). However, his presentation is not ideal. Figure 4 does not itself illustrate the collinearity condition, nor does the paper contain any in‑situ photographs with overlays indicating such alignments, or indeed any photographs at all.

At first sight this is puzzling, as Fig. 4 clearly lacks the stated property. However, I now believe that Macmillan was referring to the possibility of such an occurrence, rather than presenting Fig. 4 as an exemplar of it. This is the crucial point. The diagram is therefore not in error per se. Rather, it establishes the mathematical condition under which collinearity occurs, even if that feature is not illustrated as clearly as might be desired.

Collinearity Claim

To examine this claim, I overlay appropriate lines on photographs showing the pentagon in two settings: a square matrix and a single pentagon (Figures 2a and 2b).

A possible objection is that these perspective photographs do not portray the tiles in their true orthogonal form. However, I was reassured by an early collaborator in my investigations, the mathematician John Sharp, that such overlays remain valid regardless of the viewing angle.
Furthermore, because the tiles are physical objects rather than the geometer's theoretical "perfect" lines, and because they were laid by workmen not necessarily striving for perfect alignment, particularly in the square-format type, determining collinearity is made more difficult. Indeed, in some instances the proposed collinearities are not apparent for precisely this reason. Consequently, one could reasonably hesitate to be wholly categorical regarding the existence of collinearity.
Nevertheless, these objections can be overcome. As can be seen, the tiling does indeed possess the collinearity described by Macmillan.

Collinearity Angle Analysis
Below, I show a drawing of the pentagons satisfying the collinearity condition, together with their angles (Fig. 3). However, at first glance, the angles appear not to match. Macmillan gives 108°26′ and 143°8′, whereas I give 108°43′ and 143°.13′. Is there an error here?
For a long time, this remained an unresolved problem. However, I am pleased to say that the issue was finally resolved on 11 May 2020. I posted the apparent discrepancy on Twitter, and Daniel Piker kindly and swiftly provided the answer. He took ten minutes to resolve what I had been unable to explain in ten years!
What he pointed out was that the discrepancy arose from the use of different angular formats. Macmillan expressed his angles in degrees and arcminutes, whereas GeoGebra displayed them in decimal degrees. In my ignorance, I had not appreciated the distinction. Once this was recognised, and the values converted into a common format, the figures matched perfectly. Thanks, Daniel!
I had effectively been comparing two different systems of measurement, hence the apparent inconsistency. Therefore, Macmillan's pentagon and my own are identical, differing only in their mode of presentation. There is thus only one pentagon possessing this particular line of collinearity.

https://rechneronline.de/winkel/degrees-minutes-seconds.php
108°26’ converted to decimal = 108.43333. GeoGebra (decimal) gives 108.43 (two decimal places)
143°8' converted to decimal = 143.13333. GeoGebra (decimal) gives 143.13 (two decimal places) 

Line Drawings

Further to the above, I also examined other tilings (erroneously) said to be of the Cairo tiling, namely the equilateral pentagon (Macmillan Fig. 4a), and the Archimedean dual 3.3.4.3.4 (Fig. 4b) The equilateral pentagon and 3.3.4.3.4 clearly do not possess such collinearity, and so can this easily be dismissed by simple visual inspection. Of course, due to their respective properties, they remain of more interest than other generic Cairo tilings, but they are not of the Cairo tiling. 

Some Examples with Alignment of the Intersection of Vertices
I also examined other basic instances, with the Cordovan Pentagon (derived from the dissection of a regular octagon) (Fig. 5a), as well as another, of my own devising, 'Bailey Pentagon' with foresight, from my studies (Fig. 5b). The Cordovan Pentagon and Bailey Pentagon can be seen to possess a degree of alignment (and not collinearity), with the lines meeting on a vertex, albeit in different ways, of adjoining pentagons. All are more easily seen than described. However, both lack the collinearity aesthetics of the Cairo tiling.

Summary

This article has examined the collinearity properties of the Cairo tiling, focusing on observations first reported by Robert H. Macmillan in 1979. It argues that Macmillan's paper contains the earliest known reference to the in-situ collinearity of the tiling, a feature overlooked by earlier and later authors who worked from incorrect equilateral or related pentagonal models.

The author reviews Macmillan's contribution, notes shortcomings in the presentation of his paper, and recounts unsuccessful attempts in 2011 to obtain further clarification directly from Macmillan shortly before his death. Although the circumstances of Macmillan's visit to Cairo remain unknown, it is inferred that it probably took place shortly before publication of his paper.

A detailed examination is then made of Macmillan's famous statement that the Cairo tessellation is especially pleasing when certain edges are collinear. The author argues that Macmillan's figure was not intended to illustrate collinearity itself, but rather the mathematical condition under which collinearity occurs. This resolves an apparent contradiction between Macmillan's text and diagram.

To test Macmillan's claim, the author overlays lines on photographs of actual Cairo paving. Despite possible objections arising from photographic perspective and imperfections in the laying of the tiles, the overlays show that the pavements do indeed exhibit the collinearity described by Macmillan.

