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Personal Account

A personal account of the discovery, in which I wrote for my Facebook page, in which the emphasis was on the thrill and chase of the discovery (rather than the mathematical aspect) and the interaction between myself and Helen Donnelly, my co-discoverer in this.

A near 40-year-old mathematical mystery is solved!

Some quite fantastic news to report: in association with Helen Donnelly, we have almost certainly made a mathematical discovery, and cleared up a long-standing mystery! Admittedly, this is not in the same league as in solving some longstanding, technical, high-end problem, but rather of a different type altogether. Instead, it’s a historical discovery, involving a tiling consisting of pentagons, which has its roots in CairoEgypt. That said, although the discovery is ‘only’ of a historical nature, nonetheless the achievement in this is most gratifying.
    It’s a story of two artists (although I myself have long left the art world behind for mathematics, albeit the former is indeed my background) in the north of England, from Grimsby and Scarborough, coincidentally more or less on opposite sides of a river, but who neither knew of each other, or their worlds, despite such close proximity. Indeed, despite being geographically so close, one of us had to travel very long distances to make contact with the other, of no less than 2 000+ miles to Egypt. From this unlikely background and situation we then combine to make a mathematical discovery. How amazing
    To the outsider, looking at the discovery (see photo pages), it may seem unimportant, and perhaps even trivial. After all, it is only a tiling. It may even leave some of my non-mathematical friends here cold. But that said, the prosaic nature of the tiles only adds poignancy to the story. Even for those of a non-mathematical bent, there is a great deal of human interest and appeal to retain one’s attention, and the account below has a great deal of warmth to it. Furthermore, it’s a great story of adventure, and derring-do, with risk to personal safety (admittedly, all on my part vicariously). It revives a great British tradition of exploring Egypt, and indeed, of shamelessly plundering their antiquities (of which I might add we do so with a clear conscience, in contrast to previous British explorers). And as is that is not enough, we then run fast and loose with the authorities by smuggling the precious antiquities through customs. Perhaps the one thing it has lacking is romance, but it very nearly had that as well (and with it one could almost say it’s a story for Hollywood). How sorely tempted were Helen and I to leave everyone in England behind and return to Egypt and spend the rest of our days amongst the pentagons. Indeed, for one heady moment we very nearly decided to go, we were that obsessed about the pentagons; it was all we could talk about. But how could we hope to remain undetected, where family and friends would know immediately where we were, and indeed, to the very street of where to find us, before we would be brought back to our senses. We eventually thought better of it, but dear reader, do indulge us that pipe dream.
    It may be rather long, but the story deserves a proper treatment, so stay with it. Pull up a comfortable chair, pour yourself a glass of your favourite drink, and take ten minutes out of your day, and wallow in the spirit of human goodness and kindness. At the end if you don’t have a warm glow for the people involved, I tell you dear reader, there is no hope for you as a human being.
    The story of our interaction began on the 7 November 2010, when I received an email from Helen, in relation to my website request that I had made (detailed below), but the roots of this go much, much further back in time, back to the late 80s, which is where I’ll begin…

Some Background Details

The background to the adventure and discovery involves the well-known (at least amongst mathematicians) and picturesque Cairo pentagon tiling, with many aesthetic properties, of which it in comparison it rises above other tilings. Many mathematicians have waxed lyrical over the years over this, with flowery descriptions such as ‘This beautiful tessellation…’ (Martin Gardner) and ‘… the tessellation is particularly pleasing to the eye…’ (Robert H. Macmillan), amongst many others.

