Symmetry groups

Planar isometries

Talking about planar symmetries is a bit simplistic, as we deal with isometries, geometric operations that we are going to describe.

An isometry is an operation that preserves distances: if F is an isometry, then for any points M et N in the plane:

d(F(M), F(N)) = d(M, N)

An isometry transforms a straight line into a straight line, and a circle into another circle with the same radius. Four families of planar isometries exist:

Translations, which move objects in straight line to a constant distance, always in the same direction.
Rotations, which rotate objects with a constant angle around a fixed point (its centre).
Orthogonal symmetries, which mirror objects around a constant straight line (its axis).
Glide symmetries, which combine an orthogonal symmetry and a translation along the axis of the symmetry.

Groups

Groupes are very general mathematical objects, with no a priori link with geometry. A group is a set of properties that a set has when it is given an "internal law", like addition. A group is a set that:

• is stable through its law: composing two elements of the set always results in an element of the set.
• Has a neutral element: like zero for addition, composing the enutral element with any other element results in the second element.
• Every element of the group has an inverse, like the opposite for addition. Composing an element with its inverse results in the neutral element.
• The law is associative: a+(b+c) = (a+b)+c. Therefore parentheses are useless. Note that commutativity (a+b = b+a) is not mandatory.

So why is this useful? For many things, among which arabesques. Let's think about translations. Translations form a group, though not the most exciting one. Just considering a subset of translations will not always give a group structure. Another example is the set of all the rotations around the origin of the plane. It is a group, while the set of all the rotations is not a group, as composing two rotations of opposite angles and different origins will result in a translation...

Consequences for Arabeske

We aim at "paving" the plane, that is to find a pattern which, once translated, turned, mirrored... will not globally change. A checkered plane is a good example. Inside this pattern, a smaller one can be isolated, which will result in the whole patterned plane once it is applied all the symmetries of the group.

Smart people, i.e. not me, can demonstrate that there are exactly 17 groups of planar symmetries. The first nine share the property that their patterns are not necessarily of immutable proportions. This is why they are not used in Arabeske. There are 8 left, which are described below, and which are the only ones that feature rotations other than half-turns. Their exotic names are those used in Arabeske, which come from geometric cristallography.

Talking about rotation, a little word about symmetry "orders". The plane can only be paved using regular polygons with 3, 4 or 6 sides. It is one of the meanings of the numbers in the groups names. It is not possible to do so using pentagons or heptagons. This is why groups like p5 or p7 do not exist. Pentagons have been extensively used in islamic architecture, as they are linked with the "golden number", but they cannot pave the plane. You will notice that stars having 20, 40 or 100 arms are always located at the center of rectangular, not pentagonal patterns.

The following table describes the features of the 8 symmetry groups used in Arabeske. It does not deal with translations (which are quite obvious), and uses the letter "P" as a base pattern for better understanding.

Group name	Features	Illustration
p3	3 sets of 1/3 turn rotation centers
p31m	3 sets of 1/3 turn rotation centers Symmetry axes linking "primary" rotations centers
p3m1	3 sets of 1/3 turn rotation centers Symmetry axes linking primary axes to the other ones
p4	2 sets of 1/4 turn rotation centers 1 set of half-turn rotation centers
p4g	1 set of 1/4 turn rotation centers 1 set of half-turn rotation centers Glide symmetry axes linking "primary" rotation centers Symmetry axes linking secondary centers
p4m	2 sets of 1/4 turn rotation centers 1 set of half-turn rotation centers Symmetry axes linking 1/4 turn rotation centers Symmetry axes linking 1/4 turn rotations centers to the closest half-turn rotation centers.
p6	1 set of 1/6 turn rotation centers 1 set of 1/3 turn rotation centers 1 set of half-turn rotation centers
p6m	1 set of 1/6 turn rotation centers 1 set of 1/3 turn rotation centers 1 set of half-turn rotation centers Symmetry axes linking 1/6 turn rotation centers to every neighbouring rotation centers

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