Final answer:
To show that 2, 3, 4, and 6-fold rotational symmetry are the only possible discrete rotation symmetries of a 2D lattice, we can use the Problem Solving Strategy for Rotational Dynamics. By considering a point P on the lattice, we can show that the distance of P to the origin is invariant under rotations of the coordinate system. This demonstrates that the lattice has rotational symmetry, but only certain angles result in unchanged coordinates after rotation.
Step-by-step explanation:
The question asks us to show that 2, 3, 4, and 6-fold rotational symmetry are the only possible discrete rotation symmetries of a 2D lattice. To do this, we can use the Problem Solving Strategy for Rotational Dynamics.
- Let's start by considering a 2D lattice and a point P on this lattice. We want to show that the distance of point P to the origin is invariant under rotations of the coordinate system.
- Using the distance formula, we can find the distance of point P to the origin, which is d = sqrt(x^2 + y^2), where (x, y) are the coordinates of point P.
- Now, let's rotate the coordinate system by an angle of theta. The new coordinates of point P after rotation can be obtained using the formulas x' = x * cos(theta) - y * sin(theta) and y' = y * cos(theta) + x * sin(theta).
- Substituting these formulas into the distance formula, we get d' = sqrt((x * cos(theta) - y * sin(theta))^2 + (y * cos(theta) + x * sin(theta))^2).
- Simplifying this expression, we find d' = sqrt(x^2 + y^2) = d, which shows that the distance of point P to the origin is invariant under rotations of the coordinate system.
- Since the distance of point P to the origin is invariant, we can conclude that the 2D lattice has rotational symmetry. However, only 2, 3, 4, and 6-fold rotational symmetry are possible because these are the only angles theta for which the coordinates x' and y' remain unchanged when rotated.