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authorLukaszogg <82384106+Lukaszogg@users.noreply.github.com>2021-07-08 20:10:11 +0200
committerLukaszogg <82384106+Lukaszogg@users.noreply.github.com>2021-07-08 20:10:11 +0200
commit14033ca595b5c933caea3b214d2246529e6845b8 (patch)
tree0d6d2b2eb34e5ef5df3c517be5c1c9d803fa066c /vorlesungen/punktgruppen/crystals.py
parentUpdate teil1.tex (diff)
parentOnly include buch.ind if it exists. (diff)
downloadSeminarMatrizen-14033ca595b5c933caea3b214d2246529e6845b8.tar.gz
SeminarMatrizen-14033ca595b5c933caea3b214d2246529e6845b8.zip
Merge remote-tracking branch 'upstream/master'
Diffstat (limited to 'vorlesungen/punktgruppen/crystals.py')
-rw-r--r--vorlesungen/punktgruppen/crystals.py611
1 files changed, 611 insertions, 0 deletions
diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py
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+from manim import *
+
+import math as m
+import numpy as np
+import itertools as it
+
+# configure style
+config.background_color = '#202020'
+config.tex_template.add_to_preamble(
+ r"\usepackage[p,osf]{scholax}"
+ r"\usepackage{amsmath}"
+ r"\usepackage[scaled=1.075,ncf,vvarbb]{newtxmath}"
+)
+
+# scenes
+class Geometric2DSymmetries(Scene):
+ def construct(self):
+ self.wait(5)
+
+ self.intro()
+ self.cyclic()
+ self.dihedral()
+ self.circle()
+
+ def intro(self):
+ # create square
+ square = Square()
+ square.set_fill(PINK, opacity=.5)
+ self.play(SpinInFromNothing(square))
+ self.wait()
+
+ # the action of doing nothing
+ action = MathTex(r"\mathbb{1}")
+ self.play(Write(action))
+ self.play(ApplyMethod(square.scale, 1.2))
+ self.play(ApplyMethod(square.scale, 1/1.2))
+ self.play(FadeOut(action))
+ self.wait()
+
+ # show some reflections
+ axis = DashedLine(2 * LEFT, 2 * RIGHT)
+ sigma = MathTex(r"\sigma")
+ sigma.next_to(axis, RIGHT)
+
+ self.play(Create(axis))
+ self.play(Write(sigma))
+ self.play(ApplyMethod(square.flip, RIGHT))
+
+ for d in [UP + RIGHT, UP]:
+ self.play(
+ Rotate(axis, PI/4),
+ Rotate(sigma, PI/4, about_point=ORIGIN))
+
+ self.play(Rotate(sigma, -PI/4), run_time=.5)
+ self.play(ApplyMethod(square.flip, d))
+
+ self.play(FadeOutAndShift(sigma), Uncreate(axis))
+
+ # show some rotations
+ dot = Dot(UP + RIGHT)
+ figure = VGroup(square, dot)
+
+ rot = MathTex(r"r")
+ self.play(Write(rot), Create(dot))
+
+ last = rot
+ for newrot in map(MathTex, [r"r", r"r^2", r"r^3"]):
+ self.play(
+ ReplacementTransform(last, newrot),
+ Rotate(figure, PI/2, about_point=ORIGIN))
+ self.wait(.5)
+ last = newrot
+
+ self.play(Uncreate(dot), FadeOut(square), FadeOut(last))
+
+
+ def cyclic(self):
+ # create symmetric figure
+ figure = VGroup()
+ prev = [1.5, 0, 0]
+ for i in range(1,6):
+ pos = [
+ 1.5*m.cos(2 * PI/5 * i),
+ 1.5*m.sin(2 * PI/5 * i),
+ 0
+ ]
+
+ if prev:
+ line = Line(prev, pos)
