Colouration

Under Construction 7 August 2026
Of mathematical interest are the colouring possibilities of the Cairo tiling—both map‑colourings and what I term “shared colours”, including the “Macmillan colouring” and, more recently, combinations of these. My primary concern here is with the map‑coloured instances. The literature on this topic is extremely scant. Generally, the tiling, when coloured, is shown as a single arbitrary example, typically of four colours, with the tiles all in the same orientation, this being more readily imagined, reflecting the four orientations. So far as I am aware, only Rinus Roelofs and George Baloglou have examined these colourings in a considered way, though to different extents: Baloglou’s work is by far the more detailed, while Roelofs treats the matter essentially in passing. However, neither study can be described as “all‑encompassing”. For any number n, the colourings are not developed from first principles—specifically, from the foundational cases of three and four colours. Baloglou concentrates on six‑colourings, whereas Roelofs presents three arbitrary four‑colour examples. I would be indebted to any reader who can refer me to any work in this field, or indeed, may be interested in studying this aspect themselves. 

In contrast to the people above, I begin here from first principles, with the three‑ and four‑colour cases. It will be seen that three colours admit only a single instance, whereas four colours yield at least four distinct possibilities (likely all, although proving it is another matter). Although four-colouring, with four orientations of the tile, may appear at first sight to be a straightforward problem, it is anything but. Complications arise from the tiling’s mixture of three‑ and four‑valent vertices. For colouring purposes, the tiling is perhaps best considered initially in its simplest form—a double basketweave (Fig. 1)—before transposition to the Cairo tiling itself, which is the approach I adopt. For the pentagonal form, I have used the in situ model. 

For the study, I use an 8×8 format. This is sufficient to reveal the underlying structure without requiring a disproportionate amount of time to colour by hand (or, more accurately, with the fill bucket), as would be the case with, say, an incremental 16×16 (or larger) tiling.

A seemingly obvious time‑saving idea is to use AI. However, the readily available models—Copilot and Gemini—are, in this respect, a complete embarrassment. As a trial, I asked them to map‑colour an 8×8 grid of squares using four colours, explicitly stating that no adjacent tiles were to share a colour. Both failed miserably. I therefore dismiss this avenue entirely. Another AI possibility was numbering the individual tiles. Adding numbers by hand, in Photoshop, was a time-consuming trial, not to mention accurately aligning them (centring). Simply stated, with Copilot and Gemini, it was more trouble than it was worth, with an amateur appearance. I thought better of it. I then tried ChatGPT, which went better, but after initial success with attempts one and two, it rapidly tailed off! The third attempt had a minor omission, whilst the fourth attempt was ridiculous. Woe is me. 

The Wireframe Outlines
For the study, I use an 8×8 format. This is judged sufficient to reveal the structure without requiring a disproportionate amount of time to colour by hand (or, more accurately, with the fill bucket), as would be the case with, say, a 16×16 tiling.

A seemingly obvious time‑saving idea is to use AI. However, the readily available models—Copilot and Gemini—are, in this respect, a complete embarrassment. As a trial, I asked them to map‑colour an 8×8 grid of squares using four colours, making it explicit that no adjacent tiles were to share a colour. Both failed miserably. I therefore dismiss this avenue entirely.

Fig. 1. Wireframe Cairo tiling and Basketweave equivalent 

Three Colours

Let us begin with colouring an arbitrary patch of contiguous tiles (Fig. 2).

Fig. 2. Colouration of a contiguous patch of tiles

I begin with red, then green, and then blue is forced. However, the continuation is then not forced. For example, the tiles either side of blue can be coloured with red or green. And so on (not forced) for all the other colours. Although an initial continuation may be found, a distinct possibility is that a conflict may arise. Therefore, I put this patch aside. Instead, I now begin with another 3-colour patch (Fig. 3).

Fig. 3. Another 3-colour patch

As is immediately apparent, the continuation is forced at each suitable location. For example, in the space above the red tile only a green tile can appear, and in the space below it only a blue tile is allowed. Once these colours are fixed, the remainder of the tiling follows in the same forced manner (Fig. 4).

Fig. 4. Forced continuation

To more easily see the forced options, instead of colouring the whole tiling, I show an arbitrary patch. For example, the “hole” in the tiling at right can only admit a red tile. Then once that is in place, the other colours are then forced. 

From this, the construction is more readily seen as a series of translated parhexagons, as below, Figs. 5a-c.

