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The absolute minimum symmetry is effectively none at all beyond

In one standard notation, that minimal type of symmetry in a repeat is called p1 - which means a

As touched on above, one of the possible symmetry elements is rotation of the design to match up with itself again. Whereas a one-off mandala pattern can have any number of rotationally repeating spokes, the constraint of translation across a two-dimensional surface limits the rotation numbers which work in plane symmetry. The first number which works is 2 - which is handy for creating bidirectional fabrics. A rotation 2 centre is called a

The type of symmetry element with which people are most familiar in real life (albeit imperfectly!) is the

The simplest type of mirror reflection, when aligned with cell axes, looks like this pm example (ie still a primitive cell but with an m for the addition of the mirror). But there's another type of reflection more commonly seen (albeit again imperfectly) in plants than animals. This is the

When the mirror or glide reflection is aligned with the cell diagonals instead of the axes, it automatically induces the other type of reflection too. For my non-centred example of this I'm using a very special kind of rhombus called a square (because it's so much easier to do this with bitmapped images on Spoonflower if I stick to cartesian axes!). There you can see that the reflection axes are really diagonal ones. But the conventional view of cm would be the centred cell one (c for centred and m for mirror).

In the same way that mirror and glide reflections automatically induce each other when placed on the diagonals of a unit cell (for the centred examples above), combining any two of diads, mirrors and glides in various ways automatically induces another symmetry element. By (naming) convention, we'll assume we started with the diad set and then added a reflection to those. p2mm has two sets of mirrors at right angles to each other. p2gg has two sets of glides at right angles. p2mg has one set of mirrors against one set of glides. c2mm has sets of alternating mirrors and glides on both diagonals, which become the axes in centred view c2mm.

Temporarily skipping over a rotation number of 3 (because it's hard!), a 4-fold rotation centre is called a

Getting back to 3-fold rotations, the

The next valid rotation under the constraint of repeating across an infinite plane is 6-fold, ie the

No other types of rotation are compatible with infinite plane repeats. Any 5-fold rotational patterns you may think you have seen are the result of clever cheating in hiding the symmetry breakage from you.

See also:

• design index