+ figure.add(line)
+
+ dot = Dot(pos, radius=.1)
+ if i == 5:
+ dot.set_fill(RED)
+
+ prev = pos
+ figure.add(dot)
+
+ group = MathTex(r"G = \langle r \rangle")
+ self.play(Write(group), run_time = 2)
+ self.wait(3)
+
+ self.play(ApplyMethod(group.to_edge, UP))
+
+ actions = map(MathTex, [
+ r"\mathbb{1}", r"r", r"r^2",
+ r"r^3", r"r^4", r"\mathbb{1}", r"r"])
+
+ action = next(actions, MathTex(r"r"))
+
+ self.play(Create(figure))
+ self.play(Write(action))
+ self.wait()
+
+ for i in range(5):
+ newaction = next(actions, MathTex(r"r"))
+ self.play(
+ ReplacementTransform(action, newaction),
+ Rotate(figure, 2*PI/5, about_point=ORIGIN))
+ action = newaction
+
+ self.wait()
+ newaction = next(actions, MathTex(r"r"))
+ self.play(
+ ReplacementTransform(action, newaction),
+ Rotate(figure, 2*PI/5, about_point=ORIGIN))
+ action = newaction
+ self.wait(2)
+
+ self.play(Uncreate(figure), FadeOut(action))
+
+ whole_group = MathTex(
+ r"G = \langle r \rangle"
+ 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))
+ self.wait(5)
+
+ cyclic = MathTex(
+ r"C_n = \langle r \rangle"
+ 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)
+
+ cyclic.scale(1.2)
+ cyclic_title.scale(1.2)
+
+ self.play(ReplacementTransform(whole_group, cyclic))
+ self.play(FadeInFrom(cyclic_title, UP))
+
+ self.wait(5)
+ self.play(FadeOut(cyclic), FadeOut(cyclic_title))
+
+ def dihedral(self):
+ # create square
+ square = Square()
+ square.set_fill(PINK, opacity=.5)
+
+ # generator equation
+ group = MathTex(
+ r"G = \langle \sigma, r \,|\,",
+ 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(5)
+
+ self.play(ApplyMethod(group.to_edge, UP))
+ self.play(FadeIn(square))
+ self.wait()
+
+ # flips
+ axis = DashedLine(2 * LEFT, 2 * RIGHT)
+ sigma = MathTex(r"\sigma^2 = \mathbb{1}")
+ sigma.next_to(axis, RIGHT)
+ self.play(Create(axis), Write(sigma))
+ self.play(ApplyMethod(square.flip, RIGHT))
+ self.play(ApplyMethod(square.flip, RIGHT))
+ self.play(Uncreate(axis), FadeOut(sigma))
+
+ # rotations
+ dot = Dot(UP + RIGHT)
+ rot = MathTex(r"r^4 = \mathbb{1}")
+ rot.next_to(square, DOWN * 3)
+
+ figure = VGroup(dot, square)
+
+ self.play(Write(rot), Create(dot))
+ for i in range(4):
+ self.play(Rotate(figure, PI/2))
+ self.play(FadeOut(rot), Uncreate(dot))
+
+ # rotation and flip
+ action = MathTex(r"(\sigma r)^2 = \mathbb{1}")
+ action.next_to(square, DOWN * 5)
+
+ dot = Dot(UP + RIGHT)
+ axis = DashedLine(2 * LEFT, 2 * RIGHT)
+ self.play(Create(dot), Create(axis), Write(action))
+
+ figure = VGroup(dot, square)
+
+ for i in range(2):
+ self.play(Rotate(figure, PI/2))
+ self.play(ApplyMethod(figure.flip, RIGHT))
+ self.wait()
+
+ self.play(Uncreate(dot), Uncreate(axis), FadeOut(action))
+ self.play(FadeOut(square))
+
+ # equation for the whole
+ whole_group = MathTex(
+ r"G &= \langle \sigma, r \,|\,"
