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! ….
Analysis corrected at May 2012!
The pale blue segment of the pattern at top left and centre is the fundamental region. The whole of the rest of the pattern is made up of repeats and reflections of this segment. So the lower half of column 1, in the centre of the image, is just a dropped repetition of the pale blue segment. Column 2 is similarly made up of repetitions of the segment stacked on top of one another, but reflected to flip the design left to right, and then with the whole column dropped by one half of the height of the pale blue, original segment. Column 3 then repeats column 1, but dropped by one quarter of its height. Column 4 then repeats column 2, but once again dropped by a quarter of its height. And that’s it. The area of the pattern within the yellow rectangle is the unit cell of the pattern, repeating like a rectangular bathroom tile.
The clever bit is designing the fans of white flowers, foliage and birds so that they keep joining up in different combinations across the reflection axes. This kind of design, known as a drop design, was the most common way for 19th century designers to compose wall-paper symmetry designs. Today we would all the drops glide reflections.
I’m interested in getting my mind round all this (not my forte) because I want to explore the effect of animated tessellations, and that requires a lot more expertise.