In the mathematical literature, the tiling is extensively discussed, but there is one curious thing that has always struck me about this, in that despite the occasional first-hand reference, from the very beginning of its reporting (in 1971), it is never portrayed as in the most obvious sense i.e. as a in situ picture, as seen on the streets of Cairo, but instead rather as a line drawing. This is something I have been aware of for more than twenty years (indeed, with no picture, at one point I even thought it might be a mathematical urban myth). Indeed, aside from me, only two other people seem to have been aware of this situation, but either kept their interest to themselves, or didn’t pursue this, at least in an active sense. Indeed, I myself put this matter broadly on the back burner, but never lost interest. Books and articles were examined through the years for an in situ picture, but without success, and so the situation remained. Other resources were then used. In the internet age, picture databases were sourced, but still no success. This being so, somewhat frustrated with still nothing found, in November 2009 I created a ‘Cairo pentagon’ page on my website (Cairo Pentagon – Truly Named?) with a request, asking if anyone has seen an in situ picture. Many months passed by without anyone reporting anything. Again, somewhat frustrated by the lack of response, I then, in August 2010, decided to change tack, and adopt a more direct, proactive approach, trying to track down the authors of the few first-hand references to question them, and making contact with ‘likely’ mathematicians and Islamic tiling authorities with an interest in tiling (mostly of a professor level), who may know something of this. Some of them even joined with me in the chase, but we all met with a lack of success. Still no one, despite their different respective backgrounds in tiling could find an in situ picture! And so the situation remained until one Sunday night, of the 7 November 2010, when I had a totally unexpected email from the then unknown to me Helen Donnelly…

The Discovery!

Then, the moment I had been waiting for more than twenty years finally arrived, a red-letter day indeed, a day likely to go down in mathematical history, when I had an email from Helen who was visiting Cairo. Upon her arrival, her attention was attracted by the (as detailed above) pentagon tilings that were indeed on the pavements after all (in the Dokki district), and looked it up on Google. Upon finding the site and web page request, she then sent me a picture of what she had seen. A Eureka moment if there ever was one; at long last, an in situ picture! But does the story end there, with the picture finally in my grasp, as may readily be imagined? No, it doesn’t; but it gets even better, with much more drama to come!

More Pictures!

In her initial mail, Helen told me about another sighting (actually the first), but without a picture, promising to send me more pictures when she found it again. This additional sighting was quite a surprise; given the lack of previous pictures, I thought the tiling must be incredibly rare, perhaps only appearing once, maybe twice, and that would be it. But on the contrary, it isn’t; unbelievably, it’s quite common! Additional sightings were made in other locations in quick succession, not just one more, but many others; in total eight sightings (actually nine, with an additional one not pictured)! Incredibly, on one occasion, there were two sightings in a single day! Unbelievable! Furthermore, and most unexpectedly, the pictures did not essentially simply repeat themselves as may be thought. The tiles themselves were clearly of different periods (with obviously new tiles, being pristine; and old, with considerable wearing), along with innovations as regards colouring and texture, all of which added to the excitement. The thrill of seeing the pictures coming in was most exhilarating. Indeed, upon opening my inbox, at times I felt a like a vicarious Howard Carter in all this! Discovery after discovery ensued.

Nearly 40 years since the first reference to the Cairo pentagon in print, and all without a single in situ picture, and then Helen provides eight/nine in the matter of a few weeks!


On one occasion Helen’s personal safety was put at risk whilst taking the pictures, when she told me that she had been accosted by four strange Arabic men, raising their voices, asking her what she was doing. But even this upsetting incident did not daunt Helen in the task. No, Helen simply gave them as good as she received and sent them packing and continued taking the pictures. What an admirable attitude; the pictures simply had to be taken, come what may. How brave.

The Plundering of the Antiquities!

And then, as if the series of pictures were not good enough, we decided that we simply had to bring a sample back to England, whatever the cost and risk! To say the least, this was a precipitous decision. Would Helen succeed in her task, and upon her return be acclaimed by the mathematics community? Or, alternately, would she be rumbled at Cairo customs with the antiquities in her possession, and thrown unceremoniously into some third world Egyptian jail without trial for goodness knows how many years? Just think what was at stake here. It was literally death or glory. The pressure must have been enormous. But do you doubt the capabilities of our heroine here for one moment? This is Helen Donnelly we are talking about. Thankfully, Helen retained her composure and managed to distract the officials and pass the antiquities successfully through customs. Just ask yourself, dear reader, how you would have coped under such stresses and strains. Almost certainly, you would have given the game away and failed miserably, and met the fate described above, consigned to the vagaries of the Egyptian justice system. But, dear reader, don’t be too judgemental on your own fallibility, just ask yourself, are you of the stature of a Helen Donnelly? Well, are you? I thought not.