+ r"\sigma^2 = r^4 = (\sigma r)^2 = \mathbb{1} \rangle \\"
+ r"&= \left\{"
+ 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))
+ self.play(ReplacementTransform(group, whole_group))
+ self.wait(2)
+
+ dihedral = MathTex(
+ r"D_n &= \langle \sigma, r \,|\,"
+ r"\sigma^2 = r^n = (\sigma r)^2 = \mathbb{1} \rangle \\"
+ r"&= \left\{"
+ 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")
+ dihedral_title.next_to(dihedral, UP * 2)
+
+ dihedral.scale(1.2)
+ dihedral_title.scale(1.2)
+
+ self.play(ReplacementTransform(whole_group, dihedral))
+ self.play(FadeInFrom(dihedral_title, UP))
+
+ self.wait(5)
+ self.play(FadeOut(dihedral), FadeOut(dihedral_title))
+
+ def circle(self):
+ circle = Circle(radius=2)
+ dot = Dot()
+ dot.move_to(2 * RIGHT)
+
+ figure = VGroup(circle, dot)
+ group_name = MathTex(r"C_\infty")
+
+ # create circle
+ self.play(Create(circle))
+ self.play(Create(dot))
+
+ # move it around
+ self.play(Rotate(figure, PI/3))
+ self.play(Rotate(figure, PI/6))
+ self.play(Rotate(figure, -PI/3))
+
+ # show name
+ self.play(Rotate(figure, PI/4), Write(group_name))
+ self.wait()
+ self.play(Uncreate(figure))
+
+ nsphere = MathTex(r"C_\infty \cong S^1 = \left\{z \in \mathbb{C} : |z| = 1\right\}")
+ nsphere_title = Tex(r"Kreisgruppe")
+ nsphere_title.next_to(nsphere, 2 * UP)
+
+ nsphere.scale(1.2)
+ nsphere_title.scale(1.2)
+
+ self.play(ReplacementTransform(group_name, nsphere))
+ self.play(FadeInFrom(nsphere_title, UP))
+
+ self.wait(5)
+ self.play(FadeOut(nsphere_title), FadeOut(nsphere))
+ self.wait(2)
+
+
+class Geometric3DSymmetries(ThreeDScene):
+ def construct(self):
+ self.improper_rotation()
+ self.tetrahedron()
+
+ 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)
+
+ # initial square
+ square = Square()
+ square.set_fill(PINK, opacity=.5)
+
+ self.play(SpinInFromNothing(square))
+ self.wait(2)
+
+ for i in range(4):
+ self.play(Rotate(square, PI/2))
+ self.wait(.5)
+
+ self.move_camera(phi= 75 * DEGREES, theta = -80 * DEGREES)
+
+ # create rotation axis
+ axis = Line3D(start=[0,0,-2.5], end=[0,0,2.5])
+
+ axis_name = MathTex(r"r \in C_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()
+
+ # 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))
+
+ cyclic_sphere = VGroup(*cyclic_slices)
+ for i in range(4):
+ self.play(Rotate(cyclic_sphere, PI/2))
+ self.wait()
+
+ new_axis_name = MathTex(r"r \in D_4")
+ # move to yz plane
+ new_axis_name.rotate(PI/2, axis = RIGHT)
+ new_axis_name.next_to(axis, OUT)
+ self.play(ReplacementTransform(axis_name, new_axis_name))
+
+ # 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.wait()
+
+ condition = MathTex(r"(\sigma r)^2 = \mathbb{1}")
+ condition.next_to(square, DOWN);
+
+ self.play(Write(condition))
+ 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.play(FadeOut(condition))
+
