Examples of the 17 plane symmetry groups collected together - complete with a key to the symmetry elements and the designations of the plane groups. This would make a banner poster on a wallpaper swatch (or on each half of a large decal or yard of fabric).
The word plane
in this context refers to the concept of the design being repeated over and over in a self-similar way across an infinite 2-dimensional space. It's a key difference between a picture that one might frame and a viable pattern for printing rolls of fabric. This underlying requirement for endless repetition of the design necessarily limits the types of symmetry which work.
The absolute minimum symmetry is effectively none at all beyond translation
- ie that act of repeating the design indefinitely in two different directions. These axes
do not have to be perpendicular to each other, as per cartesian co-ordinates, let alone repeat at the same rate. The most basic repeating unit cell is a parallelogram (ie any non-zero angle and different side lengths). However, for the purposes of bitmapped images uploaded to Spoonflower, the design does need to come back in line with cartesian co-ordinates at some point! So, for mere convenience only, my examples here use a special type of parallelogram called a rectangle - which has the benefit of a right angle but still doesn't insist on equal length sides.
In one standard notation, that minimal type of symmetry in a repeat is called p1
- which means a primitive
unit cell with a pattern which only matches up with itself 1
time, ie the original orientation, when you rotate it around. The half-drop and half-brick Spoonflower repeat types do nothing to alter this lack of symmetry unless the design itself is already built to have a different repeat symmetry. They are merely mechanisms for inducing a rhombus parallelogram instead of a rectangle one. So this superficially centred version of p1
has no more real symmetry than the previous example. Nor does this stepped version of p1
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 diad
. Adding this rotation doesn't even require any further conditions on the unit cell - it can still be any parallelogram. However, as you can see from this annotated example of p2
, adding one diad immediately induces a whole set of them. They always turn up in 4-packs (although some can be turned into tetrads or hexads).
The type of symmetry element with which people are most familiar in real life (albeit imperfectly!) is the mirror
reflection. There's the horizontal mirror at the surface of water and the vertical mirror between the visible sides of the body of many animals. Introducing a mirror does affect the unit cell though. It forces a right angle - either between the sides, ie a rectangle, or (less obviously) between the diagonals, ie a rhombus (which equalises the sides instead). However, because rectangles are so much easier to handle, there is a convention to reorient and double all rhombus cells into rectangles and refer to them as centred cells. This type of centredness actually means something, unlike the half-drop and half-brick versions. Meanwhile, the underlying requirement to repeat across the plane means that reflections always turn up in pairs.
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 glide
reflection. It mixes up a simple mirror with half a translation. So, whereas some plants do put out pairs of leaves opposite each other (or have leaf veins which branch exactly opposite each other), many plants will put out a leaf (or a vein) on one side and then the other and then the first side again, alternating all the way as one glides along the stem. Here is an example of pg
(ie still a primitive cell but with a g for the addition of the glide).
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 tetrad
. When repeated across a plane, a tetrad automatically induces a second tetrad (centred between the original ones) and a pair of diads (in the same positions as the remainder of a diad 4-pack). Now the unit cell has to be right-angled and equal-sided (equilateral
) - ie a square. The simplest option, p4
(primitive cell and a tetrad set), has no further symmetry elements. But adding any reflections at all will induce even more of them. p4mm (p4m)
has mirrors cutting both sides and diagonals, relegating glides to just the diagonals. p4gm (p4g)
has glides cutting the sides and diagonals, relegating mirrors to just the diagonals.
Getting back to 3-fold rotations, the triad
does work in a plane repeat but it always comes in a 3-pack (one of which may be a hexad) and it requires a special type of rhombus unit cell formed from 2 equilateral triangles. I like to call this the trombus. Again it's handy to show this on site as a centred rectangle - though it's not labelled that way. p3
(primitive cell and a triad set), has no further symmetry elements. Adding one type of reflection automatically induces the other type and there are a couple of ways of doing this. p3m1
has the mirrors perpendicular to the trombus sides. p31m
has the mirrors parallel with the trombus sides.
The next valid rotation under the constraint of repeating across an infinite plane is 6-fold, ie the hexad
. This simultaneously occupies the position of one of a set of 3 triads and one of a set of 4 diads. The unit cell is still a trombus. The simplest option, p6
, has no further symmetry elements beyond these automatically induced ones. If you try to add any reflections at all to this you inevitably get the full p6mm
set of mirrors and glides.
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.
• design index