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 exactly the same recipe, but it’s more complicated.  In fact, it’s fiendish …..

It’s easiest to think of this design as made up of small rectangles, as above.  Notice the eight tinted rectangles with a red line round them.  If you imagine that block of rectangles flipped over (reflected) horizontally, and then shifted downwards, you will end up with exactly the block of eight rectangles underneath the ones in the red outline.  So the blue tinted rectangle top left, for example, ends up reversed left to right, and one row up from the bottom, on the right.  And that is the global symmetry transformation for this design, which means that if you took the entire pattern, flipped it horizontally, and shifted it down to the extent of two of the tinted rectangles, it would appear completely unchanged.

But what makes this global pattern shift hard to spot is that there are local symmetries embedded within it.  Look at the yellow tinted rectangle, top row.  Now note that it is repeated one row down, to the right, but reflected.  In fact the whole column of tinted rectangles under the yellow one in the top row is repeated to the right, but reflected, and shifted one rectangle down. But that’s only a local symmetry transformation – if you reflected the whole pattern like that, and then shifted it down one rectangle, it would no longer look the same. That gives the pattern unexpected effects. Check out the blue tinted bird again, top left.  Note the red tinted rectangle immediately to the right, the one the bird is looking at, as it were.  But now check out the identical blue tinted bird in the second row down.  The rectangle that bird is looking at is completely different.  It’s not just tinted yellow, instead of red, the design is quite different.  So whichever of these two blue birds you look at, the identical design just ahead of it seems to flow flawlessly into a completely different design in the next rectangle.  That takes a bit of organising.

And there are other even more local symmetries too.  Note upper right, where I’ve tweaked up the contrast a little.  The pale leaves top left in that patch are repeated exactly immediately to the right, but as dark leaves. Then note that the leaf pattern and the bird immediately below the pale leaves are also repeated to the right, still within that contrasty patch.

These repeats of the pattern are all varieties of symmetry transformation, but because local and global ones are all mixed up it took me a lot longer than I’d like to admit to puzzle out how this pattern works.

Why bother?  Because I want to explore the effect of animated tessellations, and that requires a lot more expertise than I’ve yet got with this stuff.