+ 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))
+
+ for i in range(2):
+ self.play(Rotate(dihedral_sphere, PI/2))
+ self.play(Rotate(dihedral_sphere, PI, RIGHT))
+ self.wait()
+
+ self.wait(2)
+ self.play(*map(FadeOut, [dihedral_sphere, plane, plane_name, new_axis_name]), FadeIn(square))
+ self.wait(3)
+ self.play(*map(FadeOut, [square, axis]))
+ self.wait(3)
+
+ def tetrahedron(self):
+ tet = Tetrahedron(edge_length=2)
+ self.play(FadeIn(tet))
+
+ self.move_camera(phi = 75 * DEGREES, theta = -100 * DEGREES)
+ self.begin_ambient_camera_rotation(rate=.1)
+
+ axes = []
+ for coord in tet.vertex_coords:
+ axes.append((-2 * coord, 2 * coord))
+
+ lines = [
+ Line3D(start=s, end=e) for s, e in axes
+ ]
+
+ self.play(*map(Create, lines))
+ self.wait()
+
+ for axis in axes:
+ self.play(Rotate(tet, 2*PI/3, axis=axis[1]))
+ self.play(Rotate(tet, 2*PI/3, axis=axis[1]))
+
+ self.wait(5)
+ self.stop_ambient_camera_rotation()
+ self.wait()
+ self.play(*map(Uncreate, lines))
+ self.play(FadeOut(tet))
+ self.wait(5)
+
+
+class AlgebraicSymmetries(Scene):
+ def construct(self):
+ self.wait(5)
+ self.cyclic()
+ # self.matrices()
+
+ def cyclic(self):
+ # show the i product
+ product = MathTex(
+ r"1", r"\cdot i &= i \\",
+ r"i \cdot i &= -1 \\",
+ r"-1 \cdot i &= -i \\",
+ r"-i \cdot i &= 1")
+ product.scale(1.5)
+
+ for part in product:
+ self.play(Write(part))
+ self.wait()
+
+ self.play(ApplyMethod(product.scale, 1/1.5))
+
+ # gather in group
+ group = MathTex(r"G = \left\{ 1, i, -1, -i \right\}")
+ self.play(ReplacementTransform(product, group))
+ self.wait(2)
+
+ # show C4
+ grouppow = MathTex(
+ r"G &= \left\{ 1, i, i^2, i^3 \right\} \\",
+ r"C_4 &= \left\{ \mathbb{1}, r, r^2, r^3 \right\}")
+ self.play(ReplacementTransform(group, grouppow[0]))
+ self.wait(2)
+
+ self.play(Write(grouppow[1]))
+ self.wait(4)
+
+ self.play(ApplyMethod(grouppow.to_edge, UP))
+
+ # define morphisms
+ morphism = MathTex(r"\phi: C_4 \to G \\")
+ morphism.shift(UP)
+ self.play(Write(morphism))
+ self.wait()
+
+ # show an example
+ mappings = MathTex(
+ r"\phi(\mathbb{1}) &= 1 \\",
+ r"\phi(r) &= i \\",
+ r"\phi(r^2) &= i^2 \\",
+ r"\phi(r^3) &= i^3 \\")
+ mappings.next_to(morphism, 2 * DOWN)
+
+ self.play(Write(mappings))
+ self.wait(3)
+ self.play(FadeOutAndShift(mappings, DOWN))
+
+ # more general definition
+ homomorphism = MathTex(
+ r"\phi(r\circ \mathbb{1}) &= \phi(r)\cdot\phi(\mathbb{1}) \\",
+ r"&= i\cdot 1")
+ homomorphism.next_to(morphism, DOWN).align_to(morphism, LEFT)
+ for part in homomorphism:
+ self.play(Write(part))
+ self.wait()
+
+ hom_bracegrp = VGroup(morphism, homomorphism)
+
+ self.play(
+ ApplyMethod(grouppow.shift, 3 * LEFT),
+ ApplyMethod(hom_bracegrp.shift, 3 * LEFT))
+
+ hom_brace = Brace(hom_bracegrp, direction=RIGHT)
+ hom_text = Tex("Homomorphismus").next_to(hom_brace.get_tip(), RIGHT)
