A Dream of the Number Six


As I’ve often had cause to mention here and elsewhere, I’ve been blessed with vivid dreams and nightmares ever since I was a child. I’ve recorded the details of many of these nocturnal visitors in my diaries, perhaps most notably when I experienced a series of lucid dreams a few years ago.

There have been other dreams wherein I’ve heard music and poetry that I was able to remember and commit to paper on waking, so it’s inevitable that I regard these things as an enduring source of fascination. Last night, however, I had one of the most striking dreams I can ever remember experiencing, so I’ll record it here while the details are still fresh in my mind. Due to what it contained, I’m certain that I know how it came about, by which I mean that I know it was a dream in its purest form, rather than a purported visit to the astral planes, contact by supernatural entities or the like.

If any explanation is necessary, I’ll provide it towards the end of this post, but for now, here’s the story of what I experienced last night, which left me with a sense of irreplaceable loss, as well as soaring joy and wonderment, the moment I awoke.

I found myself sitting in a large wooden chair or throne, such as you’d find in a magnificent cathedral for the comfort of a bishop or a pope. The experience made me blink in surprise, just as we do when we suddenly awake, but on this occasion, I was entering a dream rather than waking and I instantly realised that this was the case.

I was seated with my back to the stone wall of a large, circular room that I somehow new to be at the top of a tall tower; this was my first impression, which was confirmed when I was able to see out of one of the elegant windows to gaze over an undulating, star-lit rural landscape, such as we might see on the front of a Christmas card.

Four men were looking at me in an indulgent, patient fashion, taking into consideration my disorientation and momentary confusion as if they were aware I were a visitor from another time and place, as well as being an honoured guest in their company. To my left, seated behind an oaken table or desk, sat Leonardo Bonacci, or Fibonacci, dressed in the elegant raiments I’ve come to associate with Renaissance Italy. Further across, to my right, Niccolo Machiavelli and Leonardo da Vinci reclined in chairs similar to mine, regarding me with faint smiles on their faces, while the imposing figure of Cesare Borgia stood by a window alcove, lost in contemplation of the silent vista beyond the confines of the tower, but nonetheless aware of my presence and listening to the exchanges.

In other dreams, unlikely scenarios appear to be perfectly natural and unworthy of comment, but such was the vivid nature of this particular experience that I was stunned to be in such a wonderful setting with four such eminent men and they seemed to realise this. Cesare Borgia studied me for a few seconds, then glanced to his side and nodded to two young women who both started to sing some soft, enchanting song, one strumming beautiful, resonant chords on a lute, the other weaving her fingers across the strings of a harp to conjure notes and counter-melodies that spilled into the room like a thousand tiny rainbow-coloured fish, such as you might see in the shallow water close to the golden shore of some exotic coral island.

At this, I started to relax. A beautiful, dark-haired girl appeared from behind me, handing me a crystal goblet containing purple wine, then she smiled and sat on some cushions on the floor to one side of the chair, looking forward to the exchange that she seemed to know would follow. The melodies from the voices, the harp and the lute played through the candlelit atmosphere inside this magical room and as I glanced around, it was pleasurable and satisfying to see portraits, tapestries, framed sketches, inkwells, parchments and all the other paraphernalia of an alchemist’s sanctum sanctorum.

The assembled company were pleased to see that I was finally at ease, so Fibonacci began by telling me how much he’d enjoyed reading a previous post on this site, in which I’d speculated that our world would have been changed for the better if the sequence named after him had been discovered and printed in a book in 2013, rather than coming to light in various forms around two thousand years earlier. I was amazed to learn that he’d read what I had to say and I said as much, blushing furiously, but the other three men in the room assured me that they agreed with their learned friend.

Fibonacci then told me that another such sequence or theorem was awaiting discovery, one that would enable men to perform what had hitherto been regarded as miracles, even as late as 2015. Machiavelli gravely told me that it had the power to end wars, disease, famine and poverty, then Leonardo da Vinci added that it would allow men to build fantastic structures on Earth and also to voyage effortlessly to distant stars and even galaxies, if they wished. Cesare Borgia informed me that this formula or sequence couldn’t confer physical immortality on men, but that he knew it could do just about everything else, after which he returned to staring out of the shining window at the cold, distant stars.

Anticipating my next question, Fibonacci promised me that he would indeed reveal the secret of this formula to me, but first he asked me to think about what I’d written. In the post in question, I’d speculated that the most sought-after book in the world would come into existence as a result of observations of an everyday thing such as fire, water or music, but he said that the formula or sequence was suggested by something else in nature and he invited me to guess or work out what that thing might be.

