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-rw-r--r--vorlesungen/punktgruppen/crystals.py79
1 files changed, 76 insertions, 3 deletions
diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py
index 0f97a0f..d0ff205 100644
--- a/vorlesungen/punktgruppen/crystals.py
+++ b/vorlesungen/punktgruppen/crystals.py
@@ -135,8 +135,8 @@ class Geometric2DSymmetries(Scene):
self.wait()
cyclic = MathTex(
- r"C_n = \langle r \rangle"
- r"= \left\{1, r, r^2, \dots, r^n \right\}")
+ r"Z_n = \langle r \rangle"
+ r"= \left\{1, r, r^2, \dots, r^{n-1} \right\}")
cyclic_title = Tex(r"Zyklische Gruppe")
cyclic_title.next_to(cyclic, UP * 2)
@@ -266,4 +266,77 @@ class Geometric3DSymmetries(ThreeDScene):
class AlgebraicSymmetries(Scene):
def construct(self):
- pass
+ # title
+ title = Tex(r"Algebraische \\ Symmetrien")
+ title.scale(1.5)
+ self.play(Write(title))
+ self.wait()
+ self.play(FadeOut(title))
+ self.wait()
+
+ self.cyclic()
+
+ def cyclic(self):
+ root_powers = MathTex(
+ r"\left( e^\frac{\pi i}{2} \right)^0 &= 1 \\",
+ r"\left( e^\frac{\pi i}{2} \right)^1 &= i \\",
+ r"\left( e^\frac{\pi i}{2} \right)^2 &= -1 \\",
+ r"\left( e^\frac{\pi i}{2} \right)^3 &= -i \\")
+
+ root_more_powers = MathTex(
+ r"\left( e^\frac{\pi i}{2} \right)^4 &= 1 \\",
+ r"\left( e^\frac{\pi i}{2} \right)^5 &= i \\",
+ r"\left( e^\frac{\pi i}{2} \right)^6 &= -1 \\",
+ r"\left( e^\frac{\pi i}{2} \right)^7 &= -i \\")
+
+ root_powers.shift(LEFT * 2)
+ root_more_powers.shift(RIGHT * 2)
+
+ for line in root_powers:
+ self.play(Write(line))
+
+ for line in root_more_powers:
+ self.play(Write(line))
+
+ self.wait()
+ self.play(
+ ApplyMethod(root_powers.to_edge, LEFT),
+ FadeOutAndShift(root_more_powers, RIGHT))
+
+ groups = MathTex(
+ r"G &= \left\{ 1, i, -1, -i \right\} \\",
+ r"&= \left\{",
+ "1,", "i,", "i^2,", "i^3",
+ r"\right\} \\",
+ r"Z_4 &= \left\{",
+ "1,", "r,", "r^2,", "r^3"
+ r"\right\}")
+ groups.shift(UP)
+
+ self.play(Write(groups[0]))
+ self.play(FadeOutAndShift(root_powers, LEFT))
+ for part in groups[0:]:
+ self.play(Write(part))
+ self.play(ApplyMethod(groups.to_corner, UP + LEFT))
+ self.wait()
+
+ isomorphism = MathTex(
+ r"\varphi : G &\to Z_4 \\",
+ r"\mathrm{ker}(\varphi) &= \emptyset \\",
+ r"G &\cong Z_4")
+
+ iso_it = iter(isomorphism)
+ self.play(Write(next(iso_it)))
+ self.wait()
+ self.play(Write(next(iso_it)))
+ self.wait()
+
+ # TODO: show in group
+
+ self.play(Write(next(iso_it)))
+ self.wait()
+
+ self.play(ApplyMethod(isomorphism.to_edge, UP))
+
+ self.wait(5)
+