# Escher’s “Waterfall” Explained

August 11th, 2008 by david

Here’s a demonstration of one of M.C.Escher’s famous pictures, the Waterfall. (Just put Escher Waterfall into Google Images to see his version).

First of all, you need to understand how a famous “impossible figure” called the tribar produces its effect.

One the left, in the picture above, we see the tribar as an impossible figure. The top of the vertical bar to the left seems to be at both the nearest point in the image and the furthest point at the same time. In the image to the right, seen from a slightly different viewpoint, there’s no problem. The top of the vertical bar really is nearest to us. But seen as to the left, with the arm exactly aligned with the end of the arm to the rear, our brains go for the option that bars are connected as the most probable configuration – even though it’s impossible.

If you are good at fusing stereo picture pairs without a viewer, you’ll find these two images will show the tribar in 3-D.  For a guide to how to view 3D picture pairs without a viewer, in “cross-eyed” mode, try:

http://www.3dphoto.net/text/viewing/technique.html

There are other sites if you search on “viewing 3d picture pairs” or similar, and also animated guides on Youtube.

Now for Escher’s Waterfall. On the right below is my stripped down version. The water seems to be flowing uphill, and then pouring down to the bottom again. But then compare the right hand image with the small middle image: the configuration is just two tribars, one on top of the other. And on the left, with the vertical posts sawn off, so that our brains don’t have to connect them to the zig-zagging channels, the whole configuration seems to recede horizontally as it should, instead of stacking up impossibly.

# Roger Shepard’s Arc de Triomphe revisited

December 29th, 2013 by david

In 1990 the psychologist and artist Roger Shepard published a cartoon version of this effect, captioned “I stand corrected”, in his book Mindsights (page 91).  I wanted to try a photo processed version of it and here’s my second attempt.  When I tried before, back in 2008, I somehow couldn’t get my mind round what Shepard had done, and produced an even more twisted version.

M.C.Escher’s lithograph Belvedere from 1958 is famous variant on the theme.  Subsequent investigators have presented animated 3D versions of it that help explain the effect.

# Cheating with tessellations

January 3rd, 2012 by david

Tessellations are patterns whose repeat motifs fit together like jig-saw pieces, with no gaps and no repeats.  For an introduction, see our earlier animation. They can be abstract patterns, but the most intriguing are the ones devised by tessellation maestro M.C.Escher in the middle of the last century, which show representational motifs, such as animals, as tessellating patterns.

Designing abstract patterns that tessellate successfully is just a matter of getting the hang of some rules.  Discovering representational motifs that tessellate is much, much harder.  There are no procedures, or none that I know anyway.  It’s all trial and error, mostly error for me, and really hard!  Escher was brilliant at it.  My efforts are pretty feeble.

But fortunately, you can at least include representational motifs within your tessellations with a little trickery.  The pattern above, based on Leonardo’s famous Vitruvian Man, is an example.  The secret is to use segments of the outline of the representational motif for part of the outline of the tessellating pattern cell.

You do need to be up to speed with making abstract tessellations, and also pretty expert with Photoshop or an equivalent graphics package.  But if you’ve reached that point, or are just curious, here are stages in the development of the pattern shown above….

# Perspective Errors and the Best Visual Illusion of the Year Contest

May 11th, 2010 by david

This is a detail from British artist William Hogarth‘s print made in 1754, to demonstrate mistakes in perspective.  For example, the sheep lower left get larger with distance, not smaller, and the woman top right is leaning out of a window offering a light to a man in the distance.

However I’m really showing it because a brilliant new animated demo of perspective anomaly, by Kouchiki Sugihara, has just won the first prize in the international Best Illusion of the Year Contest.  Don’t miss it, the ten best entries are shown, and there is some brilliant new stuff.

Coming back to Hogarth, his print was way before its time.  It was over a hundred years later, late in the nineteenth century, that illusion and puzzle picture books became common.  Then artists took up the challenge, Magritte and Escher for example.

Want to see the whole of Hogarth’s print?

