## Wednesday, July 7, 2010

### Ever we fall, ever we get up again...

Dear Maurits,

I've recently been staring at a book of your prints, moved to sorrow for all those folks who have to keep climbing nowhere for all eternity. I almost fool myself that I feel some sympathy for them; and maybe they some for me. But the truth is that they don't really feel anything, being static, and I'm not stuck on a twisted bundle over $\mathbb{S}^1$; I'm not always climbing, because I keep falling. However much it might feel like an ongoing upward trudge, that feeling only persists so long as I forget having slid down that well-worn slope by the fascinating little rocky outcrop that you have to clamber through the briars and nettles to get at... and the rocks really aren't so nifty-looking up-close. It's just a heap of dead rocks.

Some people call your strange prints "illusions"; but that's really not right: they're properly paradox, a contradiction in pictures. What I occasionally suffer is illusion. You're just weird, and fascinating because of it.

I don't recall if you're the sort, but I'll ask you anyways, pray for me!

a sinful geometer

Paul Stilwell said...

My math and data-processing teacher in high school had his prints up in the room.

I was always fascinated as well (though not exactly a fan), trying to see all the contradictions together as a whole after viewing them separately; it was like being on the verge of seeing some new dimension.

But it was always on the verge.

Belfry Bat said...

A clever friend gave me a coffee-table-book full of them one Christmas. Now what would be even more clever is to obtain a house with space for a coffee table!

I suppose I should graduate and find a posting somewhere, first...

Belfry Bat said...

Getting back to the other thing...

I think it useful to recognize two sorts of weirdness you're likely to find in Escher; I'm not sure which is weirder... the one is exemplified in Ascending and Descending, where small bits of the picture are quite sensible, but they glue together all wrong. Belvedere and Convex and Concave are other examples.

In the other sort there isn't really any contradiction as such, but the geometry you see definitely isn't the geometry we're used to walking around in. Examples of this are House of Stairs, Up and Down, and the circular divisions like Angels and Demons (unconnected to Dan Brown pulps); I'm also tempted to include Gallery and Relativity, although it's admittedly much weirder.

Now, the really odd thing about, say, Belvedere is that it's not so much the belvedere itself that's impossible --- Escher draws it very nicely as a perfectly describable and consistent geometric object, a manifold-with-boundary (the boundary being what we can see) --- it's that it couldn't be embedded in our space the way he's drawn it, without any distortion. Again, the steps themselves in Ascending and Descending make a fine geoemtric object, but they couldn't be sitting on a house the way he's drawn that house --- and he's aided by the fact that the architectural cues supporting the uniform ascent can't be drawn in their entirety either, especially the moulding rail running under all the top level of windows. The way the building obscures the side you can't see lets him avoid showing how the rails can't meet-up properly.

Paul Stilwell said...

Ah, that moulding rail! The longer side gives it away rather badly. I googled all the images you referenced and see exactly what you're saying. For me, the weirder ones are of the latter category.

Of the ones that "glue together all wrongly", I think your statement, "that it couldn't be embedded in our space the way he's drawn it, without any distortion" is precisely what gave me that "on the verge" feeling; I was trying to reconcile in my common sense of space, in my own space, the contradiction in geometry that was apparently being supported in the picture plane.