diff options
Diffstat (limited to 'vorlesungen')
| -rw-r--r-- | vorlesungen/punktgruppen/crystals.py | 171 |
1 files changed, 105 insertions, 66 deletions
diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py index 70e1f89..cbae3d0 100644 --- a/vorlesungen/punktgruppen/crystals.py +++ b/vorlesungen/punktgruppen/crystals.py @@ -15,7 +15,6 @@ config.tex_template.add_to_preamble( # scenes class Geometric2DSymmetries(Scene): def construct(self): - # title self.intro() self.cyclic() self.dihedral() @@ -269,75 +268,115 @@ class Geometric2DSymmetries(Scene): class Geometric3DSymmetries(ThreeDScene): def construct(self): - self.symmetric() - - - @staticmethod - def get_cube(): - verts = np.array(list(it.product(*3 * [[-1, 1]]))) - edges = [ - (v1, v2) - for v1, v2 in it.combinations(verts, 2) - if sum(v1 == v2) == 2 - ] - corner_dots = Group(*[ - Sphere().set_height(0.25).move_to(vert) - for vert in verts - ]) - corner_dots.set_color(GREY_B) - edge_rods = Group(*[ - Line3D(v1, v2) - for v1, v2 in edges - ]) - - faces = Cube(square_resolution=(10, 10)) - faces.set_height(2) - faces.set_color(BLUE_E, 0.3) - # faces.add_updater(lambda m: m.sort(lambda p: np.dot(p, [np.sign(self.euler_angles[0]) * 0.2, -1, 0.2]))) - - cube = Group(corner_dots, edge_rods, faces) - cube.corner_dots = corner_dots - cube.edge_rods = edge_rods - cube.faces = faces - return cube - - def symmetric(self): - self.renderer.camera.light_source.move_to(3*IN) # changes the source of the light - self.set_camera_orientation(phi=60 * DEGREES, theta=5 * DEGREES) - - cube = Cube() - self.play(GrowFromCenter(cube)) - - axes = list( - map(lambda v: v / np.linalg.norm(v), - map(np.array, [ - [0, 0, 1], - [0, 1, 1], - [1, 1, 1], - ]) - )) - angles = [ PI, PI, PI * 2/3 ] - lines = list(map(lambda x: Line(-2 * x, 2 * x), axes)) - - camera_thetas = list(map(lambda x: x * DEGREES, [10, 100, 110])) - for axis, line, angle, camera_angle in zip(axes, lines, angles, camera_thetas): - self.move_camera(theta=camera_angle) - self.play(Create(line)) - self.play(Rotate(cube, angle, axis=axis, run_time=3)) - - self.wait(7) + self.improper_rotation() + def improper_rotation(self): + # changes the source of the light and camera + self.renderer.camera.light_source.move_to(3*IN) + self.set_camera_orientation(phi=0, theta=0) -class AlgebraicSymmetries(Scene): - def construct(self): - # title - title = Tex(r"Algebraische \\ Symmetrien") - title.scale(1.5) - self.play(Write(title)) + # initial square + square = Square() + square.set_fill(PINK, opacity=.5) + + self.play(Create(square)) + self.wait() + + self.play(Rotate(square, PI/2)) + self.wait() + + self.move_camera(phi= 75 * DEGREES, theta = -80 * DEGREES) + + # create sphere from slices + cyclic_slices = [] + for i in range(4): + colors = [PINK, RED] if i % 2 == 0 else [BLUE_D, BLUE_E] + cyclic_slices.append(ParametricSurface( + lambda u, v: np.array([ + np.sqrt(2) * np.cos(u) * np.cos(v), + np.sqrt(2) * np.cos(u) * np.sin(v), + np.sqrt(2) * np.sin(u) + ]), + v_min=PI/4 + PI/2 * i, + v_max=PI/4 + PI/2 * (i + 1), + u_min=-PI/2, u_max=PI/2, + checkerboard_colors=colors, resolution=(10,5))) + + self.play(FadeOut(square), *map(Create, cyclic_slices)) + + axis = Line3D(start=[0,0,-2.5], end=[0,0,2.5]) + + axis_name = MathTex(r"r \in Z_4") + # move to yz plane + axis_name.rotate(PI/2, axis = RIGHT) + axis_name.next_to(axis, OUT) + + self.play(Create(axis)) + self.play(Write(axis_name)) + self.wait() + + cyclic_sphere = VGroup(*cyclic_slices) + self.play(Rotate(cyclic_sphere, PI/2)) self.wait() - self.play(FadeOut(title)) + + # reflection plane + self.play(FadeOut(cyclic_sphere), FadeIn(square)) + plane = ParametricSurface( + lambda u, v: np.array([u, 0, v]), + u_min = -2, u_max = 2, + v_min = -2, v_max = 2, + fill_opacity=.3, resolution=(1,1)) + + plane_name = MathTex(r"\sigma \in D_4") + # move to yz plane + plane_name.rotate(PI/2, axis = RIGHT) + plane_name.next_to(plane, OUT + RIGHT) + + self.play(Create(plane)) + self.play(Write(plane_name)) self.wait() + self.move_camera(phi = 25 * DEGREES, theta = -75 * DEGREES) + + self.play(Rotate(square, PI/2)) + self.play(Rotate(square, PI, RIGHT)) + + self.play(Rotate(square, PI/2)) + self.play(Rotate(square, PI, RIGHT)) + + self.move_camera(phi = 75 * DEGREES, theta = -80 * DEGREES) + + # create sphere from slices + dihedral_slices = [] + for i in range(4): + for j in range(2): + colors = [PINK, RED] if i % 2 == 0 else [BLUE_D, BLUE_E] + dihedral_slices.append(ParametricSurface( + lambda u, v: np.array([ + np.sqrt(2) * np.cos(u) * np.cos(v), + np.sqrt(2) * np.cos(u) * np.sin(v), + np.sqrt(2) * np.sin(u) + ]), + v_min=PI/2 * j + PI/4 + PI/2 * i, + v_max=PI/2 * j + PI/4 + PI/2 * (i + 1), + u_min=-PI/2 if j == 0 else 0, + u_max=0 if j == 0 else PI/2, + checkerboard_colors=colors, resolution=(10,5))) + + dihedral_sphere = VGroup(*dihedral_slices) + + self.play(FadeOut(square), Create(dihedral_sphere)) + + self.play(Rotate(dihedral_sphere, PI/2)) + self.play(Rotate(dihedral_sphere, PI, RIGHT)) + + self.play(Rotate(dihedral_sphere, PI/2)) + self.play(Rotate(dihedral_sphere, PI, RIGHT)) + + self.wait(5) + +class AlgebraicSymmetries(Scene): + def construct(self): self.cyclic() self.matrices() @@ -349,7 +388,7 @@ class AlgebraicSymmetries(Scene): r"-1 \cdot i &= -i \\", r"-i \cdot i &= 1") product.scale(1.5) - + for part in product: self.play(Write(part)) |