VW - CVM 1.1

# 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 . 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 . 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.

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.

 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 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' cm cmm only sin(nX) cos(mY) terms appear; M, A positive, Mv, Av negative cmm cmm cmm only cos(nX) cos(mY) terms appear; M, A, Mv, Av positive c'mm cmm pmm cmm condition and m,n odd; M, A, Mv, Av positive; new cell half as large

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
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