# Twisted stairs

Here are some crazy stairs.  Look at the stairs leading up to the balcony, and at the foot of the stairs you’re looking down on them from above, whilst up by the balcony you’re looking up at them from below. Nothing wrong with that, in a perspective view, but these steps are in a parallel projection, which forbids it. As a result, the side of the steps that’s furthest from us, next to the outer wall down at ground level, has somehow twisted to become the side that’s nearest to the viewer up by the balcony, and vice versa.  There’s a different twist to the stairs at the foot of the image.  On those, if you start, say, on the right, you’ll find that the flat step surfaces have become vertical risers once you’ve passed the half way mark.

Update May 2012! There’s also a later version of this scene.  I wanted to give it another go, because these stairs do have an extra twist to them …..

M.C.Escher is the artist most famous for crazy staircases, but he never did twisted ones like the two leading up to the balcony in my picture. Although the scene kind of makes sense, the twist in the stairs depends on fudged perspective, and the scene could never be constructed as a three-D model.  Escher was a lucid guy, who didn’t do fudge. In his pictures the perspective is often completely consistent, and you can actually construct some of his scenes in three dimensions. Check out his picture Relativity, and an ingenious version of it made with Lego bricks.  As your eye travels across that picture, it’s just the orientation of up and down that can swing around, but only the figures and creatures indicate which direction is up.

(The easiest way to view Escher’s pictures is just to input Escher and the titles in Google Images, but for a comprehensive look at Escher, there’s an official M.C.Escher website.)

The lateral stairs in my picture are based on the famous Schroder Staircase.  Here’s an illustration of it.

We’re looking down on the staircase to the left, and looking up from below the stairs on the right.  It can be quite hard to see the right hand stairs as from below, however, because in spite of the shading and the figures, they can tend to “flip” into a view from above.  From above is how we usually see stairs, and our brains keep reverting to what seems the more probable viewpoint. Not that that stops the picture to the left sometimes flipping into a view from below, as our brains struggle with the ambiguities. In the picture at the start of the post, I’ve just used a longer staircase, and made it twist with some fudged perspective context, plus the figures.

Oh yes – the acrobats.  I borrowed those from some five hundred year old prints supposedly (nobody is sure) by an artist called Juste de Juste.  The character in the demo picture is from a print by Peter Breugel the elder.