# The Twisted Stairs (version 2)

January 15th, 2010 by david

I’ve been wanting to do a new version of my earlier post of The Twisted Stairs.  That’s partly because the way I placed the figures in the original posting, they got in a bit the way of seeing the twist in the lateral flights of stairs. I reckon you can see the twist effect better now, as they transform from stairs seen from below (at the top by the balcony), to stairs seen from above (down at floor level). I wanted to see if I could get it right, because this is an impossible stair effect that maestro M.C.Escher never used. Sometimes his staircases as a whole can be seen either as from above or from below, but they don’t twist from one viewpoint to the other half way up. As I mentioned in the earlier post, I reckon that’s because the twist effect depends on fudging the perspective, and Escher didn’t do fudge. His perspective is almost always miraculously lucid.

Another reason for a new version is that I wanted to produce a high resolution version, suitable for giant 35 x 23 inch posters. As ever, you are welcome to use downloads of the image here for any private purposes, but if you wanted to think about buying a framed print, or giant poster, here’s where to take a look.

There are more technical details on the original post. I borrowed the figures for this new version from Durer, Pieter Brueghel the elder, and Hogarth.

# Animated Illusion Cartoons – re-posting of Chicken and Leaf

January 6th, 2010 by david

Woops, slight technical glitch with the original post of this, just before Christmas. So this is a re-posting of the third of our animated illusion cartoons, Chicken and Leaf. It may still run jerkily on first run through, should be OK second time around.

These cartoons are meant to work just like movie versions of a three- or four-frame cartoon in a newspaper – each one presents a situation that ends with a punch-line.  The cast of characters are all illusion figures of different kinds, but each cartoon depends on a particular illusion effect.

The main illusion effects to watch out for in this movie are tessellations, and especially the final transformation, which transforms across the image at the same time as it transforms locally:

You can also view Chicken and Leaf as a
Quicktime Movie

You can also see our this cartoon along with the previous ones in our Animated Illusion Cartoon category.

I’m fascinated by the effect that the movie ends with – a tessellation that transforms in space and in time. Tessellation (or tiling) wizard M.C.Escher was brilliant at these transforming patterns, as in his Metamorphosis prints, but of course couldn’t do animations.  I’m sure he’d have done the animations if he could, but without a computer they’d have taken years. In my animation there are two sequences of transformations, first where the pattern morphs in sync all over the screen – a number of people have done those – and then the one that morphs across the image as well as in time.  I’m not aware anyone else has done one of those.  Please let me know if so, I’d love to see it – and otherwise, I hope if you’re an animator you’ll be provoked into doing a better one than mine.

# Pinna’s Intertwining Illusion

December 5th, 2009 by david

This is a brilliant illusion discovered by Baingio Pinna of the University of Sassari in Italy.  The circles appear to spiral and intersect, but are in fact an orderly set of concentric circles. The illusion is due to the way the orientation of the squares alternates from circle to circle, and that contrast alternates from square to square within each circle. The illusion is related to the movement illusions of Akiyoshi Kitaoka and to twisted cord illusions.

What’s going on is suggested by this next version, with the edges enhanced, plus a bit of blurring.

This image approximates (with false colour) the data transmitted within the brain once the image has been filtered by cell systems early in the visual pathway, including centre-surround cell assemblies (a bit technical, that link). The role of these is to enhance edges, so that bright edges are now emphasised by dark  fringes and vice versa. Note that between the little stacks of alternating light and dark fringes, along the line of the circles, the dark fringes of bright squares align with the dark edges of adjacent squares and vice versa. The scale and spacing of the squares is just right to get that alignment, and as a result the effect enhances the inward turning, spiralling effect due to the orientation of the squares. The fringes combine to give an effect a little like interfering waves. The illusion seems to be bamboozling processes that are usually superbly effective at filtering out the key information about edges and their orientation in the visual field.

However, showing that centre-surround cell outputs could be enhancing the inward turning character of the lines forming the large circles doesn’t explain why the brain integrates the local effects into the perception that the large circles as a whole are spiralling inwards. I guess that’s because, to a much greater extent than we realise, we infer global configurations from what we see just in the central, foveal area of the field of view. That also seems to be the case with impossible 3 dimensional shapes, as in the impossible tribar.

# A tessellation pioneer

April 2nd, 2009 by david

You probably know the tiling patterns of M.C.Escher.  But how about Koloman Moser?  Here are a couple of his designs.

