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The Square Lattice

In the square lattice we continue with the same notation [Back], adding a few isometries that are not present with rectangular cells.

We refer to the reflection about the main diagonal of the square as Mm, the main diagonal mirror, and to reflection about the line joining midpoints of adjacent sides as Me, 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.

[isometries in square]
The square lattice cell

There is a bewildering variety of relationships among these isometries, but we list only a few useful ones: d Me = Mm, Mm 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 [Back]. 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' [Image] pmm p4m only C- terms;
R4 negative, M, Mv positive
p4'm'm [Image] cmm p4m only S- terms;
R4 negative, Mm and related mirrors positive
p4'gm' [Image] pgg p4g n + m odd with S- terms;
n + m even with C- terms;
R4, Me negative, A, Av positive
p4'g'm [Image] cmm p4g n + m even with S- terms;
n + m odd with C- terms;
R4 negative, A, Av positive
p4 [Image] 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' [Image] 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 [Image] 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 [Image] 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


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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: Sep 30, 2003 9:51:04 AM
Comments to: CVM@maa.org