Perhaps some of you here, of a clearly less than charitable nature, may call into question our conduct in raiding their antiquities. If so, even though we are in this (pardon the pun) as thick as thieves, than place the blame squarely on my shoulders, and not Helen. I was the one who put the suggestion to Helen. In one of our emails, I made a throwaway suggestion as to the possibility of bringing a sample back, bearing in mind the risks involved, as detailed above, along with the practical difficulties of bringing an essentially heavy stone slab all the way back to England, with every ounce of baggage weight at a premium. As alluded to above, we did this with a clear conscience. In truth, the tiles are essentially uncared for. Some have become loose around the edges. Indeed, some have been broken. Helen tells me that Cairo is undergoing extensive renovation; in all likelihood the tiles would in due course end up on the rubbish tip. How scandalous. Without our well-intentioned intervention, in years to come the Cairo pentagon may very well have been a mathematical urban myth, with no trace whatsoever of its existence. The people of Cairo may not care for their treasures, but is that any excuse for archaeological abandonment? I think not. Who would begrudge us a tile?

To Who Goes the Glory of Discovery?

Perhaps with the photographs and tiles now in our possession petty disputes would surface as to who is to be regarded as the true discoverer. When teams of people are involved in a discovery, disputes as to priority can surface. For example, in another field, think of the scientists Francis Crick, James Watson and Rosalind Franklin who discovered the double helix structure of DNA, of which Crick and Watson then promptly and shamefully erased Franklin from their account. Would a like situation arise here? Would we fall out in bitter rancour as to who was the true discoverer?

There is plenty of scope here for dispute. Both of us could make a claim for the lion’s share of the glory. Myself, I’ve looked at this for more than twenty years. It’s been me in the vanguard all the way in this, with my own research, a web request for the missing picture, and lately I’ve been more proactive, chasing after people who could possible supply the elusive picture. Short of going to Cairo myself to search for the tiles, I’ve done everything in my powers I could to find this. Perhaps Helen may think that she should have the glory, as, after all, she indisputably does indeed possess the first (and more) pictures. But after reading the above, do you now doubt for one moment the admirable characters of the people involved here? No, we are of a like mind here. We are of the opinion that we share this 50/50. Each of us needed the other, as otherwise as individuals we could not have achieved the discovery. In short, without my web request, the pictures would have remained as a personal souvenir, shown amongst family and friends, but without anyone realising the underlying significance. Without Helen, I would have no pictures to bring to the attention of the mathematical community. Hence, an equal share of the spoils. Crick and Watson, you may be better scientists than Bailey and Donnelly, but as people you have much to learn from us…

The First Picture Claim

Can we (in reality myself, given that I have been in the vanguard of searching for the pentagon) really claim in all honesty that this is indeed almost certainly the first picture of the Cairo pentagon? To what degree has this been researched? I state categorically that this has been exhaustively researched. Note that when I say that this is almost certainly likely to be the first picture, I am emphatic; it really is likely to be the first picture! I have been most thorough in this search, it lasting for more than 20 years, and with a variety of different sources:

§         Books and articles: I have examined every ‘likely’ maths book and article (i.e. on mathematical tiling) that may contain a picture. Furthermore, I also examined (but admittedly not to the same degree) books on Egypt. No picture

§         Internet: In the internet age, I scoured deep into picture databases. No picture

§         Personal contact: Lately, I widened the search to contacting people who, due to their specialised knowledge may have something to add, such as Islamic tiling experts. Indeed, some of whom had visited Cairo, and yet had no knowledge of the tiling (amazingly so, given the now inconvertible proof). I also tried an even more direct route, with citizens of Cairo, where I contacted the Cairo university mathematics department, but to no avail. No picture

§         Academic Forums: As a last resort, in connection with the desire not to make an erroneous claim for a proposed article (detailed below) as regards the first pictures (after Helen had sent me the photographs), I contacted the Tiling Listserv, a high-end platform for mathematical tiling frequented by professors to see if they could refute our claim as to these being the first pictures. At this last stage, I must admit that I didn't want them to find a picture, which may sound strange, given my searching for so many years. Personally, I now wanted the glory for Helen, who surely deserves the kudos. However, it would have been poor scientific practice to make a claim without a final check, even if it would destroy Helen’s priority, and so this was undertaken. No picture

All inquiries were of the same outcome: no picture!

