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-rw-r--r--vorlesungen/punktgruppen/crystals.py114
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)