The article also investigates an apparent discrepancy between Macmillan's published pentagon angles (108°26′ and 143°8′) and later GeoGebra constructions. A long-standing puzzle was resolved in 2020 when Daniel Piker pointed out that Macmillan used degrees and arcminutes whereas GeoGebra displayed decimal degrees. After conversion, the values were found to be identical, confirming that there is only one pentagon satisfying the collinearity condition.
Finally, the author compares the Cairo pentagon with several related or purported Cairo-tiling models, including the equilateral pentagon, the Archimedean dual 3.3.4.3.4 tiling, the Cordovan pentagon, and a "Bailey pentagon". The study concludes that only the true Cairo tiling exhibits the distinctive collinearity aesthetic identified by Macmillan, while the other examples either lack collinearity entirely or possess only weaker forms of vertex-based alignment. 

References

James Dunn, ‘Tessellations with Pentagons’, The Mathematical Gazette, 1971, pp. 336–369.

Martin Gardner, ‘On tessellating the plane with convex polygon tiles’, Scientific American, 1975, pp. 112–117.

Doris Schattschneider, ‘Tiling the Plane with Congruent Pentagons’, Mathematics Magazine, 1978, pp. 29–44. 

Robert H. Macmillan, ‘Pyramids and pavements: some thoughts from Cairo’, The Mathematical Gazette, December 1979, pp. 251–255.

Robert H. Macmillan Biography (the Wikipedia page is derived from his obituary page below).

https://en.wikipedia.org/wiki/Robert_Macmillan

Obituary by Sebastian Macmillan (son-in-law), in Emmanuel College Magazine, Vol. XCVII 2014–2015.
https://www.emma.cam.ac.uk/publications/College%20Magazine%202015.pdf

Afterword
Another interesting aspect of Macmillan's work is a possible indirect reference to the Cairo tiling in Geometric symmetry (1978), co-authored with E. H. Lockwood. On page 88, the authors state:

...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.

The proximity in date to Macmillan's 1979 paper raises the possibility that the Cairo tiling is being referred to here. However, although the statement is specific regarding the types of tiling, it is frustratingly vague regarding their location.

Searching for examples of such street paving has proved unproductive. Given the relative simplicity of the tilings concerned, one might expect examples to be commonplace. The reference to "Moslem countries" is particularly imprecise. Was Cairo intended? Such a supposition would not be unreasonable, given that Macmillan had clearly visited the city.

Oddly, no direct reference is made to the in-situ pentagons discussed in “Pyramids and pavements” (1979). Although Macmillan refers there to the 3².4.3.4 tiling as a "special case", the discussion appears entirely abstract. This seems difficult to reconcile with the earlier statement.

One possible point in favour of the conjecture is Macmillan's remark concerning the Cairo pentagon:

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

The operative word is often. He could have written always. The choice perhaps hints at the existence of alternative pentagonal paving forms. However, this may be reading too much into a passing remark.

Indeed, all currently known in-situ examples appear to use the same pentagon, albeit arranged in two different formats. No examples of the reciprocals mentioned in Geometric Symmetry have come to light. The most likely explanation is therefore that the statement was intended only as a broad observation in a book whose treatment is considerably less detailed than that of the 1979 paper.

Nevertheless, the matter remains something of a mystery. If no such examples existed, why did Macmillan mention them? Without further evidence, the question must remain open..
*According to Macmillan's Wikipedia entry, E. H. Lockwood had written six chapters of Geometric symmetry before seeking Macmillan's assistance. Macmillan subsequently wrote two further chapters, and the book was published under their joint names. The specific chapters are not named. There are 27 chapters, in two parts. But from what we know, Macmillan's writing is to be found in Chapter 13, “Making patterns”.

Page History

27 October 2026. Although the same broad format has been retained, with the same structure and illustrations (apart from the addition of a banner image), a number of notable improvements have been made. These include (a full listing is given in the working Google Doc):

(i) Following AI-assisted review, numerous passages were identified where the text could be improved, many of which have been adopted.

(ii) Upon reflection, I added the relevant pages from The Mathematical Gazette to provide greater clarity regarding Macmillan's statement on collinearity.

(iii) Previously, several illustrations were placed one beneath another. Where these form a natural theme, they are now presented side by side. This saves space and facilitates comparison. Also, in general, I have manually reduced the images. Previously, for a printout, these have been shown “over large”.

(iv) The angle notation was not as accurate as I had thought. Previously, I wrote angles such as 108° 26' using both a space between the degree and minute values and an apostrophe for the minute symbol. On 26 October, I learned that both usages were incorrect. There should be no space, and the prime symbol (′) should be used instead of an apostrophe. In my defence, GeoGebra renders the notation in this manner, but I have deferred to the standard convention used by Macmillan and in mathematical practice generally.
(v) Added a summary and references. 

16 July 2025. New Sites improvement. Grammarly checked and corrected pending a more extensive reappraisal, with more errors than I would like to admit, in common with nearly all pages. That said, the content has stood the test of time; no wholesale rewrites were necessary, just minor adjustments of words, and not sentences.

22 June 2020. In light of the Piker finding, upon reflection, I decided to drastically rewrite the page, omitting much speculative material regarding Macmillan aspects that had become outdated, especially with regard to what were believed to be two different pentagons, albeit of slight difference. Further, I removed some line diagrams involving the thickness of the grooves drawn in GeoGebra. And added 'Angles' to the title, to better reflect the dual nature of the study. However, although much changed, the page is still broadly recognisable as the first incarnation.

15 May 2020. Clarification and correction to why I believed the Macmillan and Bailey pentagon were 'different' added as a single paragraph, effectively appended to the page upon the Piker finding.

7 October 2011. Page created.