So, when I say that this is likely to be the first picture, it really is likely to be the first picture!

That said, I still find it unbelievable that this is the first picture (or more accurately, a series of pictures), in that with it appearing in so many different places (albeit clustered around Dokki district), and with its obviously long history (with the tiles being obviously relatively old; speculating, 50, perhaps 100 years?). How can it be that with Cairo, a capital city with a population of nearly 8 million people, and Greater Cairo, with 18 million people that not a single picture would have been taken! Has not a single citizen, or indeed a visitor, had the wit and imagination to take a picture of this, at the very least, obviously ‘interesting’ tiling before Helen? It seems almost unbelievable that no one has thought to. But the evidence (or more precisely, the lack of) is there before you! If it does exist, it can only be in some minor capacity, in an obscure publication of no significance, of which Helen's series of sightings and pictures must surely put any putative picture in the shade.

Further Developments – A Paper to Mark the Finding

To mark this finding, I considered that it should be marked with a paper, to establish it in print, rather than an essentially transient web page. Indeed, the paper on this finding would ideally commemorate the 40th anniversary of the apparent first use of the term Cairo in association with the tiling, by James Dunn (who incidentally upon contact joined in the chase), in the Mathematical Gazette, of December 1971. However, whether this is a practical proposition given article peer review and turnaround time remains to be seen. Nonetheless, it will be published in due course, even if the commemoration side is not practical given the time frame.

To Helen

Helen, I salute you. You have been magnificent in all this. The mathematical community owes you a great deal. You pursued this above and beyond the call of duty. Anyone would have been satisfied with just a single picture, but you went the extra mile, and more, on this. Picture after picture after picture followed. Not once did you gripe about many additional requests for more details. In conjunction with some of my associative that had joined me in the chase, we asked you for more photographs, sometimes of a specific nature. We asked you to measure the tiles. We sent you to revisit seen places. We asked you to enquire about the age of the tiles. Some of my associatives in this contacted you independently and asked you for further aspects regarding their own interests. Right at the end of your visit, when I told my associatives you were getting ready to come home, with return arrangements to make and when likely a little tired, you still found the time to accommodate their last minute requests, and all with good grace. Not only that, you risked your personal safety, and the wrath of the customs for us, as detailed above. Half way through all this, you told me that you were enjoying the adventure as much as we were. But even so, to do all this for people you have never met or even known about previously speak volumes on your behalf. What admirable character you undoubtedly are.

The End of the Story (For Now)

And so the story concludes with the ending of Helen’s visit, and her safe return, with the pentagons now safely ensconced in Scarborough, where they will be put on display, and treated with the care and respect they deserve, all so lacking from the citizens of Cairo.

But fear not everyone; although this is the end of the story for now, this is not the end for all time, as Helen tells me that she has every intention of one day returning! What treasures awaits us all upon her next visit we can only look forward to with anticipation. Only Helen seemingly has the eye for this, no one else.

It’s been a truly fantastic adventure and excitement, one of discovery after discovery. A mathematical discovery (albeit historical) to one’s name is something to be mightily proud of, and it is made even more poignant, given that we both come from an arts background and not the more obvious mathematical one. The mathematicians have undoubtedly been caught sleeping. But, more importantly, from the human aspect, what fun it has been between us; what a blast we had. That’s what I will remember the most, our personal interaction, rather than the impersonal pentagons, as proud as I am of the discovery. The cold light of history will judge the importance of the discovery, but it would never be able to record the sheer joy of our interaction, of which only we know what we shared. Together, we found the Cairo pentagon and brought it to prominence, and to the victors the spoils and kudos. I think we deserve it.

Created 15 March 2011