Fig. 5a. Red (2), blue, green, parhexagons 

Fig. 5b. Blue (2), red, green, parhexagons 

Fig. 5c. Green (2), blue, red parhexagons 

Fig. 5d. The assembled 3-colouring

From this, it is obvious that no other 3-colouring is possible.

4 Colours

With four colours, matters naturally become more complicated than with three. It quickly becomes clear that a carefree approach — colouring without an overall plan — is inefficient, and chaos soon follows. A systematic method is therefore required, though it is not immediately obvious how best to proceed. The arrangement of the tiles is unusual, meeting at both three‑ and four‑valent vertices, and the analysis is largely visual. Although one can represent colours numerically, this lacks the immediacy of colour itself, where the overall arrangement is instantly apparent. So how to go about this? A natural first observation is that the tiling contains two rectangular tiles subdivided into a square, set at right angles to each other to form a 1×2 unit (Fig. 6). 

Fig. 6. 1 x 2 unit 

This unit block forms the foundation of the study, and it is repeatedly translated and coloured in various ways. The first task, when viewing the block as two rectangles within a square, is to determine all two‑colour possibilities. A straightforward count shows that there are twelve such distinct arrangements (Fig. 6).

Set of 24

Fig. 7. Set of 24 rectangular units of four colours 

Having thus determined all possibilities, I now investigate their arrangements in combinations, i.e., block 1+1, 1+2, 1+3, etc., up to 1+24. The same with 2+2, 2+3, 2+4, etc., up to 2+24, and so on, ending on 24+24. Or at least in theory! Calculation shows that there are 300 possibilities(!), (see appendix), with most being inadmissible (same colour adjacency), and only three are distinct in each row. After the first two rows (1+1, 1+2…1+24; 2+2, 2+3.... 2+24, I bailed out, as, analysing by hand, the time involved was taking way too long for the intrinsic worth, with a foreseeable analysis of many days, if not weeks. This is judged impractical. Likely these omissions repeat earlier findings.

4-Colouring A

Fig. 8a. 4-Colouring A
No ribbons. All colours appear in one orientation. All four tiles/colours in the same orientation.
The simplest and most obvious colouring. This colouring arrangement is shown in Roelof, Fig. 1c. This is frequently to be seen among the in situ pavings, likely being the most ‘intuitive’ colouring of all, including the 3-colouring, in that one simply places tiles of the same colour in the same orientation, and the colouration repeats as above, no "thought" as such required.

4-Colouring B

Fig. 8b. 4-Colouring B . "Ribbons"
4 Ribbons, back-to-back, vertical and horizontal. This is shown in Roelof, Figure A.  

4-Colouring C

Fig. 8c.  4-colouring C
No ribbons. Each colour appears in two orientations

4-Colouring D

Figure 8d. 4-colouring D
Yellow and red ribbons. Blue and green appear in one orientation each

4-Colouring E

Fig. 8e.  4-colouring E
Partial ribbons for all colours , green, red, blue, and  yellow. all  colour tiles appear at 180 to each other

5 Colours
Five colours become a little difficult to analyse. Likely there is more than that shown here, of which these are the work of George Baloglou.

 Not seen in the in situ pavings.

Not seen in the in situ pavings.

Six Colours
Six colours becomes even more difficult to analyse!

After George Baloglou. Not seen in the in situ pavings.
For a more in-depth analysis of six-colouring possibilities, see Baloglou's blog, with 38 examples!:
http://crystallomath.wordpress.com/2013/10/18/cairo-six/

Contiguous (Fused)  Colourings
With the restriction of map colouring relaxed, a whole host of colourations of contiguous colours, or what I term as ‘fused’ below, are possible. Indeed, there are so many possibilities here that they soon become trivial, so much so that I largely gloss over any in-depth analysis here. Undoubtedly, these are weaker than the map-colouring condition, in that the individual pentagons are not so readily seen. Consequently, I show just a few of the possibilities, with occasional comments. Reference is made to two different types of fused pentagon, with (a) ‘Macmillan’ and (b) ‘90°’; this serves for ease of description. ‘Macmillan’ is of the pentagons placed ‘back to back’ (as first described by Macmillan of the in situ pavings, of 1979), whilst ‘90°’ describes the ‘other’ meeting possible, at the 90° angle. To clarify, I show below:

(a) Macmillan Types

Macmillan type, two colours. This is undoubtedly the most intuitive of the fusion possibilities. This is frequently to be seen among the in situ pavings (a notable sighting being at the American University in Cairo, and as alluded to above, the most ‘intuitive’ fused instance of all.

