diff options
| -rw-r--r-- | vorlesungen/punktgruppen/crystals.py | 92 |
1 files changed, 67 insertions, 25 deletions
diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py index d0ff205..894199d 100644 --- a/vorlesungen/punktgruppen/crystals.py +++ b/vorlesungen/punktgruppen/crystals.py @@ -2,6 +2,7 @@ from manim import * import math as m import numpy as np +import itertools as it # configure style config.background_color = '#202020' @@ -34,7 +35,7 @@ class Geometric2DSymmetries(Scene): self.wait() # the action of doing nothing - action = MathTex(r"1") + action = MathTex(r"\mathbb{1}") self.play(Write(action)) self.play(ApplyMethod(square.scale, 1.2)) self.play(ApplyMethod(square.scale, 1/1.2)) @@ -110,7 +111,10 @@ class Geometric2DSymmetries(Scene): self.wait() self.play(ApplyMethod(group.to_corner, UP)) - actions = map(MathTex, [r"1", r"r", r"r^2", r"r^3", r"r^4", r"1"]) + actions = map(MathTex, [ + r"\mathbb{1}", r"r", r"r^2", + r"r^3", r"r^4", r"\mathbb{1}"]) + action = next(actions, MathTex(r"r")) self.play(Create(figure)) @@ -128,7 +132,7 @@ class Geometric2DSymmetries(Scene): whole_group = MathTex( r"G = \langle r \rangle" - r"= \left\{1, r, r^2, r^3, r^4 \right\}") + r"= \left\{\mathbb{1}, r, r^2, r^3, r^4 \right\}") self.play(ApplyMethod(group.move_to, ORIGIN)) self.play(ReplacementTransform(group, whole_group)) @@ -136,7 +140,7 @@ class Geometric2DSymmetries(Scene): cyclic = MathTex( r"Z_n = \langle r \rangle" - r"= \left\{1, r, r^2, \dots, r^{n-1} \right\}") + r"= \left\{\mathbb{1}, r, r^2, \dots, r^{n-1} \right\}") cyclic_title = Tex(r"Zyklische Gruppe") cyclic_title.next_to(cyclic, UP * 2) @@ -158,9 +162,9 @@ class Geometric2DSymmetries(Scene): # generator equation group = MathTex( r"G = \langle \sigma, r \,|\,", - r"\sigma^2 = 1,", - r"r^4 = 1,", - r"(\sigma r)^2 = 1 \rangle") + r"\sigma^2 = \mathbb{1},", + r"r^4 = \mathbb{1},", + r"(\sigma r)^2 = \mathbb{1} \rangle") self.play(Write(group), run_time = 2) self.wait() @@ -168,7 +172,7 @@ class Geometric2DSymmetries(Scene): self.play(FadeIn(square)) axis = DashedLine(2 * LEFT, 2 * RIGHT) - sigma = MathTex(r"\sigma^2 = 1") + sigma = MathTex(r"\sigma^2 = \mathbb{1}") sigma.next_to(axis, RIGHT) self.play(Create(axis), Write(sigma)) self.play(ApplyMethod(square.flip, RIGHT)) @@ -177,7 +181,7 @@ class Geometric2DSymmetries(Scene): # rotations dot = Dot(UP + RIGHT) - rot = MathTex(r"r^4 = 1") + rot = MathTex(r"r^4 = \mathbb{1}") rot.next_to(square, DOWN * 3) figure = VGroup() @@ -190,7 +194,7 @@ class Geometric2DSymmetries(Scene): self.play(FadeOut(rot), Uncreate(dot)) # rotation and flip - action = MathTex(r"(\sigma r)^2 = 1") + action = MathTex(r"(\sigma r)^2 = \mathbb{1}") action.next_to(square, DOWN * 5) dot = Dot(UP + RIGHT) @@ -212,9 +216,9 @@ class Geometric2DSymmetries(Scene): # equation for the whole whole_group = MathTex( r"G &= \langle \sigma, r \,|\," - r"\sigma^2 = r^4 = (\sigma r)^2 = 1 \rangle \\" + r"\sigma^2 = r^4 = (\sigma r)^2 = \mathbb{1} \rangle \\" r"&= \left\{" - r"1, r, r^2, r^3, \sigma, \sigma r, \sigma r^2, \sigma r^3" + r"\mathbb{1}, r, r^2, r^3, \sigma, \sigma r, \sigma r^2, \sigma r^3" r"\right\}") self.play(ApplyMethod(group.move_to, ORIGIN)) @@ -223,9 +227,9 @@ class Geometric2DSymmetries(Scene): dihedral = MathTex( r"D_n &= \langle \sigma, r \,|\," - r"\sigma^2 = r^n = (\sigma r)^2 = 1 \rangle \\" + r"\sigma^2 = r^n = (\sigma r)^2 = \mathbb{1} \rangle \\" r"&= \left\{" - r"1, r, r^2, \dots, \sigma, \sigma r, \sigma r^2, \dots" + r"\mathbb{1}, r, r^2, \dots, \sigma, \sigma r, \sigma r^2, \dots" r"\right\}") dihedral_title = Tex(r"Diedergruppe: Symmetrien eines \(n\)-gons") @@ -245,22 +249,60 @@ 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=-20 * DEGREES) + self.set_camera_orientation(phi=60 * DEGREES, theta=5 * DEGREES) cube = Cube() - axes = ThreeDAxes() - - axis = np.array([2,2,2]) - line = DashedLine(-axis, axis) - - self.play(Create(axes), Create(cube)) - self.play(Create(line)) - - self.play(ApplyMethod(cube.rotate, PI, axis / abs(axis))) + 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.begin_ambient_camera_rotation(rate=0.1) self.wait(7) |