Moser was working in Vienna, Austria, a hundred years ago.  (He died in 1918).  I don’t know where he would have learned to do tessellating designs, that is, designs with motifs that repeat the way jigsaw puzzle pieces fit together, with no gaps or overlaps.  If you have checked out our tessellation tutorial, you’ll know that the secret of these designs is that the edge of each “tile” of the pattern must be able to be snipped into pairs of identical line segments.  Here’s how it works with Moser’s fish design.

To the right you can see that the fish outline can be divided into three pairs of segments, a yellow pair, a red pair and a blue pair.  In the yellow pair, the top line is just repeated lower down to make the pair, in a move called a translation.  The red and blue pairs are a bit more complicated.  In each pair, the lower line segment is a mirror reflection of the upper segment, but shifted downwards.  That kind of shifted reflection is called a glide reflection.  It’s a fact that any motif whose edges can be snipped into one pair of segments that repeat by translation, connected as here to two parallel pairs whose edges repeat by glide reflection, will tessellate perfectly.  And that’s just one of 28 recipes for motifs that tessellate.

The second Moser design is based on a superficially simpler recipe, but it’s very, very clever.  In fact, it’s fiendish …..

So much so that I’ve had to correct the commentary at 13 May 2012! ….

# Bad Day in the Art Museum

March 28th, 2009 by david

Interesting things can happen when you have pictures within pictures. Not so much, for example, with an everyday photo of an art gallery, if all the pictures are behaving well and staying in their frames. But sometimes it’s not possible to tell when the picture of a picture ends, and the picture of the real world begins. Here’s an example, in which the paintings in an art gallery are definitely not behaving how paintings should.

M.C.Escher did some brilliant pictures in which the boundary between the real world and the graphic world breaks down in the same way. The most famous is his print Drawing Hands of 1948, but you’ll find lots of others. Amongst contemporary artists, Rob Gonsalves has done some really clever paintings, such as Unfinished Puzzle. It’s the kind of issue that interested Picasso and Braque too, in their cubist paintings. In one by Braque you can see a cubist palette hanging on an illusionistic nail.

Here’s another famous example, a painting from a bit over a century ago, by Pere Borrell del Caso.  It’s called Escaping Criticism. I guess the artist felt hard done by at the hands of critics, and did this as a demonstration of virtuosity.  (I believe the original painting is in the Banco de Espana Madrid   –  Spain’s national bank).

The guys climbing back into a painting in my image are borrowed from a copy of Michelangelo’s lost study for the Battle of Cascino. The shipwreck is from a 200 year old painting by English Romantic painter J.M.W.Turner in the art museum Tate Britain in London, of a bad day in the English Channel.

# Tessellation Animation

The Dutch tessellation whizz M.C.Escher was fascinated by transformations from one tessellation to another, for example in his series of prints Metamorphosis. I’m sure he would have explored animated versions if it had been practical in the 1940′s. So I’ve borrowed a couple of his motifs and animated them. I showed an animated transformation in an earlier post, but that was between two designs that shared the same kind of symmetry. (See the earlier tessellation tutorial for how these tessellations work. If you like technical detail, my earlier animation was of two motifs based on Heesch tessellation no. 11). Sticking to just that one kind of tessellation meant that the corners of each cell of the design had to remain stationary, and only the edges of the cells transformed. This new transformation is a bit different, because it’s not just a transformation from one motif to another, but between two different kinds of symmetry pattern – Heesch nos 17 and 18 in the tutorial – and the corners of the cells of the pattern are not fixed.

In the earlier transforming animation, the design transformed in space, across the image, as well as transforming in time. If I’ve got it right, (I’m not 100% sure about this), that kind of time plus space transformation is not possible in an animation if the corners of the cells of the tessellation change position, as in my new tessellation above. So in this new animation, there’s no change from cell to cell across the design, and all the cells transform together.

I’m fascinated by the artistic possibilities of these kinds of animation, and one aspect of it is to do with what you might call the dance rhythms of the animation. Here’s a variation on the new animation, speeded up and with an added wave that gives a quite different kind of pulse to the design.

These animations are bit monochrome for the moment – colour is on the way, but I’m on a steep learning curve with file sizes, compression etc.