diff options
| -rw-r--r-- | vorlesungen/punktgruppen/crystals.py | 79 |
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) + |