Macmillan type, three colours. Not seen in the in situ pavings

Macmillan type, four colours. Not seen in the in situ pavings 

(b) 90° Types

A simple two-colour diagonal. As seen in the in situ pavings, at Heliopolis, outside Caesar’s Place hotel, as reported by Robin Wilson. Whether this is still extant is not known.

90° type, with four colours used. One interpretion is of the parhexagon of order two symmetry. Of the fused type, this is perhaps better than most, it retaining ‘integrity’. Not seen directly in the in situ pavings, alhough the fusion can be seen

Combinations - Macmillan and 90° fusion of two pentagons.

Macmillan and 90° fusion of two pentagons.  A balanced composition. Not seen in the in situ pavings, likely due to relative complexity, especially if supplied without a guide sheet.

Four Pentagons in a Parhexagonal Block
Each par hexagon unit is coloured in a single colour. I have omitted the simplest one colour. These lack any real interest. Essentially, they can be said to lack imagination.

Columns, in 2 colours. Not seen in the in situ pavings

Columns, in 3 colours. Not seen in the in situ pavings

 Distinct parhexagons.  Arguably the best of the genre, with each colour block non contigous. Not seen in the in situ pavings.

Part 2

In Situ Colourings
Upon having studied the colouring possibilities in an abstract sense, I now examine the in situ pavings, with the above analysis in mind, and see how these are arranged.

Map Colourings

4 Colours

Typical 4-colouring of Fig 2a. This is by far the most frequently seen colouration, likely due to its sheer ease of placement, in that to achieve this all one needs to consider and follow is a ‘same orientation, same colour’ rule. However, again, one would have to ask if the layman would be aware of this. If there were a picture provided, then this could explain its relative frequency. Possibly not, which would explain instances of contiguous colour, likely out of either lack of interest or lack of success with trying to repeat in a regular way, leading to frustration, and so placing the tiles ‘any which way’. 

5 Colours

A single instance has been found, but the properties of the colouring remain uncertain, due to a small patch, from which one cannot tell if it is regular or not. Upon examination, isolating each colour to see if there is regularity, it would appear not. However, for reasons as above, this is a provisional statement, subject to change. As few tiles are shown, certainly not enough to determine if the pattern continues, whether this is a regular colouring is uncertain. Upon preliminary investigation, it would appear not. If a regular five colouring was indeed discerned, this would be significant, as it would suggest a mathematician is behind this; beyond all reasonable doubt, a lay person would not go to the time and trouble for such an involved colouring. 

Random Colourings
Some instances defy analysis, as upon an apparent attempt at an innovative colouring scheme, the instigator apparently loses his way, and the colouring becomes random.

Contiguous Colourings
2 Colours

'Butterfly', as defined by Macmillan, in black and white.  AUC campus outside the administration building.

Fig. 20: 'Butterfly' as defined by Macmillan, in burgundy and white.

Fig. 21: As defined by Macmillan, but burgundy and dunn, at the Old Cataract hotel. The sighting is no longer extant.

Random Arbitrary Contiguous Colourings

Somewhat hard to describe! Likely the designer was attempting a more 'involved' Macmillan colouring, of three colours, but lost his way.

Occasionally seen are colourings where no pretence is made as to regularity, with colours appearing ‘any which way’, but with contiguity. Although such a sight may grate on those who prefer order, one should not forget that most people would have no interest in the subject per se, with the intention of simply paving a given space in the least amount of time and effort. Indeed, the very nature of the tiling, with many variations as to colour, would lead this way, with the user giving up in frustration. Such examples may therefore lead to the supposition that no guidance was given to colour arrangement at the time of purchase, but as with much of the study, one cannot be too certain of this hypothesis. Even within this, some attempts at order are made. 

Page History
23 July 2026. Added improved diagrams, with larger-scale ones replacing smaller-scale ones, which made for uncomfortable viewing.

17–18 July 2025. New Sites 'adjustments'. The conversion had left the page with various shortcomings in presentation. This included large spaces between texts, captions not attached to pictures, inconsistent text boxes, with titles and discussion disjoint. All are now corrected. All the usual errors were corrected in Grammarly, pending a more extensive review later.  An attempt was made at a more consistent text style, resulting in a slight improvement, albeit marrying different aspects was not straightforward.  Nonetheless, the page is broadly ordered.

Created: 8 February 2013. Last update 21 October 2013