Test if a point on a hexagonal lattice falls on a specified superlattice?

Question

Based on previous answers (1, 2, 3) integers $i, j$ produce a hexagonal lattice using

$$x = i + j/2$$ $$y = j \sqrt{3} / 2.$$

From a point $k, l$ I can make a superlattice from integers $I, J$ using

$$i_{sup} = I k + J (-l)$$ $$j_{sup} = I l + J (k+l)$$

and

$$x_{sup} = i_{sup} + j_{sup}/2$$ $$y_{sup} = j_{sup} \sqrt{3} / 2.$$

If I have a point $m, n$ on the lattice, is there a simple test I can apply to find out if it is on the superlattice as well?

Example for $k, l = 3, 1$

Python script

score 1 · Accepted Answer · answered May 29 '20 at 14:54

1

This one's all linear algebra all the time: your two lattices can be considered as transformations of a square lattice, taking integer $(i,j)$ through a linear transformation.

The two lattices coincide at a particular point if the $(i,j)$ for a particular lattice point, passed through the transform of that lattice and then the inverse transform of the other lattice, give integer results.

So in this particular case:

the blue lattice is the transformation of the unit square lattice through

$$A = \begin{pmatrix} 1 & \frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} \\ \end{pmatrix} $$

and the red lattice is the transformation of the unit square lattice through

$$B = \begin{pmatrix} \frac{7}{2} & -1 \\ \frac{\sqrt{3}}{2} & 2\sqrt{3} \\ \end{pmatrix} $$

To get the test matrix, we find $$B^{-1}A = \frac{\begin{pmatrix} 4 & 3 \\ -1 & 3 \\ \end{pmatrix}}{15}$$

Now, to discover if a particular point from the blue lattice is also on the red lattice, we simply apply this new matrix to the $(i,j)$ coordinates, and see if the result is in the integers: $(3,1)$ gives $(1,0)$ so is on the red lattice; $(4, -7)$ gives $(\frac{1}{3}, -\frac{5}{3})$ which is not integer so it is not on the red lattice.

answered May 29 '20 at 14:54

Dan Uznanski

11,025

Thank you for the speedy answer! I'll dig in in the morning after coffee; this is extremely helpful to me because I've been away from math for more decades than I'd like to admit. I was hoping there would be a test that would return let's say zero if on-red-lattice and non-zero if off-red-lattice. If the result is $x, y$ for example I suppose I could apply a test like $|mod(x, 1)| + |mod(y, 1)|$ to see if it is a pair of integers or not. – uhoh May 29 '20 at 15:03
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well, if you're working in python, int(x) == x works to check for integerness. – Dan Uznanski May 29 '20 at 15:04
If there was a test that involved integer-only math it would be better, mixing floats and ints always runs the risk of roundoff weirdness. Maybe that's better-asked as a next question. – uhoh May 29 '20 at 15:07
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Well - use that matrix on top of the fraction and then check for divisibilty by 15 – Dan Uznanski May 29 '20 at 15:08
There was a typo in my third equation, it should read $$i_{sup} = I k + J (-l).$$ Does that change your $B$ to $$B = \begin{pmatrix} \frac{7}{2} & +1 \ \frac{\sqrt{3}}{2} & 2\sqrt{3} \ \end{pmatrix}?$$ – uhoh May 30 '20 at 05:19
How to best format the question mark that immediately follows an equation in a question sentence? :-) – uhoh May 30 '20 at 05:21
No, I couldn't follow your formula and figured the two matrices by hand. – Dan Uznanski May 30 '20 at 05:24
I think the +1 in $B$ is what you meant: https://i.stack.imgur.com/cq3nb.png Both +1 and -1 generate the same points, but +1 is "nicer" somehow. – uhoh May 30 '20 at 06:15
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The left one is the image as it was when I made the answer. It looks like that changed too – Dan Uznanski May 30 '20 at 06:18
ya it was a typo, it wouldn't make sense for $j$ to appear on the right hand side of that equation, and for some reason I'd carried it into the plotting script as well. With that corrected, putting a +1 into $B$ brings everything back into alignment. – uhoh May 30 '20 at 06:21
Do you want me to roll it back? – uhoh May 30 '20 at 06:23
If $k, l = (3, \ 1)$ and $$F = \begin{pmatrix} k & -l \ l & k+l \ \end{pmatrix}$$ generates the red dots, then $$F^{-1} = \frac{1}{k^2+l^2+kl}\begin{pmatrix} k+l & l \ -l & k \ \end{pmatrix}$$ applied to $i, j = (13, \ 0)$ should yield $(3, -1)$ and it does. This is exactly what I needed. Thank you for helping me through this! https://i.stack.imgur.com/DZm5n.png – uhoh Jun 01 '20 at 06:07
still plugging away at this, but this question is more specific: Test to determine if a coincident point in a pair of rotated hexagonal lattices is closest to the origin or not? – uhoh Jun 04 '20 at 14:36

Test if a point on a hexagonal lattice falls on a specified superlattice?

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