diff options
Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/punktgruppen/crystals.py | 114 |
1 files changed, 86 insertions, 28 deletions
diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py index b29d7e4..4a9836a 100644 --- a/vorlesungen/punktgruppen/crystals.py +++ b/vorlesungen/punktgruppen/crystals.py @@ -15,6 +15,8 @@ config.tex_template.add_to_preamble( # scenes class Geometric2DSymmetries(Scene): def construct(self): + self.wait(5) + self.intro() self.cyclic() self.dihedral() @@ -33,6 +35,7 @@ class Geometric2DSymmetries(Scene): 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) @@ -41,9 +44,7 @@ class Geometric2DSymmetries(Scene): self.play(Create(axis)) self.play(Write(sigma)) - self.play(ApplyMethod(square.flip, RIGHT)) - self.wait() for d in [UP + RIGHT, UP]: self.play( @@ -53,9 +54,7 @@ class Geometric2DSymmetries(Scene): self.play(Rotate(sigma, -PI/4), run_time=.5) self.play(ApplyMethod(square.flip, d)) - self.play( - FadeOutAndShift(sigma), - Uncreate(axis)) + self.play(FadeOutAndShift(sigma), Uncreate(axis)) # show some rotations dot = Dot(UP + RIGHT) @@ -69,7 +68,7 @@ class Geometric2DSymmetries(Scene): self.play( ReplacementTransform(last, newrot), Rotate(figure, PI/2, about_point=ORIGIN)) - self.wait() + self.wait(.5) last = newrot self.play(Uncreate(dot), FadeOut(square), FadeOut(last)) @@ -99,12 +98,13 @@ class Geometric2DSymmetries(Scene): group = MathTex(r"G = \langle r \rangle") self.play(Write(group), run_time = 2) - self.wait() + 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^3", r"r^4", r"\mathbb{1}", r"r"]) action = next(actions, MathTex(r"r")) @@ -119,6 +119,14 @@ class Geometric2DSymmetries(Scene): 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( @@ -127,7 +135,7 @@ class Geometric2DSymmetries(Scene): self.play(ApplyMethod(group.move_to, ORIGIN)) self.play(ReplacementTransform(group, whole_group)) - self.wait() + self.wait(5) cyclic = MathTex( r"C_n = \langle r \rangle" @@ -158,10 +166,13 @@ class Geometric2DSymmetries(Scene): r"(\sigma r)^2 = \mathbb{1} \rangle") self.play(Write(group), run_time = 2) - self.wait() + 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) @@ -250,6 +261,7 @@ class Geometric2DSymmetries(Scene): # 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\}") @@ -264,12 +276,13 @@ class Geometric2DSymmetries(Scene): self.wait(5) self.play(FadeOut(nsphere_title), FadeOut(nsphere)) + self.wait(2) class Geometric3DSymmetries(ThreeDScene): def construct(self): self.improper_rotation() - self.icosahedron() + self.tetrahedron() def improper_rotation(self): # changes the source of the light and camera @@ -289,6 +302,18 @@ class Geometric3DSymmetries(ThreeDScene): 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): @@ -306,22 +331,17 @@ class Geometric3DSymmetries(ThreeDScene): 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 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() - 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( @@ -340,12 +360,18 @@ class Geometric3DSymmetries(ThreeDScene): 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) @@ -375,14 +401,45 @@ class Geometric3DSymmetries(ThreeDScene): self.play(Rotate(dihedral_sphere, PI, RIGHT)) self.wait() - self.wait(5) + 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 + ] - def icosahedron(self): - pass + 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() @@ -406,7 +463,7 @@ class AlgebraicSymmetries(Scene): self.play(ReplacementTransform(product, group)) self.wait(2) - # show Z4 + # 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\}") @@ -414,7 +471,8 @@ class AlgebraicSymmetries(Scene): self.wait(2) self.play(Write(grouppow[1])) - self.wait() + self.wait(4) + self.play(ApplyMethod(grouppow.to_edge, UP)) # define morphisms @@ -429,7 +487,7 @@ class AlgebraicSymmetries(Scene): r"\phi(r) &= i \\", r"\phi(r^2) &= i^2 \\", r"\phi(r^3) &= i^3 \\") - mappings.next_to(morphism, DOWN) + mappings.next_to(morphism, 2 * DOWN) self.play(Write(mappings)) self.wait(3) |