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-rw-r--r--vorlesungen/punktgruppen/crystals.py72
1 files changed, 38 insertions, 34 deletions
diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py
index cbae3d0..7eef9b8 100644
--- a/vorlesungen/punktgruppen/crystals.py
+++ b/vorlesungen/punktgruppen/crystals.py
@@ -130,7 +130,7 @@ class Geometric2DSymmetries(Scene):
self.wait()
cyclic = MathTex(
- r"Z_n = \langle r \rangle"
+ r"C_n = \langle r \rangle"
r"= \left\{\mathbb{1}, r, r^2, \dots, r^{n-1} \right\}")
cyclic_title = Tex(r"Zyklische Gruppe")
@@ -237,7 +237,7 @@ class Geometric2DSymmetries(Scene):
dot.move_to(2 * RIGHT)
figure = VGroup(circle, dot)
- group_name = MathTex(r"S^1")
+ group_name = MathTex(r"C_\infty")
# create circle
self.play(Create(circle))
@@ -252,7 +252,7 @@ class Geometric2DSymmetries(Scene):
self.play(Rotate(figure, PI/4), Write(group_name))
self.play(Uncreate(figure))
- nsphere = MathTex(r"S^1 = \left\{z \in \mathbb{C} : |z| = 1\right\}")
+ 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)
@@ -269,6 +269,7 @@ class Geometric2DSymmetries(Scene):
class Geometric3DSymmetries(ThreeDScene):
def construct(self):
self.improper_rotation()
+ self.icosahedron()
def improper_rotation(self):
# changes the source of the light and camera
@@ -279,11 +280,12 @@ class Geometric3DSymmetries(ThreeDScene):
square = Square()
square.set_fill(PINK, opacity=.5)
- self.play(Create(square))
- self.wait()
+ self.play(SpinInFromNothing(square))
+ self.wait(2)
- self.play(Rotate(square, PI/2))
- self.wait()
+ for i in range(4):
+ self.play(Rotate(square, PI/2))
+ self.wait(.5)
self.move_camera(phi= 75 * DEGREES, theta = -80 * DEGREES)
@@ -306,7 +308,7 @@ class Geometric3DSymmetries(ThreeDScene):
axis = Line3D(start=[0,0,-2.5], end=[0,0,2.5])
- axis_name = MathTex(r"r \in Z_4")
+ 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)
@@ -316,8 +318,9 @@ class Geometric3DSymmetries(ThreeDScene):
self.wait()
cyclic_sphere = VGroup(*cyclic_slices)
- self.play(Rotate(cyclic_sphere, PI/2))
- self.wait()
+ for i in range(4):
+ self.play(Rotate(cyclic_sphere, PI/2))
+ self.wait()
# reflection plane
self.play(FadeOut(cyclic_sphere), FadeIn(square))
@@ -367,18 +370,21 @@ class Geometric3DSymmetries(ThreeDScene):
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))
+ for i in range(2):
+ self.play(Rotate(dihedral_sphere, PI/2))
+ self.play(Rotate(dihedral_sphere, PI, RIGHT))
+ self.wait()
self.wait(5)
+ def icosahedron(self):
+ pass
+
+
class AlgebraicSymmetries(Scene):
def construct(self):
self.cyclic()
- self.matrices()
+ # self.matrices()
def cyclic(self):
# show the i product
@@ -391,30 +397,31 @@ class AlgebraicSymmetries(Scene):
for part in product:
self.play(Write(part))
+ self.wait()
- 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()
+ self.wait(2)
# show Z4
grouppow = MathTex(
r"G &= \left\{ 1, i, i^2, i^3 \right\} \\",
- r"Z_4 &= \left\{ \mathbb{1}, r, r^2, r^3 \right\}")
+ r"C_4 &= \left\{ \mathbb{1}, r, r^2, r^3 \right\}")
self.play(ReplacementTransform(group, grouppow[0]))
- self.wait()
+ self.wait(2)
self.play(Write(grouppow[1]))
self.wait()
self.play(ApplyMethod(grouppow.to_edge, UP))
# define morphisms
- morphism = MathTex(r"\phi: Z_4 \to G \\")
+ morphism = MathTex(r"\phi: C_4 \to G \\")
morphism.shift(UP)
self.play(Write(morphism))
+ self.wait()
# show an example
mappings = MathTex(
@@ -425,34 +432,34 @@ class AlgebraicSymmetries(Scene):
mappings.next_to(morphism, DOWN)
self.play(Write(mappings))
- self.wait()
+ self.wait(3)
self.play(FadeOutAndShift(mappings, DOWN))
# more general definition
homomorphism = MathTex(
- r"\phi(r\circ \mathbb{1}) &= i\cdot 1 \\",
- r"&= \phi(r)\cdot\phi(\mathbb{1})")
+ 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, 2.5 * LEFT),
- ApplyMethod(hom_bracegrp.shift, 2.5 * LEFT))
+ 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}(G, Z_4)").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()
- # self.play(FadeOut(hom_brace), FadeOut(hom_text_short))
-
# add the isomorphism part
isomorphism = Tex(r"\(\phi\) ist bijektiv")
isomorphism.next_to(homomorphism, DOWN).align_to(homomorphism, LEFT)
@@ -462,7 +469,7 @@ class AlgebraicSymmetries(Scene):
iso_brace = Brace(iso_bracegrp, RIGHT)
iso_text = Tex("Isomorphismus").next_to(iso_brace.get_tip(), RIGHT)
- iso_text_short = MathTex("Z_4 \cong G").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),
@@ -478,9 +485,6 @@ class AlgebraicSymmetries(Scene):
self.play(
ApplyMethod(grouppow.to_edge, LEFT),
ApplyMethod(morphgrp.to_edge, LEFT))
- # self.play(
- # FadeOutAndShift(grouppow, UP),
- # FadeOutAndShift(morphgrp, DOWN))
# draw a complex plane
plane = ComplexPlane(x_min = -2, x_max = 3)
@@ -523,7 +527,7 @@ class AlgebraicSymmetries(Scene):
FadeOutAndShift(morphgrp, RIGHT))
modulo = MathTex(
- r"\phi: Z_4 &\to (\mathbb{Z}/4\mathbb{Z}, +) \\"
+ 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)
generated by cgit v1.2.1 (git 2.18.0) at 2024-11-23 01:52:19 +0100