From 89ff85cc982caac0a32c8f871da362c1b6970f61 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Fri, 16 Apr 2021 03:56:47 +0200
Subject: Rewrite cyclic group algebraic symmetry video
---
vorlesungen/punktgruppen/crystals.py | 165 +++++++++++++++++++++++++----------
1 file changed, 118 insertions(+), 47 deletions(-)
(limited to 'vorlesungen/punktgruppen')
diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py
index 894199d..dd55bbe 100644
--- a/vorlesungen/punktgruppen/crystals.py
+++ b/vorlesungen/punktgruppen/crystals.py
@@ -109,7 +109,7 @@ class Geometric2DSymmetries(Scene):
group = MathTex(r"G = \langle r \rangle")
self.play(Write(group), run_time = 2)
self.wait()
- self.play(ApplyMethod(group.to_corner, UP))
+ self.play(ApplyMethod(group.to_edge, UP))
actions = map(MathTex, [
r"\mathbb{1}", r"r", r"r^2",
@@ -168,7 +168,7 @@ class Geometric2DSymmetries(Scene):
self.play(Write(group), run_time = 2)
self.wait()
- self.play(ApplyMethod(group.to_corner, UP))
+ self.play(ApplyMethod(group.to_edge, UP))
self.play(FadeIn(square))
axis = DashedLine(2 * LEFT, 2 * RIGHT)
@@ -319,66 +319,137 @@ class AlgebraicSymmetries(Scene):
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 \\")
+ # show the i product
+ product = MathTex(
+ r"1", r"\cdot i &= i \\",
+ r"i \cdot i &= -1 \\",
+ r"-1 \cdot i &= -i \\",
+ r"-i \cdot i &= 1")
+ product.scale(1.5)
+
+ for part in product:
+ self.play(Write(part))
- root_powers.shift(LEFT * 2)
- root_more_powers.shift(RIGHT * 2)
+ self.wait()
+ self.play(ApplyMethod(product.scale, 1/1.5))
- for line in root_powers:
- self.play(Write(line))
+ # gather in group
+ group = MathTex(r"G = \left\{ 1, i, -1, -i \right\}")
+ self.play(ReplacementTransform(product, group))
+ self.wait()
- for line in root_more_powers:
- self.play(Write(line))
+ # 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\}")
+ self.play(ReplacementTransform(group, grouppow[0]))
+ self.wait()
+ 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.shift(UP)
+ self.play(Write(morphism))
+
+ # show an example
+ mappings = MathTex(
+ r"\phi(\mathbb{1}) &= 1 \\",
+ r"\phi(r) &= i \\",
+ r"\phi(r^2) &= i^2 \\",
+ r"\phi(r^3) &= i^3 \\")
+ mappings.next_to(morphism, DOWN)
+
+ self.play(Write(mappings))
self.wait()
+ 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})")
+ homomorphism.next_to(morphism, DOWN).align_to(morphism, LEFT)
+ for part in homomorphism:
+ self.play(Write(part))
+
+ hom_bracegrp = VGroup()
+ hom_bracegrp.add(morphism)
+ hom_bracegrp.add(homomorphism)
+
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)
+ ApplyMethod(grouppow.shift, 2.5 * LEFT),
+ ApplyMethod(hom_bracegrp.shift, 2.5 * LEFT))
- 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))
+ hom_brace = Brace(hom_bracegrp, direction=RIGHT)
+ hom_text = Tex("Homomorphismus").next_to(hom_brace.get_tip(), RIGHT)
+
+ self.play(Create(hom_brace))
+ self.play(Write(hom_text))
self.wait()
- isomorphism = MathTex(
- r"\varphi : G &\to Z_4 \\",
- r"\mathrm{ker}(\varphi) &= \emptyset \\",
- r"G &\cong Z_4")
+ self.play(FadeOut(hom_brace), FadeOut(hom_text))
- iso_it = iter(isomorphism)
- self.play(Write(next(iso_it)))
+ # 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_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)
+
+ self.play(Create(iso_brace))
+ self.play(Write(iso_text))
self.wait()
- self.play(Write(next(iso_it)))
+
+ self.play(ReplacementTransform(iso_text, iso_text_short))
self.wait()
- # TODO: show in group
+ # create a group for the whole
+ morphgrp = VGroup()
+ morphgrp.add(iso_bracegrp)
+ morphgrp.add(iso_brace)
+ morphgrp.add(iso_text_short)
+
+ 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))
- self.play(Write(next(iso_it)))
+ 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()
- self.play(ApplyMethod(isomorphism.to_edge, UP))
+ 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]))
- self.wait(5)
+ 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)
+
+ 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(Uncreate(plane))
--
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