# Intersection coordinates in lattice

• aaaa202
In summary: The equation for the line in yellow is y = mx + c, where m and c are the same for both bases. So to reflect the line in an axis perpendicular to the line, one would use the equation y = mx + c' where c' is the vector perpendicular to the line y = mx.
aaaa202
On the drawing below is a hexagonal lattice. For the basis vectors one can choose either the set of arrows in black or the set in yellow. The intersection coordinates of the plane in green seems to be the same regardless of choosing the black or the yellow basis. Why is that? My teacher said it is due to the 6-fold symmetry of the lattice while for me it seems like sheer accident.
I know of course that a hexagonal lattice has 6-fold symmetry, which for me means that I can rotate it by 60 degrees about any point and it will look the same. But how do I translate this symmetry property into a rotation of 60 degress of the basis vectors?
I know this question might be obvious to you, but I am having a hard time exactly seeing how to apply symmetry properties. How do YOU intuitively see it?

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I'm not entirely sure I understand the question. It would help if you were to post the expressions you have for the co-ordinates and indicate the sense in which they are the same in both bases. But assuming I'm interpreting correctly:
Instead of rotations, consider another symmetry.

What otter symmetries deles the lattice have? Should I mirror it in an axis? In this case which?

aaaa202 said:
What otter symmetries deles the lattice have? Should I mirror it in an axis? In this case which?

Yes, I'm thinking of a reflection. What sort of reflection leaves the line unchanged?

In an axis perpendicular to the line. So reflection in that axis will transform from one basis to the other and the lattice is the same. But does that tell me that the choosing one basis or the other gives the same? I feel like I am still missing the crucial point.

aaaa202 said:
In an axis perpendicular to the line. So reflection in that axis will transform from one basis to the other and the lattice is the same. But does that tell me that the choosing one basis or the other gives the same? I feel like I am still missing the crucial point.
You have to ask yourself what you mean by the coordinates being the same.
Since you will be using different basis vectors, you must mean not that they are automatically the same but that they can be made to look the same by a suitable mapping between the bases. Moreover, it need not be that each individual point on the green line gets the same coordinates, so you also need to find a mapping of the line to itself.
Having understood that, I think it's fairly easy to see from the diagram that this can be done.

## 1. What are intersection coordinates in lattice?

Intersection coordinates in lattice refer to the points where two or more lattice lines intersect. These coordinates are used to identify specific locations within a lattice structure.

## 2. How are intersection coordinates determined in a lattice structure?

The intersection coordinates in a lattice structure are determined by finding the points where the lattice lines cross or intersect. This can be done using mathematical equations and geometric principles.

## 3. What is the significance of intersection coordinates in lattice?

Intersection coordinates in lattice are important because they help to identify specific locations within the lattice structure. This can be useful in understanding the overall structure and properties of the lattice.

## 4. Can intersection coordinates change in a lattice structure?

Yes, the intersection coordinates in a lattice structure can change depending on the lattice type and any external forces or conditions applied to the lattice. This can affect the overall structure and properties of the lattice.

## 5. How are intersection coordinates used in lattice simulations?

In lattice simulations, intersection coordinates are used to model the behavior of the lattice under different conditions. By changing the coordinates, scientists can predict how the lattice structure will respond to different forces or stimuli.

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