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The Rhombic Lattice

We could develop recipes for the rhombic lattice from scratch using lattice coordinates. Instead, we take a shortcut and recycle our work from the rectangular cell [Back]. This can be done because the rhombic lattice arises from a rectangular one if you introduce a diagonal half-translation.

We can reuse all the isometry notation from the rectangular cell [Back]. Now the translation h, formerly viewed as a horizontal translation, is actually the diagonal translation halfway into the new cell, though we still call it h. A new player is Mx, the reflection about the x-axis, which is the central axis of the new cell; Mv and the As and Bs are as before.

[isometries in rhombus]
The rhombic lattice cell

The general term in the sum is the same as that for the rectangular cell, but now we have the requirement that m + n should always be even. This is so the diagonal half-translation in the rectangular cell will be a symmetry of the function. One consequence is that when we require n to be odd to achieve a negation by h, m must be odd as well.

Table: Symmetry types in rhombic cells
type G E Recipe for this type and remarks;
m + n is always even
cm' p1 cm only sin(mY) terms appear;
M negative
cm'm' p2 cmm only sin(nX) sin(mY) terms appear;
R positive, M and Mv negative
cm [Image] cm cm only cos(mY) terms appear;
M and A positive
c'm cm pm only cos(mY) terms appear; m, n odd;
M and A positive
cmm' [Image] cm cmm only sin(nX) cos(mY) terms appear;
M, A positive, Mv, Av negative
cmm [Image] cmm cmm only cos(nX) cos(mY) terms appear;
M, A, Mv, Av positive
c'mm [Image] cmm pmm cmm condition and m,n odd;
M, A, Mv, Av positive;
new cell half as large

[Next] The Square Lattice
[Skip] Experiment interactively with Wallpaper Design
[Up] Recipes for Negation
[UpUp] Negating Isometries
[Prev] The Rectangular Lattice

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 9:27:08 AM
Comments to: CVM@maa.org