VW - CVM 1.1

The Square Lattice

We refer to the reflection about the main diagonal of the square as M_m, the main diagonal mirror, and to reflection about the line joining midpoints of adjacent sides as M_e, the eccentric mirror. As usual, L is used for the related glide reflections, whose meanings we hope are clear from context. R₄ will indicate rotation of a quarter turn about the origin. Some of these are indicated in the diagram.

The square lattice cell

There is a bewildering variety of relationships among these isometries, but we list only a few useful ones: d M_e = M_m, M_m M = h² R₄.

Every type for which the square lattice is needed has R₄ as either a positive or negating symmetry. Therefore every pattern has a positive half-turn and we may start with terms of the kind listed for the rectangular cell types with half-turns . Because we require combinations of these terms, let us adopt shorthand notation as follows:

C⁺(n,m) = cos(nX) cos(mY) + cos(mX) cos(nY)
C^-(n,m) = cos(nX) cos(mY) - cos(mX) cos(nY)
S⁺(n,m) = sin(nX) sin(mY) - sin(mX) sin(nY)
S^-(n,m) = sin(nX) sin(mY) + sin(mX) sin(nY)

The terms with superscript + are invariant under R₄, while the others negate when R₄ is applied.

Table: Symmetry types in square cells

type	G	E	Recipe for this type and remarks
p4'	p2	p4	C- and S- terms appear; R4 negative
p4'mm'	pmm	p4m	only C- terms; R4 negative, M, Mv positive
p4'm'm	cmm	p4m	only S- terms; R4 negative, Mm and related mirrors positive
p4'gm'	pgg	p4g	n + m odd with S- terms; n + m even with C- terms; R4, Me negative, A, Av positive
p4'g'm	cmm	p4g	n + m even with S- terms; n + m odd with C- terms; R4 negative, A, Av positive
p4	p4	p4	C+ and S+ terms appear; R4 positive
p'c4	p4	p4	C+ and S+ terms; n + n odd; R4 positive, negating quarter-turns
p4m'm'	p4	p4m	S+ terms only; R4, positive, Mm, M negative
p4g'm'	p4	p4g	n + m odd with C+ terms; n + m even with S+ terms; R4 positive, Me, L negative
p4g	p4g	p4g	n + m odd with S+ terms; n + m even with C+ terms; R4, Me, L positive
p'c4gm	p4g	p4m	n + m odd with S+ terms only; R4, Me, L positive, Mm negative
p4m	p4m	p4m	C+ terms only; R4, Mm, M positive
p'c4mm	p4m	p4m	n + m odd with C+ terms only; R4, Mm, M positive, Me, L negative

The Hexagonal Lattice Experiment interactively with Wallpaper Design Recipes for Negation Negating Isometries The Rhombic Lattice