I thought hard about this, then suggested darkness, light, shadows, the tides, the lines on the hands of men and a few others, then shook my head when I ran out of inspiration. After I’d fallen silent, Leonardo da Vinci rose from his seat, then opened a window behind him and I could see that the wonderland beyond the tower was now carpeted with snow, with the strange, almost tangible silence that follows after a blizzard. The snow, however, was still falling, so he extended his hand through the window into the cool air outside, retrieving several glistening flakes that he held out for my inspection.

As if lost in amazement himself, he muttered that certain snowflakes always form as hexagons or as hexagonal prisms, then he informed me that the likelihood of any two snowflakes being identical was extremely remote, because there are 10 to the power of 158 possible arrangements for such a thing. At this point, Fibonacci’s eye widened, adding that this was roughly twice as many possible arrangements as there are atoms in the universe, then he rose from his chair to cross to a blackboard on a frame behind his desk, where he inscribed the hexagons you can see at the top of this post.

He patiently explained to me how the most fundamental aspects of basic physical structures, as well as of space and time, are comprised of two-dimensional hexagons, inviting me to bear in mind the super-powerful graphene as an example. My grasp of mathematics is woefully limited at the best of times, but this all seemed to make sense and in any event, I wasn’t about to argue about such matters with either or both the Leonardos, Bonaccia and da Vinci, who both stared at me intently and seemed very keen that I understood what they were saying.


The part of this dream that will come as a crashing disappointment or anti-climax to any mathematicians, engineers or cosmologists who might be reading this came when the two Leonardos explained to me that there were as many aspects, permutations or manifestations of hexagons and of the number six as there are possible snowflakes; perhaps even more, because they admitted they hadn’t yet found the time to be able to estimate them all, although they added that they’d not yet worked out all the numbers in the Fibonacci Sequence either, as there was no real point in this once the validity of the original observation had been established.

They explained that the value of their discovery lay in the fact that each of these different manifestations of hexagons had a particular property – in the case of graphene, it was its combined lightness and strength, in the case of the cells of honeycomb in a bee hive, it was economy of space, all of which I could just about understand. I struggled when they went on to speak of space and its relationship with time, but I’ve read enough science fiction and science reports in my time to be just about able to grasp the concept of Spacetime, so I was able to keep up.

Their point was that by using certain hexagons in a certain way, we could construct something like graphene, but they stressed that there were a vast amount of possible hexagons or other aspects of the number six, which is the best way I can remember what these men were saying to me. Some of these thing could be used to build materials, some could be used as portals or conductors, while many were already in existence, forming a lattice structure of space and time that could be manipulated and exploited if an observer knew the precise nature of the hexagons in question.


Fibonacci then drew a simple sequence on the blackboard, which Leonardo triumphantly expressed on another part of the board by means of hexagons such as the ones at the top of this post, then the blessed light of understanding dawned and it was as if all Blake’s doors of perceptions had been simultaneously thrown wide open for me. I can remember groaning in sheer amazement, but this groan was exceeded in volume and intensity when I awoke because of course, I cannot for the life of me remember the precise figures and diagrams these two men so patiently produced for me on that blackboard in the tower.

“The clouds methought would open and show riches
Ready to drop upon me that, when I waked,
I cried to dream again.”
Shakespeare, The Tempest.

I have to say that I don’t feel any sense of loss at not being able to recall the ‘magic formula’, even though I saw it clearly in my dream and understood it as well. I’m sure I know how this dream came about, because in a previous post, I’d speculated about the idea of someone discovering what is essentially another Fibonacci Sequence in our modern era and the consequences that would arise from its publication.

I know next to nothing about mathematics, but I’ve read about graphene in many news reports since it was invented or discovered, while I also read Sir Arthur C.Clarke’s The Fountains of Paradise decades ago, a book that contains a description of a material that sounds remarkably like graphene to me. I’m interested in many subjects in addition to antiquities, mythology and the supernatural; I particularly enjoy certain brands of sci-fi such as those portrayed in the Star Trek series, which I seem to remember once featured a scene in which Lieutenant Data conferred with Leonardo da Vinci.

Finally, I’d been thinking a lot about Renaissance Italy over the last few weeks, as a result of my son buying me The Prince, and I’ve written about other material that appeared in my dream last night, so I shouldn’t be surprised by its imagery and premise. The experience was so pleasurable and frustrating at the same time that it more than repaid all the time I’ve spent wondering about the characters, settings and concepts it contained, although I have to say that dreams such as this come unbidden to me in any event.

I’m sure that this post would have had greater impact as a short story or as the script for a short film, because as it stands, it’s little more than a plain recounting of the salient features I saw and heard in my dream, rather than an evocative portrait that conveys the range of emotions and intense curiosity I experienced. Instead of aiming at perfection in some future work, however, I’m more than content for now to record the bare details as I recall them and to present them here, because the writing of this post – and others – provides me with a very welcome distraction from my pain.

“The power of thought! The magic of the mind!”
Lord Byron.

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