Tuesday, April 10, 2012

@Andrew: Escher's Penrose Waterfall

I have a question about Art. Andrew understands art. So, my hope is that he can explain something regarding this painting:



I'm reading Gödel, Escher, Bach. It contains an image of M.C. Escher's Waterfall. After staring at the picture for a while, I got online to read about it. According to the wikipedia page, the waterfall has the structure of a Penrose Triangle. The optical illusion results from its Penrosian structure.

Alright, fine.

But then I start reading about Penrose triangles and I notice a key difference that is bugging me: For Penrose objects, each line segment is only partially visible; in Escher's Waterfall, the water is always visible.

Let me explain what I mean. Take this Penrose Triangle:



Suppose we label it as having a left side, a right side, and a bottom side.

Grey appears on the Left and Right sides, but not Bottom.
Black appears on the Left and Bottom sides, but not Right.
White appears on the Right and Bottom sides, but not the Left.

This pattern seems to hold for the other Penrose objects. On the Penrose Octagon, for example, pink is only visible on the bottom right portion of the figure. But in Escher's waterfall, the water side is always visible.

So, I'm confused as to why the internet says that Escher's waterfall is similar in nature to Penrose objects, but differs from them in this respect.

I can understand the relation between Escher and Penrose in the case of Penrose Stairs, since Escher's painting and the Penrose stairs look the same. But in the case of the Waterfall, it seems to have a different quality than the Penrose triangle since one of the sides is always visible.

My suspicion is that it has to do with the point at which the waterfall meets the waterwheel. If we label the water we see in those little troughs as the "top" of the water, then at the point of the waterwheel there would have to be a transition from our seeing the top of the water to the bottom of the water. The strip of water has to flip over at that point, so to speak.

But if that's the illusion, then it's a Möbius strip, rather than a Penrose object.

So, why have the art scholars of wikipedia labeled this as a Penrosian Waterfall, rather than a Möbius Waterfall?

109 comments:

_J_ said...

This page has a thing, with some words.

Roscoe said...

It's a false perspective trick that you're stumbling on.
The Waterwheel can't actually be under the first zag, which is how it appears.

The trick's not the water, but the vertical support struts. That's where the Penrose Triangle effect you're seeing and comparing occurs.

kylebrown said...

There are three penrose triangles in the image. One at each turn of the stream.

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