VW - CVM 1.1

Functions Invariant under Positive Rotations

Once rotations are among the desired symmetries, there are fewer possibilities for the terms. The most general element is

a_nmcos(nX) cos(mY) + d_nmsin(nX) sin(mY).

We continue to use the conventions for naming isometries established above .

Table: Symmetry types with rotations

type	G	E	Recipe for this type and remarks
pm'm'	p2	pmm	sines only; M and Mv negative
pm'g'	p2	pmg	n even with sines and odd with cosines; L and Bv negative
pg'g'	p2	pgg	n + m even with sines and odd with cosines; B and Bv negative
pmm	pmm	pmm	cosines only; general parities; M and Mv positive
p'bmm	pmm	pmm	n is odd; new cell half of old; Av negative; negating half-turns
p'cmm	pmm	cmm	n + m is odd; new cell half of old; B and Bv negative; negating half-turns
pmg	pmg	pmg	m even with cosines and m odd with sines; B and Lv positive
p'bgm	pmg	pmm	m odd with sines alone; new cell half of old; M and Mv negative
p'bmg	pmg	pmg	pmg conditions and n odd; B and Bv negative
p'cmg	pmg	cmm	pmg conditions and n + m odd; L andAv negative
pgg	pgg	pgg	n + m even with cosines and odd with sines; B and Bv positive
p'cgg	pgg	cmm	n + m odd with sines alone; new cell half M and Mv negative
p'bgg	pgg	pmg	pgg conditions and n odd; Lv and A negative

The Rhombic Lattice Experiment interactively with Wallpaper Design The Rectangular Lattice Recipes for Negation Functions having Mirrors but No Positive Rotations