When I realised it was the opening gate, about three weeks ago, I knew it was full of dynamical information. It was about ten in the night, the night was dark and I was watching outside the windows with all the lights inside the house turned off. At some point, the light from outside started to alternate between bright light and deep darkness, it took me a while to figure it out what was the cause of this strange phenomena. The reason was a car with its powerful headlights on that was waiting a huge gate made from solid metal bars (of approximate 5cm x 5cm located at distance of about 10 cm of each other) to entire open. I was watching from an oblique direction.
I remember to have thought that this was something wonderful, and that somehow it must be related to a mathematical phenomena, but which one? I was entirely sure… This alternating between luminous night and dark night produced by the fact that the gate was at some times producing an overlapping between the metal bars so that the area of the surface not allowing the light from the car headlights to pass was minimal and therefore the light that I was seeing maximal, and at other times, maximising this area of the surface, so that the light that I was seeing was minimal, and even more, I was seeing no light at all at some instants, because of the oblique direction I was located from the gate.
Short time ago, I was working with iterated functions schemes on the interval with overlapping, and I could learn about a similar phenomena, the non-continuity of the Hausdorff dimension of the limit set as a function of the contraction rates of the function scheme. So this overlapping must be related to the overlap of the metal bars of the gate, and the continuity must be related to the intensity of the light, the change between intensity at some very specific points was entirely different, it was the difference between not light at all and light, this can be seen as the non-continuity of the function of light intensity. Then Pisot numbers, it happens that they are related with the distances between the metal bars of the gate… This was still to be just a bunch of formally related facts but it must be done more precisely, on there hand it was surely possible to write this phenomena more elegantly, in a simple way an general enough…
Then it happens, that I had a board game a cousin had giving me as a gift recently for my birthday. In order to play for the first time, a set of circular tokens had to be removed from two apparently equal rectangular cardboards. So that, once removed the tokens, both cardboards remained full of circular holes of the same size. So this two cardboards with circular holes could produce a similar phenomena that the car waiting the gate to open…Indeed, turning off the light of the room, pointing a flashlight at the two cardboards plates with holes, and moving them one over the other, a similar phenomenon occurred, a non continuous map of the light intensity. This time, I could realise the size of the circular holes were essential, as well as their distribution on the cardboard plate, so this seems a kind of generalisation to more complicated geometries, surely the same phenomena could be observe in more dimensions, considering projections… It seemed at this point it was something too general not to have been studied before, many unrelated mathematical results should had exhibit a similar phenomena, but which ones? How close they were to this phenomena? Or maybe, in the most optimistic case (or pessimistic if you prefer to say)… It could be written as a something new? At least in some sense new?