Rotational Symmetry Simply Set Out

Lattice Labyrinth Tessellations rarely (and inconsequentially so far as I’m concerned) exhibit mirror symmetry, that is they don’t look the same if you reflect them or flip them over to look from the other side. They are right- or left-handed you might say – which to most of us means asymmetrical. They do however exhibit beautiful rotational symmetry. Here are some flower-petal patterns that illustrate what I mean and the symbols that are used to illustrate rotational symmetry properties.


The chessboard or square lattice tessellations exhibit patterns of symmetry axes. They also exhibit mirror symmetry, but let’s ignore that to avoid complicating the picture –  only the rotational symmetry is inherited by Lattice Labyrinths tessellations.


The tetrad axes are marked in two colours as the red and black axes have different environments, the centres and corners of squares respectively. The dyad axes appear in differing orientations because that’s how they are inter-related by the tetrad and hexad or triad axes, around with they “swing”. If you now return to some of the other posts, you can see how they (so far, there will be others) share one or other of these patterns of rotational symmetry axes.


About davescarthin

After terminating academic and local government careers, long an independent bookseller/publisher at Scarthin Books, Cromford, Derbyshire, UK. An antiquarian bookseller in two senses, I now have more time to be an annuated independent post-doc, developing the long dormant topic of lattice labyrinth tessellations - both a mathematical recreation and a source of compelling practical tiling and textile designs. Presenting a paper and experiencing so many others at Bridges Seoul 2014 Mathart conference was a great treat, as were the spirited MathsJam Annual Conferences in November 2016 and 2017. I'm building up to a more academic journal paper and trying hard to find practical outlets in graphic design and landscape architecture. I submitted 8 ft square tiling designs to the Wirksworth Festival Art and Architecture Trail in 2016 and 2017. I love giving illustrated talks, tailored to the audience. Get in touch to commission or to collaborate.
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