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Functions Invariant under Positive Rotations

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

anmcos(nX) cos(mY) + dnmsin(nX) sin(mY).

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

Table: Symmetry types with rotations
type G E Recipe for this type and remarks
pm'm' [Image] p2 pmm sines only;
M and Mv negative
pm'g' [Image] 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 [Image] pmm pmm cosines only; general parities;
M and Mv positive
p'bmm [Image] pmm pmm n is odd; new cell half of old;
Av negative; negating half-turns
p'cmm [Image] pmm cmm n + m is odd; new cell half of old;
B and Bv negative; negating half-turns
pmg [Image] 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 [Image] pmg pmg pmg conditions and n odd;
B and Bv negative
p'cmg [Image] pmg cmm pmg conditions and n + m odd;
L andAv negative
pgg [Image] 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


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Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 08 Jul 1998 --- Last modified: 18 Aug 1998 23:59:59
Comments to: CVM@maa.org