diff options
| -rw-r--r-- | vorlesungen/punktgruppen/crystals.py | 165 |
1 files changed, 118 insertions, 47 deletions
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)) |