VW - CVM 1.1

The Rhombic Lattice

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 M_x, the reflection about the x-axis, which is the central axis of the new cell; M_v 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.

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	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

The Square Lattice Experiment interactively with Wallpaper Design Recipes for Negation Negating Isometries The Rectangular Lattice