+ hom_text_short = MathTex(r"\mathrm{Hom}(C_4, G)").next_to(hom_brace.get_tip(), RIGHT)
+
+ self.play(Create(hom_brace))
+ self.play(Write(hom_text))
+ self.wait()
+ self.play(ReplacementTransform(hom_text, hom_text_short))
+ self.wait()
+
+ # add the isomorphism part
+ isomorphism = Tex(r"\(\phi\) ist bijektiv")
+ isomorphism.next_to(homomorphism, DOWN).align_to(homomorphism, LEFT)
+ self.play(Write(isomorphism))
+
+ iso_bracegrp = VGroup(hom_bracegrp, isomorphism)
+
+ iso_brace = Brace(iso_bracegrp, RIGHT)
+ iso_text = Tex("Isomorphismus").next_to(iso_brace.get_tip(), RIGHT)
+ iso_text_short = MathTex("C_4 \cong G").next_to(iso_brace.get_tip(), RIGHT)
+
+ self.play(
+ ReplacementTransform(hom_brace, iso_brace),
+ ReplacementTransform(hom_text_short, iso_text))
+ self.wait()
+
+ self.play(ReplacementTransform(iso_text, iso_text_short))
+ self.wait()
+
+ # create a group for the whole
+ morphgrp = VGroup(iso_bracegrp, iso_brace, iso_text_short)
+
+ self.play(
+ ApplyMethod(grouppow.to_edge, LEFT),
+ ApplyMethod(morphgrp.to_edge, LEFT))
+
+ # draw a complex plane
+ plane = ComplexPlane(x_range = [-2.5, 2.5])
+ coordinates = plane.get_coordinate_labels(1, -1, 1j, -1j)
+
+ roots = list(map(lambda p: Dot(p, fill_color=PINK), (
+ [1, 0, 0], [0, 1, 0], [-1, 0, 0], [0, -1, 0]
+ )))
+
+ arrow = CurvedArrow(
+ 1.5 * np.array([m.cos(10 * DEGREES), m.sin(10 * DEGREES), 0]),
+ 1.5 * np.array([m.cos(80 * DEGREES), m.sin(80 * DEGREES), 0]))
+ arrowtext = MathTex("\cdot i")
+ arrowtext.move_to(2 / m.sqrt(2) * (UP + RIGHT))
+
+ square = Square().rotate(PI/4).scale(1/m.sqrt(2))
+ square.set_fill(PINK).set_opacity(.4)
+
+ figuregrp = VGroup(plane, square, arrow, arrowtext, *coordinates, *roots)
+ figuregrp.to_edge(RIGHT)
+
+ self.play(Create(plane))
+ self.play(
+ *map(Create, roots),
+ *map(Write, coordinates))
+ self.wait()
+ self.play(FadeIn(square), Create(arrow), Write(arrowtext))
+
+ for _ in range(4):
+ self.play(Rotate(square, PI/2))
+ self.wait(.5)
+
+ self.play(
+ *map(FadeOut, (square, arrow, arrowtext)),
+ *map(FadeOut, coordinates),
+ *map(FadeOut, roots))
+ self.play(Uncreate(plane))
+ self.play(
+ FadeOutAndShift(grouppow, RIGHT),
+ FadeOutAndShift(morphgrp, RIGHT))
+
+ modulo = MathTex(
+ r"\phi: C_4 &\to (\mathbb{Z}/4\mathbb{Z}, +) \\"
+ r"\phi(\mathbb{1} \circ r^2) &= 0 + 2 \pmod 4").scale(1.5)
+ self.play(Write(modulo))
+ self.wait(2)
+
+ self.play(FadeOut(modulo))
+ self.wait(3)
+
+ def matrices(self):
+ question = MathTex(
+ r"D_n &\cong \,? \\"
+ r"S_n &\cong \,? \\"
+ r"A_n &\cong \,?").scale(1.5)
+
+ answer = MathTex(
+ r"D_n &\cong \,?\\"
+ r"S_4 &\cong \mathrm{Aut}(Q_8) \\"
+ r"A_5 &\cong \mathrm{PSL}_2 (5)").scale(1.5)
+
+ self.play(Write(question))
+ self.wait()
+ self.play(ReplacementTransform(question, answer))
+
+ self.wait(3)