Using Python code below as a template, demonstrate the Rotation + Translation theorem for a two-fold rotation axis, show that the combination of a two-fold rotation, and a translation vector generates a new two-fold rotation axis at half the translation distance.

theta = np.pi/2# define theta, the angle of rotation

A1 = np.mat([[1, 0, 0],# A1 is a 3x3 rotation matrix around the x axis!

[0, np.cos(theta), -1*np.sin(theta)],

[0, np.sin(theta), np.cos(theta)]])

b =np.mat([[1],[1],[1]])# b is a column vector!

print("rotation matrix: ")

print("A1")

print(A1)

print("point vector: ")

print("b")

print(b)

print("rotated point vector afer one rotation along x - axis")

print("A1*b")

print(A1*b)

two fold rotation:

pi = np.pi

theta = pi

x = sp.Symbol('x')

y = sp.Symbol('y')

z = sp.Symbol('z')

b = sp.Matrix([[x],[y],[z]])

# Rotation matrix for 2-fold rotation along x-axis

A1 = sp.Matrix([[1,0,0],[0,np.cos(theta),-1*np.sin(theta)],[0,np.sin(theta),np.cos(theta)]])