|
|
| Samuel Verbiese | 2003-01-12 22:59:42 |
| In this new text written to explain the very purpose of this programming project as asked by a few correspondents, I take the opportunity to add some references of people who also contributed ideas and whom I forgot to mention in previous notes. I do not repeat here those already acknowledged earlier.
The point is that, among others (tilings, polyhedra, quest to infinity, etc...), Escher was working on images of impossible things and situations. Its "Print Gallery" belongs to that last kind.
In this work, a young man is looking at a print in a glasshouse gallery. That print represents a view of a river with on the opposite bank, among the buildings, the same gallery. But the reason why this print looks weird, is because Escher made the gallery in the background of the other side of the river (and which should normally appear 256 times smaller due to perspective) burst out from the right hand side of the frame of the print, and blow it up into coincidence with the real gallery that started shrinking from the place where stands the young man towards the right.
You can thus look clockwise at Escher's work, starting from the young man, entering the print on display, traverse the river and see the glasshouse coming towards you till you're inside with the young man again. So doing, you wander around a blurred white spot in the middle of the print where Escher put his signature.
To succeed that wonder, Escher needed a proper so-called conformal transformation, in which right angles are conserved during the expansion of the image. He therefore developed in a square a grid in which starting from the four corners you can follow almost along the edges series of lightly curved squares expanding clockwise to the next corner of the grid, where the sides of the squares are now 4 times bigger, and so on. See D'Genard's right figure in his site mentioned in the guestbook, www.interspace.be, under Method, Compare grids. The used transform scheme is such that the series of squares that are closer towards the center tilt in s-curves till they reach the vicinity of the blue lines. Between the blue lines, the center tends to curl up in a spiral, before emerging again, and you can imagine there is one middle green line that keeps curling, swallowed towards infinity, before getting back from infinity to reach the other side of the grid. (that pic doesn't show the continuous swirling, because it was intended for introducing another, non-mathematical, right-brain "solution").
If you now look at the lightly curved red squares you see they are representing the same picture as the preceding one but smaller and tilted clockwise. The same process screws deeper and deeper towards infinity. Each line entering a red square, quits it. Some lines enter the next smaller red square before exiting, and so on... The closer you get to the "central" green line, the more red squares you reach in a spiralling swirl, before swirling out again to the other border of the grid.
But Escher, who himself went a few steps further into that sink, as commented in the guestbook by Johan Groenen, and as can be seen on Escher's own grid, most probably knowing he was sucked there to infinity, didn't decide this was the purpose of his print, and although probably most capable of filling himself the white spot as suggested by Adriana Rainwater in the guestbook, he rather deliberately removed that central singularity region swirling to unifinity and kept only what was useful to have all parts of the drawing represented only once. Therefore his original print doesn't show the young man across the river, only the glasshouse surrounding him . If he had continued in the same systematic way in the central part, he would have kept going on, with first representing the young man in the gallery, then the print again, with its river, its gallery across, the young man again watching a print, and so on to infinity. I believe now indeed, after the comments from quite a few people in the guestbook and some thinking about it, that Escher knew that, but didn't intend here to follow that path.
What the team of mathematicians from the university of Leiden did, was precisely to figure out how the white spot would have looked like, if Escher would have kept going on, as he did a few months later with another transform in "Smaller and Smaller I" (see http://www.mcescher.com/Gallery/recogn-bmp/LW413.jpg). Therefore, departing from Escher's intentions and realisation, as pointed out by Tijn Schmits in the guestbook, they had to find the mathematical transform that the closest matches Escher's "hand made" one, and remap Escher's print on it to encompass the (small) differences and give the opportunity to computers to work out the zoom effect that finally appears in the looping clips, filling so the central part to infinity, which is a recurrent process, as at each step you encounter the young man, all starts identically again... They so realised that "mise en abyme" in French, as pointed out by Martin B. in the guestbook, what Dutch people call the Droste effect, after the images that adorn the box of the famous chocolate and the cup, which represent a nurse holding a plate with a box and a cup with an image of herself holding a plate with a box and a cup, etc...
This project in itself was a quite tremendous undertaking where a team of mathematicians, computer programmers and artists helped Prof. Hendrik Lenstra to work out his theoretical findings. Furthermore, the Internet site they built is very nice looking at, especially if you run the clips full screen : you get really swirled into it till dizziness !
Now, the main thing that bothers a few people is the unrespectful way the enterprise is sometimes marketed in the press, and reported as such in the site iself. It explicitly says that this team has finally discovered what is inside the white spot of Escher's "uncomplete" print, as if Escher certainly didn't know it, and second that he "fooled around" a little with mathematics. This isn't very fair for two reasons.
One, because he was genial enough to find this by hand (without computers) and without high level math knowledge, and turned it, as dozens of other prints, in art works with both a mindblowing content and an exquisite craftmanship in execution.
Second, this also fails to point out the present work didn't yet fully represent all plastic, artistic qualities of Escher's work ! Indeed, when we look at the result of the so-called "mathematically correct" print, we see that it is in fact plastically inferior than Escher's original, as Escher's transorm close to the corners, restituted more nicely the glasshouse with vertical pillars, and the full upper side of the frame of the print. This is why some people believe the mathematical transform could be merged with another one in that region to get a hybrid grid respecting Escher's work better.
Several people believe the present stunning work should be presented as a non-definitive addition (George W. Hart and others in the guestbook call for interactive software "to play" Escher), probably prone to improvement and admitting it might possibly going in another direction than was the purpose Escher assigned himself with his print, rather than a "completion" of an "uncomplete work", as Escher's print is often understood by people, pushed in that direction by misleading marketing...
Finally, it must be emphasised, as I did in my last note, that the filling of the white spot to infinity NOT perfectly solves Escher's quest to encapsulting completely infinity in a print, as he finally could do in "Cercle Limit III" (see http://www.mcescher.com/Gallery/recogn-bmp/LW434.jpg), something pointed out in the guestbook by Bram van Leer.
We hope the scientific team will embrace these visions, and issue the needed statements to curb unfortunate signals from propagating !
The team should be thanked for triggering our thinking on Escher and allowing this discussion with the opportunity of the guestbook that is even provided with a search engine !
Thanks to open up a section on the mathematic equations at hand !
|
|
