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-rw-r--r--vorlesungen/punktgruppen/crystals.py81
1 files changed, 49 insertions, 32 deletions
diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py
index dd55bbe..76dee1f 100644
--- a/vorlesungen/punktgruppen/crystals.py
+++ b/vorlesungen/punktgruppen/crystals.py
@@ -66,9 +66,7 @@ class Geometric2DSymmetries(Scene):
# show some rotations
dot = Dot(UP + RIGHT)
- figure = VGroup()
- figure.add(square)
- figure.add(dot)
+ figure = VGroup(square, dot)
rot = MathTex(r"r")
self.play(Write(rot), Create(dot))
@@ -184,9 +182,7 @@ class Geometric2DSymmetries(Scene):
rot = MathTex(r"r^4 = \mathbb{1}")
rot.next_to(square, DOWN * 3)
- figure = VGroup()
- figure.add(dot)
- figure.add(square)
+ figure = VGroup(dot, square)
self.play(Write(rot), Create(dot))
for i in range(4):
@@ -201,9 +197,7 @@ class Geometric2DSymmetries(Scene):
axis = DashedLine(2 * LEFT, 2 * RIGHT)
self.play(Create(dot), Create(axis), Write(action))
- figure = VGroup()
- figure.add(dot)
- figure.add(square)
+ figure = VGroup(dot, square)
for i in range(2):
self.play(Rotate(figure, PI/2))
@@ -317,6 +311,7 @@ class AlgebraicSymmetries(Scene):
self.wait()
self.cyclic()
+ self.matrices()
def cyclic(self):
# show the i product
@@ -374,9 +369,7 @@ class AlgebraicSymmetries(Scene):
for part in homomorphism:
self.play(Write(part))
- hom_bracegrp = VGroup()
- hom_bracegrp.add(morphism)
- hom_bracegrp.add(homomorphism)
+ hom_bracegrp = VGroup(morphism, homomorphism)
self.play(
ApplyMethod(grouppow.shift, 2.5 * LEFT),
@@ -384,21 +377,21 @@ class AlgebraicSymmetries(Scene):
hom_brace = Brace(hom_bracegrp, direction=RIGHT)
hom_text = Tex("Homomorphismus").next_to(hom_brace.get_tip(), RIGHT)
+ hom_text_short = MathTex(r"G \simeq Z_4").next_to(hom_brace.get_tip(), RIGHT)
self.play(Create(hom_brace))
self.play(Write(hom_text))
+ self.play(ReplacementTransform(hom_text, hom_text_short))
self.wait()
- self.play(FadeOut(hom_brace), FadeOut(hom_text))
+ 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)
self.play(Write(isomorphism))
- iso_bracegrp = VGroup()
- iso_bracegrp.add(hom_bracegrp)
- iso_bracegrp.add(isomorphism)
+ iso_bracegrp = VGroup(hom_bracegrp, isomorphism)
iso_brace = Brace(iso_bracegrp, RIGHT)
iso_text = Tex("Isomorphismus").next_to(iso_brace.get_tip(), RIGHT)
@@ -412,44 +405,68 @@ class AlgebraicSymmetries(Scene):
self.wait()
# create a group for the whole
- morphgrp = VGroup()
- morphgrp.add(iso_bracegrp)
- morphgrp.add(iso_brace)
- morphgrp.add(iso_text_short)
+ morphgrp = VGroup(iso_bracegrp, iso_brace, iso_text_short)
self.play(
- FadeOutAndShift(grouppow, UP),
- FadeOutAndShift(morphgrp, DOWN))
+ ApplyMethod(grouppow.to_edge, LEFT),
+ ApplyMethod(morphgrp.to_edge, LEFT))
+ # self.play(
+ # FadeOutAndShift(grouppow, UP),
+ # FadeOutAndShift(morphgrp, DOWN))
# draw a complex plane
- plane = ComplexPlane()
- plane.axis_config["number_scale_val"] = 1
- self.play(Create(plane))
+ plane = ComplexPlane(x_min = -2, x_max = 3)
+ 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]
)))
- self.play(
- *map(Create, roots),
- *map(Write, plane.get_coordinate_labels(1, -1, 1j, -1j)))
- self.wait()
-
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 = Group(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(FadeOut(square), FadeOut(arrow), *map(FadeOut, roots))
+ 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: Z_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 \,? \\ S_n \cong \,? \\ A_n \cong \,?")
+ question.scale(1.5)
+
+ self.play(Write(question))
+
+ self.wait(3)