CRYSTAL SYMMETRIES

symmetry group of the pattern: G = p3m1

symmetry group of the lattice: T = p6m

"point group" symmetry: G/T = d3

symmetry of the motif: d1

and now: color symmetry

a page from

Grammar of Ornament

Owen Jones, 1868

Designers of ornamental patterns everywhere use colors symmetrically.

A detail from Grammar of Ornament.

Notice that the 5-pointed star patterns do not contradict the crystallographic restriction.

Notice also the color changes in some of these patterns.

Chemists use colors to describe crystal structures.

Here the “unit cell” is primitive, not centered, because the differently colored atoms are different species.

Color symmetry is everywhere.

The blue vertical vector takes white horses to red ones and vice versa; thus it effects a color permutation.

m2'x

m1' xrr x

m2 x

r x

m1 x x

rrr xrrr x

xm1 x

r x

m2 x

rr x m1' x

m2'xm2'x

m1' xrr x

m2 x

r x

m1 x x

rrr x

The sectors with yellow circles are an orbit of a subgroup H. We say H fixes the color yellow.

G = symmetry group of the square

The sectors with blue circles corresponds to a left coset of H,and so on. Thus the number of colors equals the index of H in G.

A pattern colored by a subgroup and its left cosets is called a “perfect coloring” (G,H).

rrr x

xm1 x

r x

m2 x

rr x m1' x

m2'x

Is this a perfect coloring?

rrr x

xm1 x

r x

m2 x

rr x m1' x

m2'x

rrr x

xm1 x

r x

m2 x

rr x m1' x

m2'x

Left cosets: each symmetry operation effects a color permutation. For example,

m2 interchanges yellow and pink, and also green and brown.

Right cosets: What does m2 do?

WHY LEFT COSETS?

rrr x

xm1 x

r x

m2 x

rr x m1' x

m2'x

1. Find (and list) the color permutations effected by each of the eight symmetry operations of the square.

2. On any face of your plastic cube, draw one of the color patterns shown above. Then draw the same pattern on each cube face, in such a way that each symmetry operation either fixes black and white, or interchanges them.

3. What is the subgroup H for your colored cube?

Exercises

A crystallographic subgroup H of a crystallographic group G has a point group PH and a translation group TH. Then either

a) PH is a subgroup of P, the point group of G; or

b) TH is a subgroup of T, the translation group of G; or

c) both.

Let H be the subgroup that fixes the color black (and thus white). Which type is it?

a) PH is a subgroup of P, the point group of G; or

b) TH is a subgroup of T, the translation group of G; or

c) both.

Which type is this?

More colored cubes

Find a subgroup of index 6 and color the cube accordingly.

Find a subgroup of order 6 and color the cube accordingly.

When each coset is assigned a color, this permutation of cosets permutes the colors.

The cosets of a subgroup H of index k partition G into k sets:

G = H « g2 H « g3 H « . . . « gk H (g1 = e)

Left multiplication of the cosets by any g in G permutes them: g gi H = (g gi) H = another coset

For example, if G = the group of the square and H = {e, m1}, the cosets are rH = {r, m2}, r2H ={r2, m1’}, r3H ={r3, m2’} and r(rH) = r2H, r(r2H) = r3 H, r(r3H) = H.

That is, r corresponds to a cyclic permutation of order 4.

Thus color symmetry is not really about colors, it’s about subgroups and left cosets.

But there are subtleties.

rrr x

xm1 x

r x

m2 x

rr x

m1' x

m2'x

rrr x

xm1 x

r x

m2 x

rr x m1' x

m2'x

More about mapping one group onto another

If G is any group, and f is a mapping of G such that

f(g1g2) = f(g1) f(2),

then the set of images {f(gi)} forms a group, and the elements of G that map to the identity in that group is a normal subgroup of G, called the kernel of f.

In perfect colorings, the subgroup that fixes all the colors is the kernel of the permutation map. (This subgroup may consist only of the identity {e}.)

If H is not normal, then the kernel is a subgroup of H.

NOTE: H is NOT determined by the kernel.

m2'x

m1' xrr x

m2 x

r x

m1 x x

rrr xrrr x

xm1 x

r x

m2 x

rr x m1' x

m2'xm2'x

m1' xrr x

m2 x

r x

m1 x x

rrr x

What are the kernels of these colorings?

G = symmetry group of the square

Find the kernel N of the permutation mapping.

Analyze this pattern:

find the subgroup H that fixes black

find its cosets

find the kernel of the permutation mapping.

Atlas der Krystallformen by Victor Goldschmidt(pub. 1913 to 1923) contains 23606 crystal drawings and a short description of each.

Among them are many drawings of twins.

=+

Twinned crystals can be described by external and internal color symmetry.