# How many are the points which are $n$th nearest to a certain point in a hexagonal lattice

Suppose there are infinite points arranged as hexagonal lattice. The question is the one as the title. First we choose a point called $A$. Then when we count the $n$th nearest points to $A$, what number will I get. Actually I don't know whether this is a math problem or a more physics one.

I'm so sorry I misused the jargon hexagonal lattice. What I want to say is something arranged like graphene where points indeed make some hexagon rather than something points arranged like triangles. I'm so sorry I've made such a stupid mistake.

• No matter whether this is a physics or math question, why don't you just draw the lattice and count? – ACuriousMind Oct 4 '14 at 16:59
• @ACuriousMind "infinite points arranged" - have fun counting. – oerpli Oct 4 '14 at 17:00
• @oerpli: For fixed $n$, you can always simply draw a suffiently large lattice and count. After doing this for a few $n$, the general prinicple almost always emerges. – ACuriousMind Oct 4 '14 at 17:02
• For questions like this, you should count the numbers for small distances, then look in oeis.org. There are three at a distance of one, six at distance $\sqrt 3$, three at distance $2$, six at the next. A search for $3,6,3,6$ gives 21 pages, not too many to look over. – Ross Millikan Oct 5 '14 at 3:34
• I suggest you edit your original question to call it a "honeycomb" lattice. This is the correct terminology, especially in the context of graphene. Also change the term in the title, so people who search for it later can find the answer :) – Lachy Dec 5 '14 at 5:22

This is a pure math question, and it is not fully trivial. It is very closely related to the number of ways an integer $n$ can be represented over the integers by the quadratic form $x^2 + xy + y^2$.

More precisely the graphene structure is the difference of the hexagonal lattice and its index 3 sublattice. The squared distance of elements of this lattice from the origin are of the form $x^2 + xy + y^2$. This will often have 0 or 6 solutions, but depending on the prime factors of $n$ it can have many more (and it has a precise expression in terms of the prime factors and their residue classes modulo 3).

Here's an image of a hexagonal lattice to help you count http://www-personal.umich.edu/~sunkai/teaching/Winter_2013/hexagonal.png You can image the lattice being infinite in extent outside the image. So pick any point and count its nearest neighbors. You count six. Now pick another point and count them? Do you notice a pattern? You can be sure that this pattern will continue because of symmetry the lattice has.

• I'm so sorry I misused the jargon hexagonal lattice. What I want to say is something arranged like graphene where points indeed make some hexagon rather than something points arranged like triangles. I'm so sorry I've made such a stupid mistake. – Simon Oct 5 '14 at 2:11

Let $N(x)$ be the cumulative number of distinctive neighbours in a distance $x$ for a certain point $A$ (all points with a distance of $x$ or less) and $n(x)$ be the distributive number of neighbours (distinctive points with the distance of exactly $x$).

$N(x) = n(0) + n(1) + \ldots + n(x)$

$n(0) = 0$

$n(x) = 3*x$

$N(x) = \sum_0^x n(x) = \sum_0^x 3x = \frac{3}{2}(n^2+n)$

PS: distance in the context above means how many edges between 2 points you have to use to get on the shortest path from one point to the other.

PPS: If you want to define the point itself as it's $0$th nearest neighbour then $n(0) = 1$ and you have to add $1$ to the final result.

• I'm so sorry I misused the jargon hexagonal lattice. What I want to say is something arranged like graphene where points indeed make some hexagon rather than something points arranged like triangles. I'm so sorry I've made such a stupid mistake. – Simon Oct 5 '14 at 2:12
• I don't exactly unterstand what you mean. The formula above is for a hexagonal lattice like in this one: jameshedberg.com/img/samples/graphene-lattice-topview.jpg – oerpli Oct 5 '14 at 8:39
• @oerpli The term "hexagonal" has a special meaning in this context. Although graphene, as in the picture you just shared, is made of hexagons it is in fact known as a honeycomb lattice. An actual hexagonal lattice has an extra atom in the middle of each hexagon. A hexagonal lattice is Bravais, a honeycomb lattice is not. – Lachy Dec 5 '14 at 5:21